Supercooled Liquids for Pedestrians

I Introduction

In March 2007 I was invited at the Weizmann Institute to deliver an introductory lecture on the physics of supercooled liquids and the glass transition. Even though I had to talk about equilibrium behaviour, leaving aside anything connected to aging and off-equilibrium phenomena, the task looked quite daunting. In approximately two hours I had to cover an exceptionally vast subject in front of an unspecialized audience, who explicitly asked me to stay at an elementary level. I therefore selected very few concepts that were in my opinion the keystones in the physics of supercooled liquids, and I tried to frame them within a brief, but coherent story. The result was apparently not too bad, and at the end of my talk the Chairs invited me to turn the lecture into a review. I was given complete freedom in deciding the plan of the work, but for one key point: it had to be as elementary, incomplete and fluent as the original lecture.

The problem is that the field of supercooled liquids is already embarrassingly rich in reviews and textbooks. References angell88; goetze-review; gutzow; debenedetti-review; angell-00; tarjus-01; donth; kob-review; kob-binder; dyre-review; leuzzi; kive-tarj-08 make just a partial selection. Therefore, it seems polite to give a justification of the apparently needless new effort, and explain what is the new value of the work, beyond the belief common to all authors to be considerably better than the predecessors (which is, however, small consolation to the reader).

I believe that in the literature on supercooled liquids and the glass transition there is a certain lack of really simple reviews, aimed at a young and unspecialized audience. By this I mean reviews that can be read and understood without any specific training in liquid theory or disordered systems, but with just a background in statistical mechanics. This is probably the reason why my lecture at Weizmann was appreciated. The aim of the present review is therefore to span the whole subject in a brief and compact way, at the price of being utterly inexhaustive and somewhat superficial. The level is truly elementary. The expert reader is warned that many important topics are not covered.

Despite the similar pedestrian spirit, the plan of the present work is quite different from my previous notes, Spin-glass theory for pedestrians cavagna-review. In that case, we analyzed in full mathematical detail a single mean-field spin-glass (the $p$-spin model), as a paradigm for the entire field. A similar operation would be impossible for supercooled liquids, where it is very hard to single out one paradigmatic system and where no well-established theoretical framework is present. Therefore, I decided to proceed mainly through the phenomenology, as on a journey through the various temperature regimes that are crossed when a liquid is cooled. This journey is schematically depicted in Fig.1, which can be used as a map throughout these notes.

After a first part containing some viscoelastic preliminaries, we will try to understand what are the infamous ‘certain conditions’ under which crystallization is avoided and a supercooled liquid is formed below the melting point $T_{m}$. In this part we will focus on nucleation theory and in particular on the concept of kinetic spinodal, namely the temperature below which the crystal nucleation time becomes shorter than the liquid relaxation time. Below this temperature the only way to prevent crystallization is to push the system out of equilibrium. The kinetic spinodal represents a metastability limit for the supercooled phase and it is thus essential to discover under what conditions it can be avoided. This chapter is important to understand that the existence of supercooled liquids must not be taken for granted, and that skipping crystallization is more subtle a process than what is sometimes assumed.

Once the existence of the liquid is secured against unwanted crystallization, the temperature can be decreased so much as to meet the dynamic glass transition, $T_{g}$. At this point the relaxation time of the system grows so sharply as to exceed our available experimental time: below the glass transition the system falls inevitably out of equilibrium, and a glass is formed. This operational definition may suggest that $T_{g}$ is a purely conventional point, whose existence is solely due to the finite timespan of our experiments. Moreover, the structure of liquid configurations close to $T_{g}$ seems to remain exactly the same as at higher temperature. This fact too gives the impression that perhaps nothing crucial is going on at the famous glass transition, and that the growth of the relaxation time is just a not-too-relevant quantitative feature. We will see that this is not true, and that at least for fragile liquids some really new physics kicks in at $T_{g}$. We will focus on those phenomenological traits that distinguish at a qualitative level a liquid in its fluid phase from a liquid close to the glass transition. We will find that the two steps relaxation of the dynamical correlation function is the most insightful landmark of the proximity to the glass transition.

In order to explain what physical ingredients leads to the glass transition, we will discuss the crossover from nonactivated to activated dynamics taking place at the Goldstein’s temperature $T_{x}$. Above $T_{x}$ activation is not the main mechanism of diffusion and dynamics is reasonably described by mode coupling theory. Below $T_{x}$, however, the liquid enters in its activated phase, eventually leading to the dynamic glass transition at $T_{g}$. To better understand the nature of this crossover, we will discuss the mean-field results obtained within the $p$-spin model and show that the dynamical transition $T_{c}$ predicted by both mode coupling and the $p$-spin is conceptually the same as Goldstein’s temperature $T_{x}$. We will finally explore the role of saddles of the potential energy and explain how two steps relaxation can be the result of a system running out of unstable directions in the phase space.

After this, we will dive into the deeply supercooled phase. We will imagine that we can skip the glass transition (we cannot) and keep the liquid at equilibrium even at very low temperature. If we do this, we meet the famous Kauzmann’s temperature $T_{k}$, namely the point where the entropy of the liquid becomes equal to that of the crystal. According to some theoretical schemes, at this point there must be a phase transition, with a true divergence of the relaxation time. Whether or not such transition really occurs is perhaps not too important, since anyway the dynamic glass transition prevents us to reach this point. However, the physical mechanisms behind the phase transition may affect and regulate the behaviour of the system even at higher temperatures, giving rise to some precursor phenomena that may be also detected experimentally. For this reason studying the deeply supercooled phase is not a mere academic exercise.

Finally, we will look for a correlation length. Intuition tells us that wherever there is a large time, there should also be a large length. This makes sense: the relaxation time becomes large because the system needs to rearrange larger and larger correlated regions at low temperature. A growing lengthscale is a key ingredient of some important theoretical frameworks of supercooled liquids, like the Adam-Gibbs theory and the mosaic theory. Yet, how to define this lengthscale in supercooled liquids is not quite clear. Finding a correlation length normally requires to define a suitable order parameter and measure its relative correlation function. However, how to do this is nontrivial in glass-forming liquids, because all amorphous configurations look the same and it is difficult to detect correlated regions. Explaining how to uncover the growth of amorphous order will be the final aim of this notes.

As in my former review on mean-field spin-glasses, I do not cover the subject of aging and off-equilibrium dynamics. I cannot stress enough that this is not due to my dislike or disregard of the subject, but, on the contrary, to the fact that it is such an important and vast field that it cannot be covered by a single section of a brief review as this is. I have tried (starting from the title) to use as much as possible the term ’supercooled liquids’ rather than ’glasses’ expressly in order to avoid any confusion between equilibrium and off-equilibrium physics.

A final word of caution on the theoretical frameworks discussed in these notes. A shared theory of the glass transition does not exist and different frameworks clash continuously. As I write, a group of researchers from all over the world prepares for a workshop in Leiden, where they will kindly disagree about everything. In this context, I had two choices: either to present a more or less complete list of all theories on the market, briefly discussing the pros and cons of all of them, or to tell a single, coherent, but inevitably partial story, and discuss what is mainly my personal viewpoint. Not surprisingly, due to lack of space and overwhelming laziness, I have chosen the second solution. This was also more true to the spirit of my original Weizmann lecture. In fact, even the structure I gave to the material contained in this review reflects my own understanding of the field. Therefore, even though my view about the fundamental physical mechanisms ruling supercooled liquids is not entirely a minority one, the reader should be well aware that this is not the whole story.

II Preliminaries

It seems nice to start a review on supercooled liquids with a definition of what liquids are. This is particularly important for deeply supercooled liquids, since in this case the viscosity may be so large that, from the mechanical point of view, the distinction between liquid and solid becomes blurred. Viscoelasticity is crucial to understand the mechanical features of supercooled liquids, and it is also the ideal setting where to introduce two key tools for the rest of these notes, namely shear relaxation time and viscosity. Moreover, we will introduce the diffusion coefficient, and explain how it is connected to the viscosity through the Stokes-Einstein relation. Finally, I will clarify what I intend for ‘equilibrium supercooled liquid’, as it may seem rather contradictory to define as ‘equilibrium’ a metastable phase of matter. As we shall see, however, there are good reasons to adopt such a terminology. A good discussion of the basic concepts of rheology can be found in ferry, and in briefer form in cates-review. For a very sharp introduction about diffusion and the Langevin equation see zwanzig.

Liquids flow, solids do not. For the level of these notes this may be a good enough definition of the liquid phase. Yet, let us expand a little. First, we must forget about liquids, and focus on solids. Consider a cubic solid of side $L$, subjected to a small shear. The effect of the shear is to produce a displacement $u_{x}$ along the $x$ axis, proportional to the height $y$,

When $y=L$, the displacement $u_{x}$ becomes equal to the maximum shift of the top of the cube, let us call it $l$. In this way we have $\gamma=l/L$. The shear strain is the $(x,y)$ component of the strain tensor, $u_{xy}$, and it is given by,

The solid responds elastically to such a step shear strain, giving rise to a stress $\sigma_{xy}$ proportional to the strain,

This equation defines the elastic shear modulus $G$ of the solid: the induced stress is proportional to the small step-strain and the proportionality factor is the shear modulus. The stress does not decay in time. Solids do not flow.

Let us apply a similar small shear to a liquid sample: at time $t^{\prime}$ the liquid is subjected to a step shear given by (1). Saying that liquids flow is equivalent to say that at time $t^{\prime\prime}>t^{\prime}$ the stress will be partially relaxed. Provided that we are at equilibrium, so that time-translation invariance (TTI) holds, and provided that we are in the linear regime, we can generalize equation (4) to,

where $G_{\infty}$ is the infinite frequency (i.e. zero time) shear modulus. Within the Maxwell model there is just one exponential relaxation time. In real liquids things are more complicated, and there may be a superposition of many components,

In this case one has to extract a singly time-scale from $G(t)$ in a more or less arbitrary way (typically, one focuses on the largest relaxation time). However, the essence of the problem does not change: the shear relaxation time of a liquid rules the way the stress-relaxation function $G(t)$ decays.

Let us consider now an arbitrary time-dependent shear strain $\gamma(t)$, rather than a step-strain. In this case we can say that the stress builds up as a consequence of the many accumulated steps of strain,

If the applied shear is zero for $t<0$ and different from zero for $t>0$, we can integrate over time and obtain the total stress,

From this formula it is evident the role of the time-dependent shear modulus $G$ as a memory kernel. In the particular, but important, case of a constant shear rate $\dot{\gamma}$, we get,

In a normal liquid, the shear relaxation function $G(s)$ decays faster than $1/s$, so that in the asymptotic regime $t\to\infty$ the system develops a constant stress in response to a linear shear strain (i.e. a constant shear rate). The ratio of shear stress to linear shear strain is the very definition of the viscosity $\eta$, and thus we have,

For Maxwell liquids (equation (6)) we recover the well-known formula linking viscosity, relaxation time and shear modulus,

Let us specify the dimensions here, which will be useful later. The shear modulus has the dimensions of force per unit surface, and it is normally measured in the c.g.s. system in dyne/cm2. We conclude that the viscosity has the units of dyne sec/cm2, which is a Poise.

It is interesting to note that there are some soft materials that are neither liquids, nor solids. When $G(s)$ decays to zero slower than $1/s$, the shear relaxation function is not integrable for infinite times: the system is able to relax the shear for long times (in this sense it flows, so it is not a solid), but it has an infinite viscosity (so it is neither a liquid). From equation (10) we see that the response of such systems to a constant shear rate $\dot{\gamma}$ is a stress $\sigma_{xy}$ that grows indefinitely in time. Reversing the situation, when they are subjected to a constant stress (as, for example, gravity) these materials do flow, although slower and slower, that is at a shear rate that vanishes as time goes by cates-review. Such systems are sometimes called ‘power law fluids’, since typically $G(t)\sim 1/t^{a}$ with $a<1$, or ‘dilatant compounds’. Silly putty is a good approximation of what a dilatant compound is.

II.2 The diffusion coefficient and the Stokes-Einstein relation

The last key player in the physics of viscous liquids is the diffusion coefficient $D$. Let us consider the diffusion equation for the concentration $\rho(x,t)$ of a tagged particle in a liquid (for simplicity we stay in one dimension),

where $D$ is the diffusion coefficient. Let us compute the time derivative of the mean square displacement of the particle,

from which we derive the relation between mean square displacement and diffusion coefficient,

The LHS of equation (14) can be calculated using directly the particle velocity $v(t)$. From,

where we have used time-translation invariance. Therefore, in the long time limit, we have a relation between diffusion coefficient and velocity-velocity correlation function, which is valid in any dimension,

The next step is to link diffusion to friction. This can be done through the Langevin equation for a particle of mass $m$,

where $\zeta$ is the friction coefficient and the noise $\eta(t)$ is delta-correlated in time. Equation (19) is a linear, first-order, inhomogeneous differential equation, that can be easily solved, giving,

By computing the long-time limit of the average kinetic energy, $1/2\;m\langle v^{2}\rangle$, and by using equipartition, we can fix the balance between amplitude of the noise and temperature,

which is known as the static fluctuation-dissipation theorem. This result says that the origin of the noise $\eta$ and that of the friction $\zeta$ is the same, namely the temperature $T$ zwanzig. By using (20) and (21), we can calculate the RHS of (18), and we obtain,

which is Einstein’s relation for the diffusion coefficient. In order to make a final link with the viscosity $\eta$, we need a relation between friction coefficient $\zeta$ and $\eta$. This is provided by Stokes’ equation, which we will not derive here (see stoke). A large sphere of radius $a$ moving in a liquid with shear viscosity $\eta$, undergoes a friction whose coefficient is given by,

where $C$ is a constant depending on the boundary conditions for the fluid on the sphere’s surface. Putting together the last two equations we finally get the Stokes-Einstein relation between diffusion coefficient and viscosity einstein,

Although this relation strictly holds only for a diffusing sphere much larger than the molecules comprising the fluid, it is in fact surprisingly accurate also in describing the self-diffusion coefficient $D$ of a molecule surrounded by other molecules of equal size. Yet, the Stokes-Einstein relation sometimes fails. This typically happens when the diffusion coefficient remains finite in situations where the viscosity is either zero (as in superfluid helium), or infinite (as in elastic crystals). As we shall see, something similar happens in supercooled liquids close to the glass transition, when the diffusion coefficient $D$ decreases with decreasing temperature less steeply than $T/\eta$, so that the ration $D\eta/T$ is no longer a constant and the Stokes-Einstein relation is violated.

II.3 Metastable equilibrium?

In the physics of supercooled liquids and glasses even the terminology is not universally accepted, and this can give rise to dangerous ambiguities. Given that it is essential to avoid confusion from the very beginning, let us specify the terminology used in these notes.

I use the words ‘liquid’, or ‘supercooled liquid’, whenever I talk about the equilibrium, albeit metastable, liquid phase. One may object that a supercooled liquid is, by definition, out of equilibrium, since below the melting temperature true thermodynamic equilibrium is given by the crystal, while the liquid is metastable. This is true. However, we can experimentally equilibrate a liquid in its metastable phase, in such a way that time-translation invariance (and thus the dynamical fluctuation dissipation theorem) holds. In such situation, no experimental measurement is able to tell us that the system is metastable. Only an explicit crystallization of the sample would unveil metastability. Therefore, with a slight abuse of language, I will call this the ‘equilibrium’ liquid phase, supercooled or not depending on whether we are below or above the melting temperature. As always in first-order phase transitions, there is no way to experimentally detect the presence of the transition temperature, as long as the system remains equilibrated into one of the two phases.

On the other hand, whenever I will use the word ‘glass’ it will always be to characterize the off-equilibrium phase. By this I mean a phase where time translation invariance is broken, so that two-times correlation functions do not depends simply on the difference of times and the fluctuation dissipation theorem does not hold. Note that this is not simply a stronger way to be out of equilibrium compared to metastability; it is a completely different thing. In principle it is possible to cool so rapidly a liquid that it goes off-equilibrium even above the melting point, where it is not metastable. In an off-equilibrium glass the relaxation time of the substance is too large compared to our experimental time, whereas in the equilibrium metastable phase the crystal nucleation time is larger than our experimental time, and yet the relaxation time is shorter than the experimental time. We shall discuss these two points more in details later on.

Of course, supercooled liquids and glasses are connected: as we shall see, the deeper the degree of supercooling of a liquid, the larger the relaxation time, and the easier is to form a glass. For this reason, sometimes the words ‘glass-formers’ or ‘glass-forming liquids’ will be used in alternative to ‘supercooled liquids’.

III Negotiating the crystal

People working on supercooled liquids and the glass transition normally show little interest in the crystal. The reason is obvious: we want to study the exciting properties of glasses, not the boring crystalline phase. When pressed about the subtleties of crystallization, and the risks of disregarding it, the typical glassy physicist replies: “There are systems that do not have a crystalline ground state. In that case, we certainly should not care!”, which may be true, but leave us quite anxious about all systems that do have a crystalline ground state. Therefore, despite my full approval of supercooled liquids and glasses, I believe that few words must be spent on how crystallization can (sometimes) be avoided when cooling a substance. We often read that ‘if we are careful enough’, or ‘if we are fast enough’, we can avoid crystallization and enter in the supercooled - metastable - phase. How does this happen precisely? Answering this question is what this chapter is about.

Liquids display a first order phase transition at the melting temperature $T_{m}$. Their stable phase at low temperatures is the crystal, which has a lower free energy density than the liquid. In order to form a supercooled liquid we therefore have to cool our sample below $T_{m}$ avoiding crystallization. A supercooled liquid is thus a metastable phase. To learn how to skip the crystal, we have first to learn how it forms within a supercooled liquid. This is the aim of nucleation theory turnbull49; turnbull69. In these notes I will only discuss homogeneous nucleation, that is the formation of crystal nuclei only due to thermal fluctuations within the bulk of the sample. In fact, crystal formation is often assisted by the presence of impurities and boundaries, and it is known in these cases as heterogeneous nucleation. Before we forget about it, note that heterogeneous nucleation can make the lifetime of a metastable sample even shorter than what expected from the purely homogeneous theory.

Consider a sample in the liquid phase. Due to thermal fluctuations there is a nonzero probability that some particles form a nucleus of the ordered crystalline phase. The crystal has a lower energy than the liquid, but also a lower entropy. As long as $T>T_{m}$, the balance between energy and entropy is still in favour of the entropy and the liquid phase has a lower free energy than the crystal. In these conditions forming a crystal nucleus is of no advantage, and the nucleus will rapidly melt. On the other hand, if $T<T_{m}$ things change: now the crystal is thermodynamically favoured, so that the crystal nucleus formed by thermal fluctuations has a lower free energy than a liquid droplet with the same number of particles. However, there is a price we must pay to form the nucleus: this is the free energy cost due to the mismatch between the two different phases along the interface between liquid and crystal. This cost is proportional to the surface of the nucleus.

The total Gibbs free energy change, due to the formation of the crystal nucleus in $d$ dimensions, is therefore given by,

where we have disregarded all the dimensionless geometric prefactors dependent on the shape of the nucleus. Equation (25) is the the pillar of nucleation theory: $\sigma$ is the surface tension, i.e. the (positive) free energy cost per unit area in forming the interface between the two phases. On the other hand, $\delta g$ is the Gibbs free energy density difference between the metastable and stable phase,

Note that, by definition, $\delta g=0$ at the melting temperature $T_{m}$, and it grows when $T$ goes below $T_{m}$. In fact, $\delta g$ is often fitted to a linear form below $T_{m}$,

where $\delta h$ is the molar enthalpy of fusion, and $\nu$ is the molar volume of the crystal weimberg02. This is a handy formula, which may however break down well below $T_{m}$.

Equation (25) encodes the competition between the surface cost, $\sigma$, opposing nucleus formation, and the bulk thermodynamic drive, $\delta g$, favouring it. For small radii $R$, the surface term dominates, so that small nuclei are thermodynamically unstable and melt; on the other hand, for $R$ larger than a critical nucleus size $R_{c}$, the volume term takes over, and the nucleus is stable. By maximising (25) we find the critical nucleus $R_{c}$ and the free energy barrier to form a stable nucleus, $\Delta G(R=R_{c})$,

where once again we disregarded dimensionless geometric factors, irrelevant for the present discussion. From these relations we understand that the size of the critical nucleus, and thus the nucleation barrier, diverge at the melting point, where $\delta g=0$. This is obvious: at $T=T_{m}$ the two phases have the same free energy and there is no thermodynamic advantage in the formation of the crystal nucleus, whereas there is still a surface disadvantage. Therefore, the sample always needs a certain degree of supercooling (i.e. metastability) in order to form stable nuclei, and eventually collapse to the stable crystal phase. Note that the surface tension is normally a function weakly dependent on the temperature close to $T_{m}$, so that the largest part of the $T$ dependence of $R_{c}$ and $\Delta G$ comes from $\delta g(T)$ through equation (27).

It is very important not to make confusion between $\Delta G$ and $\delta g$: the first one is the free energy change caused by the nucleus formation, it depends on the nucleus size $R$, and, once evaluated at the critical size $R_{c}$, coincides with the barrier to nucleation; $\delta g$ is the thermodynamic drive, i.e. the free energy (density) difference between the two phases and it does not depend on $R$. In fact, equation (29) shows that these two quantities are inversely proportional: when the temperature decreases, entering in the supercooled phase, the thermodynamic drive to nucleation $\delta g$ increases, and thus the barrier to crystal nucleation $\Delta G(R_{c})$ is smaller the lower the temperature. Hence, from a thermodynamic point of view, the lower $T$, i.e. the larger $\delta g$, the easier is crystal nucleation.

III.2 Nucleation time vs. nucleation rate

After the free energy barrier to nucleation is calculated, it is common to derive directly the nucleation time, i.e. the time needed to form a critical nucleus, of size $R_{c}$. To do this one simply invokes the Arrhenius formula that rules activated processes, with $\Delta G(R_{c})$ playing the role of the activation barrier. In this way one gets for the nucleation time,

If the rate per unit volume is constant, then the time needed to form one critical nucleus in a sample of volume $V$ scales as $1/V$,

The larger the volume, the smaller the time needed to nucleate the crystal, and thus leave the metastable phase. Using small samples is indeed a well-known experimental trick when trying to avoid nucleation. Therefore, the prefactor of the nucleation time $\tau_{0}$ in (30) must contain a term inversely proportional to the sample’s volume, $1/V$. As we shall see, this is just the first of many tricky ingredients of the infamous nucleation prefactor $\tau_{0}$.

The second reason is deeper: if a stable nucleus is formed at a certain position $x$ distant from a position $y$ in the same sample, the nucleus must grow from $x$ to $y$, before metastability is lost at $y$ as well. As we shall see later, the growth of a nucleus can be slowed down significantly, sometimes even blocked. As long as the nucleus is far enough from $y$ (beyond few correlation lengths), any local measurement performed in $y$ will be unable to detect the loss of metastability taking place in $x$. The underlying reason for this is that ordinary first-order transitions (unlike second order ones) are very local in nature, without long range correlations. So, unless growth is very fast, we need many locally formed nuclei to loose metastability. We will come back to this point when we will discuss the kinetic spinodal.

III.3 Avoiding the crystal

Let us cool the system with a certain cooling rate $r$, which is the temperature variation per unit time,

If we use a linear cooling rate, that is if $r(T)$ is a constant, then we cannot cool slower than a critical value $r_{c}$, which is approximately equal to the inverse of the minimum nucleation time (Fig.2),

If, on the other hand, we cool nonlinearly, we can cool fast at higher temperatures, until the minimum of the nucleation curve is negotiated, and slow down progressively at lower temperatures to cope with the increasing relaxation time of the supercooled liquid (Fig.2).

III.4 The kinetic spinodal

In some materials the relaxation time may exceed the nucleation time below a certain temperature, as shown in Fig.3 (right panel). This temperature is the kinetic spinodal, which is thus defined by the relation,

It was Kauzmann in 1969 kauzmann who first hypothesized the existence of a kinetic spinodal in supercooled liquids. He wrote: “Suppose that when the temperature is lowered a point is eventually reached at which the free energy barrier to crystal nucleation becomes reduced to the same height as the barriers to the simpler motions. At such temperatures the liquid would be expected to crystallize just as rapidly as it changed its typically liquid structure to conform to a temperature or pressure change in its surroundings.” Kauzmann introduced the concept of kinetic spinodal as an effective way to avoid the entropy crisis, defined as the point where the (extrapolated) entropy of the liquid becomes equal to the entropy of the solid. We shall discuss this phenomenon in great detail later on in these notes.

III.5 Can the prefactor of nucleation save the day?

Up to know we have assumed that relaxation and nucleation are completely independent phenomena. In fact, this is not true. The prefactor $\tau_{0}$ in the nucleation time (30) depends on the temperature, although this dependence is quite complicated and somewhat unclear. It has become customary to approximate the $T$ dependence of $\tau_{0}$ by using the self-diffusion coefficient $D$, thus writing turnbull49; turnbull69,

Although this equation is definitely not exact, its meaning is rather clear: nucleation proceeds through an aggregation process, and it is reasonable that the rate of this process is proportional to the diffusion in the system. By using the Stokes-Einstein relation (24), one then has,

where we have used the linear relation between shear viscosity and relaxation time of normal liquids (12). From equation (30) we get,

This equation implies that the nucleation time is always larger than the relaxation time. If this were true, a kinetic spinodal would never exist: nucleation would always be slower than relaxation, so that there would always exist a time window where equilibrium properties could be measured. This is the underlying argument when it is sometimes said that once a liquid enters in its deeply supercooled phase, then crystallization is no longer an issue, since it becomes too difficult. The idea is that when relaxation is too slow, and viscosity is very high, nucleation, and thus crystallization, is kinetically inhibited.

Although the prefactor of nucleation definitely increases by decreasing the temperature, it is hard to claim that this fact can completely rule out the existence of a kinetic spinodal. First, there are systems where a kinetic spinodal exist (see ctls1, for example), so that this argument does not work in general. Secondly, even if equation (38) strictly held, it would not be sufficient to grant the survival of the liquid in a condition where the barrier to nucleation is very small. In fact, in order to measure the equilibrium properties of the liquid, we need the nucleation time to be significantly larger than the relaxation time. This means that there must be some orders of magnitudes of difference between the two times, otherwise the metastable liquid phase is only virtually, and not operationally, existent. If the exponential is of order $1$, the two times are comparable, and equilibrium is lost. Third, in the deeply supercooled phase, where the liquid becomes very viscous, the Stokes-Einstein relation is known to break down rossler-90; cicerone-96, and typically,

III.6 An elastic twist

There is a mechanism that can inhibit nucleation in supercooled liquids, in some cases so much as to suppress the kinetic spinodal. To understand what such mechanism is we have to make a small detour.

Indeed, when the system has an elastic response, as in the case of solids, there is an extra energetic price that has to be paid. Because of the long-range nature of elastic forces, the elastic price is proportional to the volume of the nucleus, and it thus corrects the bare thermodynamic drive $\delta g$ roi78; cah84; bus03. The correct formula becomes,

The elastic term is proportional to the square of the relative volume difference between the two phases, $\delta v/v$, and to the elastic shear modulus, $G_{\infty}$ elastoreview,

where we remind that $\delta h/\nu$ is the molar enthalpy of fusion per molar volume (see equation (27)), which is the drive to nucleation. The ratio $G_{\infty}/\delta h$ thus encapsulates the competition between elastic loss and thermodynamic gain in forming the nucleus. In those liquids where $\lambda$ is small, elasticity is weak compared to the thermodynamic drive to nucleation, so that nucleation is only moderately depressed, and a kinetic spinodal may still exist. On the other hand, when the parameter $\lambda$ is large, elastic effects become so strong that the kinetic spinodal can be completely suppressed attanasi1; attanasi2.

We conclude that some supercooled liquids, those with a large enough elastic response, may survive in their metastable phase even when the relaxation time becomes very large, because the kinetic spinodal is suppressed by elastic effects. Of course, the absence of a kinetic spinodal does not mean that these systems cannot crystallize: we still have to be very careful when cooling. However, knowing that, at least in principle, there are systems where the metastable phase is well-defined, no matter how low is $T$, is somewhat a consolation.

III.7 Nucleation vs. growth

The curve of the nucleation time vs. temperature is quite instructive from a theoretical point of view, but it is not of great help experimentally. What we can measure in an experiment is crystallization, rather than nucleation. In particular, we can measure the time $t_{x}$ the system takes to develop a substantial amount of crystalline order. This amount must be large enough to be detected experimentally and see that crystallization has started at that particular temperature $T$. The function $t_{x}(T)$ obtained in this way is often called Time-Temperature-Transformation (TTT) curve, and it is the only experimentally available tool to check how likely is crystal formation at a given temperature uhlmann72. The TTT curve, as the nucleation curve, displays a minimum, indicating that at this point crystallization is most likely. In the experimental context the TTT curve is normally plotted in a temperature vs. time representation, so that this minimum looks in fact like the ’nose’ of the curve, and this is the word normally used for it.

One may argue that the TTT curve is just the experimental counterpart of the nucleation curve: they both tell us how likely crystallization is, they both have a minimum, they both diverge at $T_{m}$ and $T=0$. In fact, up to know we made little difference between nucleation and crystallization. However, nucleation is a different phenomenon from crystallization, and the two curves have a fundamentally different meaning.

In order to form a substantial amount of crystal, which can be detected experimentally, the system has to first form some stable nuclei and then to grow them until they invade a large part of the sample. The first process is nucleation, the second is growth. These two processes are different, and crystallization is given by the concourse of the two of them. Nucleation is dominated by the competition between surface tension and free energy (potentially renormalized by an elastic factor). Growth, on the other hand, is largely dominated by the viscosity of the background liquid in which the nucleus must expand. Moreover, growth may be slowed down because different mismatched crystallites come in contact with each other, giving rise to a very slow process of crystal domain growth (or coarsening) bray. Therefore, growth is ruled mostly by relaxation mechanisms. As a result, crystallization proceeds at the pace of the slowest process, nucleation or growth.

Close to the melting point the viscosity of the liquid is very small, whereas nucleation time is very large. Therefore, it is always nucleation to slow down crystallization near $T_{m}$ and to make the TTT curve diverge here. Moreover, if there is no kinetic spinodal, i.e. if nucleation is always significantly slower than relaxation, then as slow as growth may be, nucleus formation is typically even slower, and thus nucleation is the real bottleneck of crystallization. In this case the TTT curve gives the same qualitative physical information as the nucleation curve (Fig.5 - left panel).

The final message of this section is the following: when trying to avoid crystallization, we must make the best use of the TTT curve, which is the only experimentally available tool; therefore, we have to cool rapidly around the nose, and cross our fingers. However, there is no way we can detect the existence of a kinetic spinodal by simply inspecting the TTT curve, as it looks always the same (a divergence at $T_{m}$ plus a nose), both when there is and when there is not a metastability limit. The difference is that when a kinetic spinodal is present the left branch of the TTT curve is ruled by growth, rather than nucleation (Fig.5); but this is something the TTT curve itself is not going to tell us. As a consequence, we have always to be very careful in ruling out the possibility that our sample has crystallized. Even after we have negotiated the nose of the TTT curve, there is always the risk that nucleation starts, but growth is so slow that it keeps the crystallites’ size below the experimentally detectable threshold. In that case our sample is a sneaky polycrystal, i.e. an off-equilibrium system: however slow growth may be, it is bound to drive away the system from its supercooled equilibrium condition. Therefore, when a spinodal is present (Fig.5 - right panel), we can only study the supercooled liquid in the regime of equilibrium arrested nucleation (right branch of the TTT curve), not in the regime of off-equilibrium arrested growth (left branch of the TTT curve) ctls1; ctls2.

Even though off-equilibrium glasses are not our main concern in these notes, it is worthwhile saying that such off-equilibrium polycrystals, as those obtained from a widespread nucleation arrested by a very slow growth, raise some conceptual issues about the nature of the glassy phase. The question is whether or not it is always possible to identify an order parameter that clearly distinguish these sneaky polycrystals from bona fide amorphous (i.e. glassy) configurations. In general, it is not at all easy to clear up the difference between the two: there are systems where it exists a grey zone where it proves very hard to qualitatively distinguish a highly disperse polycrystal from a truly amorphous glass ctls1; ctls2. This said, for the rest of these notes we will make the simplifying assumption that bona fide amorphous minima, qualitatively well separated from the crystal, can always be defined.

IV The glass transition and thereabouts

The glass transition is one of the most interesting open problems in condensed matter physics. There is first of all an issue of conceptual definition, since the glass transition is not a ’transition’ at all. But more importantly, there is the problem of whether or not a liquid close to the glass transition is in any respect qualitatively different from a normal liquid. In other words: does the glass transition mark a fundamental change in the physical properties of the liquid, or is it a mere conventional point? I will try to convince the reader that a supercooled liquid close to the glass transition is indeed different from a high temperature liquid at a qualitative level, and not simply because its relaxation time is much larger than normal.

Throughout these notes I will always use temperature, rather than density, as the control parameter triggering glassy behaviour. For a thorough comparison of the lowering $T$ vs. increasing $\rho$ effects on glass-forming liquids see tarjus-0; tarjus-rhot. Excellent brief accounts of the glass transition and related phenomena are given in angell88 and kob-review.

Once we have learned how to defend our supercooled liquid sample from crystallization, we turn to face another enemy, the glass. When we cool a liquid significantly below the melting point, the shear relaxation time and the viscosity grow sharply. This growth is very impressive and very general (see Fig.6). Data show that viscosity can increase by up to $14$ decades in a relatively narrow range of temperature angell-poole. This sharpness is very different from what happens above the melting point, where the viscosity increases in a much smoother way. This dramatic growth is common to many liquids with very different microscopic structures, including polymers.

By decreasing $T$ we are therefore bound to hit a temperature where the relaxation time exceeds the time experimentally available to our measurement. Below such temperature it is impossible to equilibrate the system: the sample is out of equilibrium on the time scale of our experiment, and we have formed a glass. We give a provisional definition of dynamic glass transition (or kinetic glass transition, or simply glass transition) as the temperature $T_{g}$ where the relaxation time exceeds the experimental time,

What happens from the experimental point of view at the glass transition? The most important practical consequence of going off equilibrium is that we do not give the system enough time to properly explore the phase space, and, in so doing, we sharply cut the number of degrees of freedom accessible to the system. This causes a sharp drop (up to a factor $2$) of the constant pressure specific heat $c_{p}$ at $T_{g}$ angell88. A schematic view of the typical behaviour of $c_{p}(T)$ is reported in Fig.7. The experimental time is smaller than the ergodicity time, i.e. the time needed by the system to explore a representative fraction of the phase space. In this dynamic sense, we can say that the system is no longer ergodic.

This phenomenon becomes all the more clear when we notice that the specific heat below $T_{g}$ drops to a value very close to that of the crystalline phase. In a crystal the motion of all particles consists of vibrations around their (ordered) equilibrium positions, without any kind of rearrangement. Ergodicity is broken and the system is confined to one (absolute) energy minimum in the phase space. The behaviour of the specific heat thus suggests that also in a glass at low temperature particles vibrate around their (disordered) equilibrium positions, with almost no structural rearrangement. Ergodicity is dynamically broken and the glass is confined to one (local) energy minimum in the phase space. For this reason, the specific heat is approximately the same in the crystal as in the low temperature glass.

Even though the view of a glass stuck in a local minimum is good enough to understand the behaviour of static quantities as the specific heat, it is unfortunately far too simplistic if we want to understand the dynamic properties of the off-equilibrium phase. This is not our focus, but we nevertheless must be a bit more precise here. A glass is something more complicated than a system vibrating around an amorphous minimum of the energy. Were this simple picture true, the glass would be at a (broken-ergodicity) equilibrium within this minimum, as it happens to the crystal. However, a glass is drastically out of equilibrium. Even though one-time quantities (as the volume or the energy) may look almost constant in the long time limit, two-time quantities (as the dynamic correlation function) show a stark off-equilibrium behaviour, in that they depend explicitly on both times, rather than on their difference. In other words, the properties of the system depend on the time elapsed from the instant the system was cooled below $T_{g}$. This is aging. The reasons for this behaviour are complex and go beyond the scope of these notes. Very nice introductions on aging and off-equilibrium dynamics can be found in leticia-review and biroli-aging. Here, let us be content in knowing that on short time-scales the most conspicuous effect of going off-equilibrium below $T_{g}$ is indeed to remain stuck in an amorphous configuration, where only vibrations contribute to the specific heat, and for this reason $c_{p}$ has a drop.

There is, of course, a crucial difference between an off-equilibrium glass and a crystal: ergodicity breaking in a glass is a purely dynamical accident, whereas in a crystal is a truly thermodynamic phenomenon. On the time scale over which we observe a glass, ergodicity cannot be restored and equilibrium is lost. In such a situation, in principle, we should not measure the specific heat at all, which is an equilibrium concept. If we stubbornly do that anyway, we inevitably record a loss in the number of degrees of freedom that the system is dynamically prevented to use. If we waited enough time to equilibrate the system, we would see the curve $c_{p}(T)$ modify drastically, and the specific heat drop would disappear. In a crystal, on the the contrary, the system is confined in a limited portion of the phase space forever, since energy barriers are infinite. In other words: a crystal is a bona fine broken ergodicity equilibrium phase of the system, whereas an off-equilibrium glass is not.

To conclude this section, imagine that a very patient researcher give us (as a gift) a liquid sample thermalized at a temperature where we would not be able to equilibrate the system, that is at $T$ smaller than our experimental glass transition $T_{g}$. The sample now is at equilibrium at very low temperature and we can start doing our equilibrium measurements. The energy will have the right value, and so the volume. But do we get the right specific heat? The answer is no. In fact, we would still give a very bad off-equilibrium experimental estimate of $c_{p}$. The reason is that, even though our system is initially into an equilibrium configuration (that is a configuration that has been correctly drawn with the Boltzmann-Gibbs distribution), in order to perform an equilibrium measurement of the specific heat we need to let the configuration explore ergodically the phase space. But we do not have enough time to do so, otherwise we could have equilibrated the sample ourselves! Of course, there are some quantities (like the volume, or the energy) whose measurement would coincide with the right equilibrium value, but this is small consolation. Most of the interesting observables, that is those involving the fluctuations, need the system to be truly ergodic in order to be measured correctly. Thus, few equilibrium configurations do not solve the problem at all.

This apparently pointless issue is in fact relevant in the field of numerical simulations: there are powerful algorithms that use a non-physical dynamics to produce very low $T$ configurations equilibrated according to the Boltzmann-Gibbs distribution. One may be tempted then to use these equilibrium configurations as a starting point to run a simulation with a more physical dynamics (as, for example, molecular dynamics) and measure the specific heat. This, however, cannot be done, since, as we have seen, it would be impossible to make the system explore ergodically the phase space, and to exploit all the relevant degrees of freedom.

IV.2 Is TgT_{g} a meaningful concept?

By differentiating both sides of (45), we obtain,

where $a$ and $b$ are two constants. Equation (47) is just a trivial consequence of (46).

For these reasons, it is sensible to give a general definition of the kinetic glass transition by fixing a conventional value for the maximum experimental time we are prepared to wait in order to equilibrate a liquid. This was first suggested in laughlin and fixed at $100-1000$ seconds, so that we can write,

By using the relation (12) between shear viscosity, relaxation time and infinity frequency shear modulus, and using the standard value for $G_{\infty}\sim 10^{10}-10^{11}$ dyne/cm2, equation (48) is equivalent to define $T_{g}$ through the following classic relation,

At the melting point a liquid’s viscosity seldom exceeds the value $\eta\sim 10^{-2}-10^{-3}$ Poise, so we see that at the glass transition the viscosity is pretty huge.

Of course, if we decrease by several orders of magnitude our experimental available time, than we do move upward significantly the glass transition. This is exactly what happens in numerical simulations of liquids, which, compared to real experiments, can cover a much smaller time interval. For this reason, the definition of ’glass transition’ in numerical simulations is ruled by the CPU time, rather than the simulated ’real’ time of the liquid, and $T_{g}$ is significantly higher than in realistic experiments.

The conclusion of this section is that, yes, the glass transition is a meaningful concept, at least from a practical point of view. Even though it is not a sharply defined transition, the increase in the relaxation time is so steep, that for all practical purposes it makes sense to mark a point where this happens. As we shall see in the next section, for some systems $T_{g}$ not only makes practical sense, but it also cries for a physical explanation.

IV.3 Fragile vs. strong liquids

The conventional definition of glass transition given in (49) allows us to report in a compact way the viscosity data of many different substances in a single plot. The benchmark for the increase of the relaxation time (and thus of the viscosity) is the Arrhenius law (45). For this reason data are normally plotted as a function of $1/T$, as done in Fig.6. In this way, however, there is a spread of the data due to the different values of $T_{g}$ of the various systems. To balance this we can plot the logarithm of the viscosity as a function of $T_{g}/T$, in such a way that all curves have the same value (that is $10^{13}$ Poise) at the point $T_{g}/T=1$. This kind of graph, conventionally called Angell’s plot laughlin; angellplot, is shown in Fig.8, and it is quite instructive.

We see that different substances may have quite different behaviours. Some systems fall on a straight line, meaning that the simple Arrhenius behaviour is a reasonable description. Other systems, on the contrary, are very far from an Arrhenius behaviour and display a smooth growth at high temperatures, which becomes however steeper and steeper as the temperature is lowered. These two extremes cases have names angellplot: the almost purely Arrhenius systems are called strong liquids, whereas those that have a sharper super-Arrhenius behaviour, displaying the largest deviation from a straight line in the $T_{g}/T$ representation, are called fragile liquids. Archetypical strong liquids are SiO2 (window glass) and GeO2, whereas o-terphenyl and toluene are two champions of the fragile class.

Of course, we have to be careful with the strong-fragile classification, because, as Fig.8 shows clearly, there is a whole spectrum of behaviours in between the two extrema. Moreover, even in the most fragile cases, it is clear that a piece-wise Arrhenius fit, with two different values of the barrier $\Delta$ at high and low temperatures, will fit very reasonably the data (as can be appreciated also by eye). The only important point we have to grasp here is that the increase of the viscosity by lowering $T$ is not equally sharp in all systems. For strong-like liquids the increase is relatively smooth, in the sense that it is “only” exponential. For strong systems it seems that at $T_{g}$ really nothing particular happens, apart the viscosity hitting $10^{13}$ P; in this case, one may argue that the definition of $T_{g}$ has purely practical implications, and nothing more.

On the other hand, in strongly fragile liquids the increase is sharper the closer we get to $T_{g}$. Fragility can be easily quantified by measuring the slope of ${\rm\;log}(\eta)$ vs. $T_{g}/T$ at the point $T_{g}/T=1$: the larger this quantity, the sharper the growth of the viscosity, the larger the fragility. Fragile behaviour suggests that something new does happens close to $T_{g}$, and therefore that the glass transition is a more fundamental quantity for fragile systems than for strong ones. For this reason, fragile liquids are the most interesting glass-formers, and many of the questions we ask about glassy physics are in fact pertinent only for this kind of systems. If a strong glass-former may be interpreted simply as a very very viscous liquid, this seems to be inappropriate for fragile liquids, which seems to require a deeper explanation.

We have provided an unambiguous definition of the glass transition, equation (48), and we have reasons to believe that the physics of fragile liquids becomes tricky close to the glass transition. Yet, we must admit that we are still somewhat unsatisfied: the $T_{g}$ we introduced is indeed well-defined, but entirely conventional. It seems a completely anthropocentric concept: give that our lifespan is $\penalty\ 10^{2}$ rather than $\penalty\ 10^{30}$ years, and given that that we get bored at doing experiments longer than $1$ day, we give a different name to samples whose relaxation time exceeds a certain threshold, and we call them glass. Even though in very fragile liquids one feels that $T_{g}$ has a deeper meaning, from the formal point of view there is little or nothing upon which to base this judgment. Fragility is a quantitative feature, not a qualitative one. What about the whole zoology of intermediate systems? What is $T_{g}$ for them? The problem is that, from what we have seen up to now, nothing qualitatively remarkable happens at $T_{g}$, apart from our inability to keep the sample at equilibrium. Let us reformulate the problem in the following way: if our lifespan were $\penalty\ 10^{30}$ years, and we were able to equilibrate a supercooled liquid well below $T_{g}$ as defined in (48), would we notice any qualitative difference compared to the high $T$ phenomenology? Fortunately for the popularity of the field, the answer to this question is: yes.

IV.4 Supercooled liquids are structurally unexciting

Before we see how to answer positively to the question above, it is important to point out that standard structural observables cannot distinguish the deeply supercooled phase at a qualitative level. The static structure of the particles in a supercooled liquid close to $T_{g}$, and even in a glass below $T_{g}$, is virtually indistinguishable from that of a liquid at temperatures well above $T_{g}$. As far as structure is concerned, a glass looks exactly the same as a liquid. Let us see in more detail what ‘structurally’ means here.

The simplest way to give a structural characterization of a homogeneous and isotropic liquid is to consider the radial distribution function, $g(r)$ hansen; allen. This quantity tells us what is the probability to find a particle at distance $r$ from a certain focal particle. Its formal definition is the following,

where $N$ is the total number of particles, $\rho$ is the density and,

This shows that the mean local density at distance $r$ from a focal particle is equal to $\rho\,g(r)$. From its definition it is clear that $g(r)$ must go to $1$ for $r\to\infty$, whereas it has a nontrivial structure at finite values of $r$.

The radial distribution function $g(r)$ is very good at distinguishing different phases (gas, liquid, crystal) of a particle system: the higher the degree of order in the system, the more structured in term of peaks is the $g(r)$. In a liquid, the typical shape of this function is showed in Fig.9: at small $r$, $g(r)$ is zero, due to the short-range repulsion that prevents particles from getting too close to each other; at larger $r$ the function steeply rises in correspondence to the first layer (or shell) of particles around the focal one; at even larger $r$ there are some weaker, although still well defined, peaks corresponding to the various shells around the focal particle. In the liquid there is no long range order, so that the peaks are weaker and weaker the larger $r$. In a crystal, on the other hand, the peaks are very sharp, and do not decay, because long range order sets in. On the contrary, in a gas there is only the drop of probability at very low $r$ due to the hard core of the particles, and no peaks at all, since there is no structure.

From the experimental point of view the easiest thing to measure is the static structure factor $S(q)$, which is experimentally accessible from inelastic neutron scattering hansen. This quantity is related to the radial distribution function $g(r)$ by a simple Fourier integral hansen; allen,

and it provides in momentum space $q$ the same kind of structural information as $g(r)$.

The weak modification of the structure factor with temperature also implies that any lengthscale that can be reasonably extracted from $S(q)$ or $g(r)$ shows a depressingly weak dependence on $T$ near the glass transition price; ernst-91; blaade; leheny. This is surprising. The common wisdom is that the presence of a sharply increasing (or diverging) relaxation time should be associated to a sharply increasing (or diverging) correlation length. The wisdom comes mainly form the theory of critical phenomena cardy, and the argument is basically that a large relaxation time derives from the need to rearrange larger and larger correlated regions. Even though common indeed, such wisdom deserve some carefulness anyway. First, we cannot stress too much the (obvious) fact that in a system with a finite number of degrees of freedom, the relaxation time must be equally finite. A real divergence can only occur in the thermodynamic limit and only in presence of a phase transition. Still, the time can be very large, typically because of the presence of large energy or free energy barriers to relaxation. In general, a barrier could arise from an external potential, and in this case there is no need to invoke the existence a large length scale. However, in all interesting systems barriers almost invariably arise from the internal interaction among the degrees of freedom. In this case, increasing barriers (and thus increasing relaxation times), are indeed due to the increasing number of degrees of freedom that must be rearranged together to relax the system. Hence: large time, large length.

This argument makes sense, but of course in order to detect a sharply growing correlation length, we need to identify a suitable correlation function. This is the crucial point: in very viscous liquids and glasses we are completely lost when it comes to identify the correct order parameter. Structural correlation functions are not up to the job and fail to provide any exciting characterization of the viscous and glassy phase. This is unfortunately true for all other standard static correlation functions.

The reason for this can be either because we are using the wrong correlation function, or simply because there is no sharply growing lengthscale (the common wisdom could be wrong!). In fact, we will see at the end of these notes that the first hypothesis is the correct one. For now, we simply note that the standard structural observables are not good enough to say something about the deeply supercooled phase. Structure remains qualitatively the same going through the glass transition. Supercooled liquids and glasses are structurally unexciting.

IV.5 Equilibrium fingerprint of glassiness: two steps relaxation

The failure of a standard static approach to find a signature of the glass transition is, in fact, not surprising. The very definition of the glass transition is purely dynamic in nature, and therefore if something new happens close to $T_{g}$, the dynamics, rather than the static structure, should detect it. In particular, the viscosity, which marks the onset of glassiness, is the integral over time of a dynamic correlation functions, namely the shear relaxation function, see equation (11). The same is true for the diffusion coefficient, related to the time integral of the velocity-velocity correlation function (18). An integral wraps up an entire function into a single number, thus loosing a lot of information. Hence, it seems a good idea to check the dynamic correlation functions, rather than their integrals, to see whether they show some qualitative signature of $T_{g}$.

Let us consider, in full generality, the dynamic correlation function,

where $\varphi_{k}(t)$ is a generic quantity relative to particle $k$, observed at time $t$. If the system is at equilibrium, then time translation invariance (TTI) holds, and the correlation function only depends on the difference of times, $t=t_{2}-t_{1}$, so that $C(t_{1},t_{2})=C(t)$, and we can write,

In liquids, a typical choice for $\varphi_{k}(t)$ is the Fourier transform of the density fluctuations of a tagged particle $k$, $\delta\rho_{k}({\bf q},t)=\exp[-i{\bf q}\cdot{\bf r}_{k}(t)]$, at fixed momentum $\bf q$. In this case, the dynamic correlation function $C(t)$ coincides with the incoherent intermediate scattering function $F_{s}(q,t)$ hansen, which is normally measured in experiments. In systems other than liquids, $\varphi_{k}(t)$ can be any meaningful observable carrying a real space label (be it particle or spin).

The correlation function $C(t)$ measures how quickly correlations within the system decay in time. At high temperatures we expect a very short-time ballistic regime (for Newtonian dynamics), where particles move freely with no mutual interactions, followed by a dissipative regime, described by a normal exponential relaxation,

Two steps relaxation is the qualitative fingerprint of approaching glassiness. By performing a purely equilibrium measurement, we have a clear way to say whether or not our sample is close to the dynamic glass transition, $T_{g}$. The relaxation time increases (more or less sharply) at low temperatures, but it is only from the nonexponential relaxation of the dynamic correlation function, and in particular from the formation of a plateau, that we can say that a system is approaching the glass transition. Of course, the precise temperature window where this non-exponential behaviour kicks in depends on the system (and much less strongly on the observable $\varphi$), but, on balance, it is fair to say that whenever the dynamic correlation function develops a plateau, the glassy phase is not far down in temperature, even though the relaxation time (and the viscosity) may still be significantly lower than their $T_{g}$ values.

The nonexponential relaxation and the plateau of the dynamical correlation function appears above the glass transition $T_{g}$, as a qualitative precursor of it. We therefore see that the glass transition $T_{g}$ does have some physical significance, after all, because something qualitatively new shows up in the equilibrium properties of a liquid close to $T_{g}$.

or a time-scale proportional to the integral of $C(t)$,

The appearance of two steps relaxation in the dynamic correlation function occurs gradually on approaching the glassy phase. For this reason it is not possible to sharply define a temperature where the departure from the high-$T$ exponential behaviour firstly appears. Under this respect, we may say that the transition from exponential to two steps relaxation, and thus from fluid to glassy behaviour, is continuous. However, under another important respect the transition can be considered as discontinuous. Indeed, the height of the plateau, i.e. the value $C(t)$ has at the plateau, is already different from zero when the plateau appears.

We can make this observation more concrete in the following way. Imagine we are at a temperature $T$ where the plateau is well defined. By using some reasonable fitting method, we can measure the height of the plateau, let us call it $C^{\star}$. This quantity is sometimes called nonergodicity parameter in the literature. If we now raise the temperature we observe that $C^{\star}$ decreases when $T$ increases sastry-00. This dependence, however, is rather weak (Fig.11). For this reason the value of $C^{\star}$ is still well above zero when we reach the temperature regime where there is no plateau anymore. In this regime the nonergodicity parameter $C^{\star}$ is ill-defined and we cannot assign a value to it. Conversely, if we decrease, rather than increase, the temperature, we observe that as soon as the plateau exists, $C^{\star}$ has a nonzero value. In this sense the transition from the fluid to the glassy phase is discontinuous. We will go back to this crucial point when discussing mode coupling theory and spin-glasses.

IV.6 The cage

What is the origin of two steps relaxation? To have a first clue we turn to a different dynamical observable: the mean square displacement (MSD) of a tagged particle,

We have already met this quantity in the Section II.B when discussing diffusion. We expect the MSD to have an early regime where $\langle r^{2}(t)\rangle\sim t^{2}$, when particles move ballistically without many collisions, followed by a diffusive regime, where $\langle r^{2}(t)\rangle\sim t$, dominated by collisions. In fact, as we have seen, the diffusion coefficient $D$ is given by the ratio between MSD and time in the asymptotic regime.

In a deeply supercooled liquid, however, when we plot the MSD as a function of the logarithm of time, the ballistic and diffusive regimes are separated by a plateau, whose length increases on lowering the temperature (Fig.12). The similarity with the dynamical correlation function (Fig.11) is obvious, but the advantage now is that the immediate physical meaning of the MSD provides a simple interpretation of what is going on. In the time region of the plateau the MSD increases very little with time. The tagged particle is definitely beyond the ballistic regime, but despite its many collisions with the other particles something prevents it from a standard diffusive motion, and the particle remains confined in a small region of space. If we look at the actual value of the MSD in correspondence of the plateau, we discover that it is quite small, well below the (square) interparticle distance kob-msd.

We can explain these facts by hypothesizing that, on the time scale of the plateau, the particle cannot exit the cage formed by its neighbours. Under this interpretation, the plateau corresponds to the vibrations of the tagged particle within the cage. If we wait long enough, though, the particle finds its way out of the cage, and a standard diffusive dynamics sets in. Beware: we are watching things in log-time. Therefore, after the particle has got out of the cage, it is effectively out of any cage! The time needed to get out of the cage is longer the lower the temperature, and it corresponds to the $\alpha$ relaxation process, whereas the $\beta$ relaxation is given by the particle vibrations within their local cages.

This seems a reasonable interpretation, and by similarity it also extends to the plateau in the dynamic correlation function. Few comments, however, are in order. First, the cage explanation is in fact not so much of an explanation: particles have neighbouring particles at all temperatures; what we would like to understand is why below a certain temperature, i.e. close to glassiness, the cage suddenly becomes stiff. Moreover, how the particle gets out of the cage after a certain time, and why this time increases by decreasing the temperature?

That of the cage is little more than a description of what is going on, but it is nevertheless a very useful one. For example, it immediately suggests that to break a stiff cage, and thus to exit the plateau and restore ergodicity, particles must in some way rearrange. This can be done by either finding a rare ‘hole’ through the many energy barriers giving rise to the cage (a process that may be entropically costly, but energetically harmless), or by climbing up these barriers (energetically costly, but entropically harmless). We will get back later to these two alternatives, but the example shows that the cage interpretation should always be kept in mind.

Note that the size of the cage, i.e. the value of the MSD at the plateau, changes very little with the temperature, and it does not approach zero when raising $T$. As soon as we start seeing the cage (i.e. a plateau of the MSD), its physical size is already different from zero. Because of this, the transition from high-$T$ simple diffusion to low-$T$ two steps diffusion is discontinuous for the MSD exactly as it was for the dynamic correlation function.

The cage description of two steps relaxation is deeply rooted in real space: it is only concerned about what the particles do in the real physical space, as opposed to phase space arguments. On one hand this is good, since real physical systems live in real space, and what happens to them is ultimately related to real space phenomena. On the other hand, this is somewhat puzzling, because there are mean-field systems that have no real space structure at all, but where the plateau of the dynamical correlation function is anyway present at low temperatures. In these cases the cage effect cannot be the right interpretation, since there is no real space, no local neighbours and no cage: each degree of freedom (particle) interacts with all the others, and a phase space description is the only one we are left with. These observations do not imply that real space arguments are incorrect. Rather, the presence of mean-field model showing two steps relaxation suggests that there may be an alternative explanation of this crucial landmark of glassiness. As we shall see in the next chapter, this alternative explanation can be given in terms of what the system does in phase space.

A final warning. Even though we described decaging as a single particle event (the particle ‘gets out of the cage’), this is a simplistic view. As we shall see later on, dynamical (as well as static) relaxation is achieved through cooperative behaviour. Many particles must be dynamically correlated in order to unstuck from their local vibrational positions. Hence, one should rather say that it is the cage that collectively breaks up to free the particles, before another cage is formed. However, at this stage of our study we still do not know what are the details of such dynamical cooperativity, so we ask the reader to simply keep this remark in the back of her/his mind whenever we will use the naive short-cut of a particle getting out of the cage. Things may become clearer when we will study the dynamical correlation length.

IV.7 Stretched exponential, dynamical heterogeneities and Stokes-Einstein violation

One may think that the departure of the dynamic correlation function from exponential relaxation is solely due to the break-up of the decay into two steps; this would imply that the late relaxation, i.e. the $\alpha$ decay from the plateau is exponential. However, this is not the case: even the late relaxation is nonexponential. In particular, it is found that the Kohlraush-Williams-Watts stretched exponential form kohl; watts,

fits reasonably well the data. The interesting point is that the exponent $\beta$ decreases when the temperature is decreased, marking a larger and larger deviation from a standard exponential relaxation, the deeper we get into the supercooled phase angell-00. On the other hand, at higher $T$ the exponent $\beta$ approaches $1$, and at the same time the whole two steps structure of the dynamical correlation function disappears, and the relaxation goes back to simple exponential.

Regarding the origin of nonexponential relaxation we can put forward two hypotheses, let us call them the heterogeneous and the homogeneous explanation. According to the first, the relaxation of the entire system is stretched because different regions have significantly different relaxation times, and in this sense dynamics is heterogeneous. When we measure the global relaxation time, we are typically averaging over many different regions, each one with a purely exponential relaxation ruled by a different relaxation time, and from this heterogeneous average one gets a global nonexponential decay. On the other hand, according to the homogeneous explanation, relaxation is equally nonexponential in all regions, so that the stretched nature of the correlation function is not due to an heterogeneous spatial average, but it is an intrinsic phenomenon also at the local level, due to the disordered environment each particle sees around itself.

These two views are not necessarily contradictory. In fact, there are reasons to believe that they are both valid to some degree. There are mean-field systems (as the $p$-spin model) where two steps relaxation and stretched decay of the dynamic correlation function are present at low $T$ just as in supercooled liquids. In these systems, due to their mean-field nature (each spin interacts equally with all other spins) there is no space structure, so that there cannot be any spatial etherogeneities. Still, these mean-field systems are spin-glass models, where there is quenched disorder in the interactions between the degrees of freedom; it is therefore reasonable that the relaxation becomes nonexponential at low $T$, even without the aid of spatially heterogeneous dynamics. In fact, even in mean-field systems, at low $T$, different spins have very different flipping frequencies ricci-03; ricci-03b.

On the other hand, it is now well established both numerically perera-96; donati-97; doliwa-98; donati-98 and experimentally vidal-00; kegel-00 that close to the glass transition the dynamics of real supercooled liquids is heterogeneous. It is possible to directly observe domains few nanometers away from each other with significantly different mobility and relaxation times. Therefore, in real liquids heterogeneous dynamics is likely to contribute substantially to the formation of an overall nonexponential relaxation. Given that heterogeneous dynamics has implications beyond the stretched exponential relaxation, it is worth considering it more carefully. The literature on this subject is vast; see sillescu-99 and ediger-00 for experimental reviews and glotzer-00 for a more numerically oriented one.

Few remarks. First, it is important to appreciate the fact that the time interval $t$ must have an intermediate value, in order to observe an heterogeneous dynamics. Such intermediate value is close to the plateau regime of the dynamic relaxation function $C(t)$ perera-96; donati-97; doliwa-98; donati-98. This observation strongly suggests that the nonexponential relaxation of the dynamical correlation function, whose main feature is indeed the plateau, is linked to heterogeneous dynamics. Each patch has a different value of the local relaxation time, and all locally exponential relaxations add up to a stretched exponential one. Second, when we raise the temperature, the dynamical correlation function and the MSD loose their plateau structure, the ballistic regime is directly followed by the diffusive one, and dynamics becomes homogeneous, irrespective of the value $t$ has. Third, we must not forget that these patches, or clusters, are dynamical, not structural. We could not detect them by using one snapshot of the system. We need two. Mobility fluctuations becomes very large at low $T$, even though nothing similar happens at the structural level.

Heterogeneous dynamics is believed to be at the basis of another important phenomenon in low $T$ supercooled liquids, namely the violation of Stokes-Einstein (SE) relation between diffusion coefficient and viscosity (24),

Close to the glass transition $T_{g}$, the diffusion coefficient $D$ may be several orders of magnitude larger than $T/\eta$ rossler-90; fuja-92; chang-94; cicerone-96. In particular, the ratio $T/(D\eta)$, far from remaining constant as the SE relation wold require, decreases sharply when the temperature is lowered close to $T_{g}$ rossler-90; fuja-92. This means that $D$ and $1/\eta$ have different functional dependence on $T$. Hence, in the temperature regime where structural relaxation becomes very sluggish, self-diffusion is still relatively vital. Why is that?

It is therefore a decoupling between diffusion coefficient and relaxation time that we are after. Now imagine an oversimplified heterogeneous dynamics: there are just two kinds of clusters, fast and slow ones, with relaxation time and diffusion coefficient respectively $\tau_{f}\ll\tau_{s}$ and $D_{f}\gg D_{s}$. In each cluster the SE relation is obeyed, so that $D_{f}\sim 1/\tau_{f}$ and $D_{s}\sim 1/\tau_{s}$. Let us also assume that fast and slow clusters are equally numerous across the system. Under these hypotheses, the global relaxation time measured in an experiment is given by,

whereas a measurement of the diffusion coefficient gives,

The violation of SE relation has been also put in direct connection with the nonexponential decay of the dynamic correlation function. More precisely, by testing the SE relation in various substances at the glass transition, it has been concluded cicerone-96 that the stretched exponent $\beta$ of (60) is smaller the larger is the measured deviation from the SE relation. Furthermore, it has been shown that the SE equation is restored when the diameter of the probe used to measure the diffusion coefficient increases cicerone-96. This makes sense in terms of heterogeneous dynamics: when the probe is large enough, i.e. larger than the size of the patches, we are averaging over many heterogeneities, thus washing out their influence. These results strongly supports the link between stretched exponential, violation of the SE equation and heterogeneous dynamics.

Even though mobility clusters do not necessarily imply cooperativity, their very existence strongly suggests that particles in these patches are in fact correlated in some way, and thus that the cluster size is a very good candidate for a correlation length, even though of a dynamical nature. Indeed, empirical data show that on lowering the temperature mobility clusters grow in size donati-97; donati-98. In the final chapter of these notes, when we will look for a growing correlation length in supercooled liquids, we will see in detail how this dynamical lengthscale can be defined.

V Theoretical views on the glass transition

As I said in the introduction, the quest for a theory of the glass transition is far from being over. Different theoretical frameworks provide different interpretations of the phenomenology, and it is hard to write about this subject without having in mind our own ideas. Here I will try to present a coherent perspective, rather than to enumerate all different approaches. The result will inevitably be partial.

Goldstein’s picture of the equilibrium dynamics of a deeply supercooled liquid is so simple, that it may seem trivial today, although it was definitely not when it was formulated back in 1969 gold. The very fact that his description seems so natural to us is proof of how vast the influence of his work has been on the understanding of this field.

Goldstein put the emphasis on the evolution of the system in the phase space, i.e. the space of all the configurational degrees of freedom. For example, for a monatomic liquid in three dimensions, this is the space of all $3N$ coordinates of the particles. Over this space it is defined the total potential energy of the system, and the surface of this function is often called potential energy landscape. Each different configuration is represented by a point in the phase space, and the dynamics of the system can be thought of as the motion of this point over the potential energy landscape.

The local minima of the potential energy correspond to locally stable configurations of the particle system. One of them is of course the crystal, and this will be the absolute minimum, i.e. the ground state. Moreover, there will be all the minima obtained by introducing defects and dislocations into the crystal; we can see these as excitations over the crystalline ground state. But apart from these crystalline, polycrystalline and defected crystal minima, there will be many local minima corresponding to particles arrangement that are completely lacking long-range crystalline order. These are amorphous, or glassy, minima, and have a potential energy that is extensively larger than the crystal one. For the rest of these notes we will not deal with crystalline and polycrystalline minima and only be interested in amorphous local minima.

Goldstein’s idea is that at low enough temperatures a supercooled liquid explores the phase space mainly through activated jumps between different amorphous minima, separated by potential energy barriers. It is important to understand that the system does not do these jumps in the attempt of reaching lower energy minima: we are considering a system at equilibrium (albeit metastable with respect to the crystal), so that the level of average potential energy where the system lives is constant in time. The system passes from one minimum to another of similar energy (on average), and in so doing it is ergodic and in equilibrium.

Even though the idea of a system globally jumping from one minimum to another is clear enough, we should not forget that the true dynamics of the system takes place in real space. Goldstein himself is very keen in giving a real space interpretation of the hopping process in phase space. The jump over a barrier leading the whole system from a minimum to another corresponds to the local rearrangement of a relatively small number $n$ of particles localised in a limited region of space. Particles far from where the rearrangement takes place are very weakly affected by this process, even though in a phase space description it is the point representing the global system that performs the transition.

Moreover, in a large system the fact that each event of rearrangement (jump over a barrier) is localized in space implies that several independent molecular rearrangements will be occurring in different regions at the same time: however small is the transition rate per unit volume, in a very large system there will always be a nonzero probability to have a rearrangement happening somewhere at a given instant of time (we met a similar argument when discussing nucleation). This, however, may seem a bit counterintuitive: a system that is always on top of a barrier does not exactly match our naive understanding of activated barrier crossing, where one is supposed to wait a long time between different jumps.

This confusion arises from the difficulty in visualizing an activated event performed by a small number of degrees of freedom embedded in a multidimensional space (the phase space), where most of the other degrees of freedom are unaffected by the transition. Once again, it is essential to keep in mind the local nature of the activated jump. Goldstein writes a very illuminating sentence about this gold: “The system will always be in the process of transition, rather than some of the time, but always near a minimum, in the sense that a sudden cooling will drop it into a minimum with relatively small changes of most of the coordinates”. The last words are the most important: even though in principle it is the entire system that performs the transition, in fact it is only a very small subpart of it that crosses the barrier. If we focus our attention on this subpart, we indeed recover our intuitive idea of activated dynamics.

but they are quite larger than that of simple liquids at their melting point, which is of order,

One of the arguments used by Goldstein in his derivation of $\tau(T_{x})$ is worth to be mentioned explicitly. We have stressed above that each activation event takes place locally, and that particles that are far only participate as a background. This background is however crucial, since it constitutes the rigid matrix that provides the potential energy barrier resisting the rearrangement. As long as the temperature is well below $T_{x}$ different events in different positions will be independent, each one of them surrounded by a rigid background. However, when the temperature is raised the activation rate increases, so that there will be more and more rearrangements taking place simultaneously and rather close to each other. In this case, different activation events will no longer be independent. As a consequence, the view of a rigid background breaks down. To locate this temperature Goldstein compares the time to complete a rearrangement with the shear relaxation time, which is a measure of the time the background matrix can retain its rigidity. In this way, using some rather arbitrary, but sensible assumptions, he finds $\tau(T_{x})\sim 10^{-9}$ seconds.

Goldstein’s scenario provides a natural description in terms of two time scales below $T_{x}$: a short time due to the vibrational relaxation within a potential energy minimum, and a long time relaxation due to the transition between different minima. The second time scale is much longer than the first one, since the hopping time is given by activation and it becomes exponentially large when lowering the temperature. The low-$T$ separation of time scales has been indeed confirmed via numerical simulations and it has been used as an explicit way to locate $T_{x}$ sastry-00.

This separation of time scales seems to fit very well the two steps relaxation of the dynamical correlation function. In particular, one may interpret the short vibrational time scale and the long activated time scale of Goldstein’s description respectively with the $\beta$ (fast) and $\alpha$ (slow) relaxation of the dynamic correlation function. Under this interpretation, vibrations in the phase space around a single potential energy minimum correspond to vibrations of particles within their cages, whereas crossing of a potential energy barrier corresponds to the local decaging of some particles. By accumulating many decaging events, i.e. many activated rearrangements, the system decorrelates and the dynamic correlation function leaves the plateau, giving rise to the $\alpha$ relaxation.

V.2 Mode coupling and the infamous pp-spin

Placing two rather extensive subjects as Mode Coupling Theory (MCT) and mean-field Spin-Glass Theory (SGT) in the same section is perhaps the most pedestrian choice of this entire review. Mode coupling theory and mean-field spin-glass theory have been developed completely independently, within rather different communities. MCT was formulated in the attempt to describe at a quantitative level the dynamical properties of supercooled liquids. SGT, on the other hand, was originally developed to study the properties of a class systems (spin-glasses), which had apparently little in common with liquids. Just to mention one thing, spin-glasses have quenched disorder, liquids do not. However, at some point the SG community stumbled upon a class of models that showed a phenomenology strikingly similar to that of supercooled liquids. The $p$-spin model is the paradigm of such class. It was proved that this particular model was described by a set of dynamical equations formally identical to those obtained by MCT. Moreover, and most importantly, these equations reproduced the two steps relaxation typical of supercooled liquids, even though none of the two theories made use of activation. Therefore, the reason for putting together these two theoretical frameworks is simple: both MCT and SGT provide a physical mechanism able to explain the two steps relaxation of the dynamical correlation function without resorting to thermal activation.

I will not give any detail about MCT and SGT, because the literature on both MCT and SGT is already extensive. Apart from the original papers on MCT bengzelius; leuth, useful reviews can be found in goetze-review; sjogren; kob-review; kob-binder. On the other hand, classic reviews on SGT can be found in binder-young; virasoro; hertz; young. For a more specific, yet elementary, account of the $p$-spin model and of its similarities with supercooled liquids see cavagna-review. For the quantitative connection between $p$-spin model and MCT dynamical equations see bouchaud-96.

By using the Zwanzig-Mori formalism zwanzig-61; mori, MCT aims to write a set of self-consistent equations for the dynamical correlation function of the density fluctuations. This program is rather complicated, and MCT needs to make a number of simplifying approximations. Even though these approaximations are the defining core of the theory bengzelius; leuth, it is unfortunately, not that easy to explain the physical intuition behind them (for a diagrammatic approach see bouchaud-96). Here we simply note that the approximations adopted made it possible for MCT to formulate a set of dynamical equations that can be handled numerically, and even analytically at the qualitative level goetze-review. The crucial point is that the input of the MCT equations is given only by the static observables, and in particular the static structure factor $S(q)$ (53). This is somewhat surprising, since we have seen that structural quantities do not show anything peculiar close to $T_{g}$: how can the MCT equations be anymore exciting, having the boring structural quantities as an input?

The answer is the very nonlinear form of the interaction vertex in the MCT equations, which gives rise to a sharp feedback between static structure and dynamics: even a tiny change in the structural properties causes a steep slow down of the dynamical relaxation goetze-review. This feedback in the MCT equations has a striking consequence: at low temperature the MCT dynamical correlation function displays two steps relaxation, developing a plateau whose length increases when the temperature is decreased. This is qualitatively the same as in real supercooled liquids. Moreover, the behaviour of the nonergodicity parameter $C^{\star}$, i.e. the height of the plateau, is as discontinuous as in liquids: as soon as the plateau can be defined, its value is already different from zero. The qualitative agreement between MCT and real liquids regarding the shape of the dynamic correlation function is an argument in favour of the theory. But it is not the only one.

MCT predicts in a quantitative fashion the way the correlation function should arrive to and depart from the plateau, and thus it gives a precise description of both the $\beta$ (fast) and $\alpha$ (slow) relaxation. These prediction agrees rather well with the experimental and the numerical evidence kob-prl-94; kob-msd; kob-corr; gotze-99. This is remarkable, and sometimes overlooked: in the general quest for a theory able to explain the long time dynamics, or $\alpha$ relaxation, most of the theories simply forget about the short time $\beta$ relaxation, which is however such a conspicuous feature of glassy relaxation: as we have seen, the only qualitative fingerprint of glassiness is two steps relaxation, not simply the sharp growth of the relaxation time.

MCT thus captures the qualitative feature of two steps relaxation and it also gives some quantitative predictions about the approach and departure from the plateau, which are (depending on the system) in more or less good agreement with the data. However, there is also a down side. The theory predicts that the length of the plateau, and thus the $\alpha$ relaxation time diverges at a finite temperature, $T_{c}$. The predicted divergence is a power law,

If the divergence located by MCT were at very low temperature, well below $T_{g}$, we could not completely rule out its existence. However, this is not the case, and there is in fact overwhelming evidence that such divergence is not present in experimental data, but it is rather an artefact of the theory.

Whenever the relaxation data are fitted to a power law, as in (70), the resulting $T_{c}$ is invariably and significantly larger than $T_{g}$, so that the $\alpha$ relaxation time is not infinite at $T_{c}$. In fact, even in those systems where MCT is most successful what happens is the following: the data follow rather well a power law, with a certain critical temperature $T_{c}$. However, the closer one gets to $T_{c}$, the larger the discrepancy between data and fit is, until right at $T_{c}$ such discrepancy is infinite angell88. Unfortunately, this means that whenever we want to perform a MCT fit to a data-set we are in troubles. Imagine we include in the fit data in the range $T_{1}\leq T\leq T_{2}$, and get a certain $T_{c}$, which is by construction lower than $T_{1}$. If we now perform again the fit using the data in the interval $T_{c}\leq T\leq T_{2}$ we are bound to obtain a lower critical temperature $T_{c}^{\prime}$, since at the former $T_{c}$ the fit will try to stick to the data, rather than locating a divergence.

Why MCT predicts an inexistent divergence? There is a common consensus that this happens because MCT in its original form does not take into account activated barrier crossings. The relaxation mechanism of MCT is something different from activation, even though it is not easy to understand its nature by a mere inspection of the MCT equations. Below $T_{c}$ the MCT dynamical mechanism is completely stuck, so that the theory cannot go beyond $T_{c}$ and it therefore locates a divergence here. A real liquid, however, can switch from the MCT mechanism to activated barrier crossing, thus keeping the relaxation time finite, although sharply increasing with lowering the temperature. When we include low $T$ data in a MCT power law fit, we are effectively asking the theory to work in an activated regime, where it is not supposed to work

This considerations suggest a key connection with Goldstein’s scenario. As we have seen above, activated dynamics only makes sense at temperatures low enough, where potential energy barriers are substantially larger than the thermal energy, i.e. for $T<T_{x}$. Above $T_{x}$ Goldstein’s scenario breaks down because activation is no longer the main mechanism of relaxation. On the other hand, MCT breaks down below $T_{c}$ because activation becomes the main mechanism of relaxation. This strongly suggests the identification,

The identification of the MCT transition temperature with Goldstein’s crossover temperature has an important implication: even though one knows there is no real divergence, it is nevertheless very useful to work out $T_{c}$ (typically via a power law fit of the data) as a reference temperature for a system approaching glassiness. In fact, from what we have discussed above one may conclude that the definition of $T_{x}\sim T_{c}$ is more fundamental than that of $T_{g}$, which depends on an arbitrarily fixed time or viscosity scale (see equation (49)). For example, the position of the dynamic glass transition $T_{g}$ is effectively very different in experiments and numerical simulations, since in the latter we can reach much shorter time scales; on the other hand, the temperature where a crossover from nonactivated to activated dynamics takes place is independent from the experimental time, so that $T_{x}\sim T_{c}$ is conceptually a better defined quantity. In particular, while in real experiments $T_{g}$ can be quite below $T_{c}$, in numerical simulations, where $T_{g}$ is pushed at much higher temperatures due to the shorter available time, the two temperatures are typically much closer. For this reason, it is somewhat customary to use $T_{c}$ as a landmark of impeding glassiness in numerical simulations.

The scenario depicted above, and in particular equation (71), relies on the hypothesis that $T_{c}$ is the limit of validity of MCT: the theory does not include activated events, and therefore at the temperature where there is a crossover from non-activated (MCT) to activated (Goldstein) dynamics, MCT locates a divergence. However, we did not provide any solid evidence in favour of this hypothesis, apart from saying that there is common consensus. In particular, given the complete lack of details in our exposition of MCT, it is far from obvious that the theory’s failure at low $T$ is the result of disregarding activation. One could argue that the spurious divergence at $T_{c}$ just means that MCT is wrong, end of story. To investigate this point we take a rather indirect road, and dive into the physics of spin-glass.

After the explosion of spin-glass physics in the early 80s, a great deal of discussion started about the possible analogies between glass-forming liquids and spin-glasses. However, there was one conceptual problem: supercooled liquids display a sort of discontinuous transition, whereas spin-glass models analyzed in the early days had a continuous nature. We have seen above what ‘discontinuous’ means in the context of the dynamical correlation function of a liquid close to $T_{g}$: when the plateau develops at low $T$, its height is already quite different from zero. The discontinuous nature of the structural glass transition is something more profound than the behaviour of the dynamic correlation function; it is a trait that pops out in very diverse contexts, including off-equilibrium behaviour. For example, when the system goes out of equilibrium at $T_{g}$ the specific heat jumps discontinuously to a smaller (crytal-like) value (Fig.7), rather than simply changing slope. This discontinuous nature of glass-forming liquids clashed with early spin-glass models, which had a continuous behaviour of all the corresponding observables. In particular, the nonergodicity parameter was zero at the (spin) glass transition virasoro.

Quite soon, however, it was discovered a new class of mean-field spin-glass models that displayed a discontinuous glass transition. These models included the random energy model derrida-81, the $p$-spin model gross-84, and the Pott’s spin-glass model gross-85. In the following we will use the $p$-spin models as a representative of this entire class of systems. It was Kirkpatrick and Wolynes who first noted in kirk-0 the close analogy between the discontinuous transition in this class of spin-glasses and that of glass-forming liquids. They also noted the similarity between dynamic spin-glass theory and mode coupling theory. In a subsequent series of remarkable papers kirk-1; kirk-2; kirk-chi4; kirk-3, Kirkpatrick, Thirumalai and Wolynes reached a deep understanding of this class of spin-glasses and of its connections with structural glasses.

The Hamiltonian of the $p$-spin model for $p=3$ is the following,

The degrees of freedom $\sigma$ are spins (they may be real or integer variables) and the couplings $J$ are quenched random variables cavagna-review. The model is mean-field because all spins interact with all other spins: the couplings $J$ do not decay with the distance between spins. This is therefore a model where there is no underlying lattice, nor space structure at all, where each spin is equally close (or distant) to every other spin (such mean-field models are also called fully connected or infinite dimensional models; for this reason, normal, non-mean-field systems are sometimes called finite dimensional systems). We wrote explicitly Hamiltonian (72) just to make it clear how far is this model from a real liquid, whose Hamiltonian looks something like,

where the degrees of freedom are the positions $\vec{r}_{i}$ of the particles and $V(r)$ is a pair-wise interaction potential.

It therefore comes somewhat as a surprise to find some impressive analogies between the phenomenology of the $p$-spin model and that of real supercooled liquids. Thanks to its mean-field nature, the Langevin dynamics of model (72) can be studied analytically kirk-2, and in particular one can write a self-consistent equation for the correlation function,

which at equilibrium only depends on the time-difference $t$. This function is large when the two configurations $\sigma(t_{0})$ and $\sigma(t_{0}+t)$ are very similar, and it decays to zero when they becomes completely decorrelated. What is found from the equations is that at low temperatures the correlation function of the $p$-spin model develops a plateau, giving rise to the (now familiar) two steps relaxation pattern. As in real liquids and in MCT, the length of the plateau increases when decreasing the temperature and the formation of the plateau has a discontinuous nature. Unlike real liquids, however, but similar to MCT, the length of the plateau (and thus the relaxation time $\tau$) diverges as a power law at a finite temperature $T_{c}$, called the dynamical transition in the spin-glass literature The dynamical transition is sometimes indicated with $T_{d}$ in the spin-glass literature. Here we call it $T_{c}$ in analogy with the MCT transition, in order to keep as small as possible the number of different labels for the same concept..

The interesting thing is that no thermodynamic (static) divergence, nor anomaly takes place at $T_{c}$: the specific heat, free energy and so on show nothing noteworthy around $T_{c}$: from a standard thermodynamic treatment of the model, it is impossible to detect the dynamical transition. What happens at $T_{c}$ is that the system remains dynamically trapped within metastable states. Due to the fully connected interaction, in a mean-field model metastable states are surrounded by infinite free energy barriers that the system cannot surpass by using thermal activation cavagna-review. For this reason in mean-field metastable states have an infinite lifetime and can be sharply defined. The system is therefore forbidden to restore ergodicity, and a true dynamical divergence occurs. Of course, metastable states are irrelevant from a thermodynamic point of view, since their individual weight is negligible. For this reason $T_{c}$ goes undetected by a standard thermodynamic investigation. I strongly encourage the reader to see the original papers by Kirkpatrick, Thirumalai and Wolynes about the nature of the dynamic transition at $T_{c}$, and in particular the illuminating discussion at the end of Ref.kirk-chi4.

The fact that two steps relaxation in the $p$-spin models ends up in a true dynamical transition at $T_{c}$ is quite similar to what happens at the corresponding $T_{c}$ in the context of MCT. Moreover, this divergence is described by a power law both in MCT and in the $p$-spin. All these coincidences are not fortuitous. In fact, the dynamical equations for the correlation function analytically found in the $p$-spin model are formally identical to those formulated by MCT kirk-2; bouchaud-96. This is what really brought together for the first time the supercooled liquids and spin-glass communities. MCT writes a set of dynamical equations that are formally identical to those describing the exact dynamics of a mean-field model, where, by definition, activation is forbidden. Even though far from a theorem, it seems reasonable to conclude that in doing its approximations MCT lost the ability to describe activated events. For this reason MCT predicts a divergence at $T_{c}$, which is formally completely equivalent to $T_{c}$ in the $p$-spin. Following the same line of thought, the nonactivated relaxation mechanism of MCT and that ruling the dynamics of the $p$-spin above $T_{c}$ are probably similar.

At this point one may ask: given that the $p$-spin model is so similar to MCT, why do we need to bother about it? After all, our dish was already full enough! The reason why it is useful to consider both these theoretical frameworks is that they originate from completely different starting points, and this helps us a lot in understanding what is going on. MCT is an approximated theory of a real system staged in real space; the cage effect is the pivotal interpretation of two steps relaxation in terms of real particles. The correlation function leaves the plateau when particles get out of their cage. The $p$-spin model, on the other hand, has no real space structure: no distance, no neighbouring spins (particles), and thus no cage, decaging, and all that. Any reasonable interpretation of the dynamical slowing down for the $p$-spin model must be staged in phase space, not in real space: activation is forbidden by construction, not because of an approximation.

Despite these differences, the phenomenology and the formal structure of the dynamical equations of the $p$-spin model are identical to MCT. It is clear that understanding what is the dynamical relaxation mechanism in the $p$-spin above $T_{c}$, can be very helpful to clarify what is the origin of two steps relaxation and decaging in MCT, and perhaps in real liquids too. Indeed, whereas MCT is somewhat the pinnacle of the analytic effort in real liquids, and it is very hard to go beyond it, the big advantage of the $p$-spin is that, being a mean-field model, many calculations can be performed exactly, and its physical behaviour can be understood in great detail.

V.3 From minima to saddles

To understand the origin of the dynamical singularity in the $p$-spin model we have to investigate the topology of the potential energy surface visited by the system at a certain temperature $T$. Roughly speaking, we shall see that Goldstein’s activated scenario corresponds to a motion from minimum to minimum in the phase space, whereas MCT/$p$-spin nonactivated dynamics corresponds to a motion from saddle to saddle. Most of the following concepts are treated in more detail in cavagna-review, where all the relevant references may be found.

Let us start below the dynamical transition, $T<T_{c}$. In the $p$-spin model we can (analytically) equilibrate a configuration in this temperature region, and compute the dynamical correlation function burioni. If we do this what we find is that for large times $C(t)$ approaches a plateau and it never leaves it. In other words, after a brief $\beta$ relaxation, the correlation function does not show any $\alpha$ relaxation, since the $\alpha$ relaxation time is infinite. This is due to the fact that ergodicity is broken, and the system never leaves the metastable state where it has been initially equilibrated. The system is at equilibrium within this particular state, but it cannot jump out, because free energy barriers are infinite. Thus, ergodicity is broken and there is no long term relaxation.

The nice thing about the $p$-spin model is that the topological properties of the minima trapping the dynamics for $T<T_{c}$ can be worked out very precisely cavagna-review. By topological properties we mean essentially two things: the height of the minimum in the landscape, i.e. the energy density $E$ of the bottom of the well, and the shape of the minimum in the $N$-dimensional phase space, where $N$ is the number of degrees of freedom. In a minimum, by definition, the gradient of the potential (i.e. the force) is zero, and thus it is the matrix of the second derivatives, the so-called Hessian, that contains much of the information about the shape. The $N$ eigenvalues of the Hessian give the curvature along the $N$ directions (eigenvectors) out of the minimum. By definition of minimum, the eigenvalues are all positive, but their values tells us how soft or stiff is the minimum: a large eigenvalue $\lambda_{i}$ means that the energy climbs very rapidly along direction $i$, whereas a small (but positive) eigenvalue $\lambda_{i}$ tells us that direction $i$ is almost flat (at least at the second order of the expansion). Such soft directions are called marginal when the eigenvalue is zero Each continuous symmetry of the system, i.e. each continuous transformation that leaves the energy invariant, generates a null (marginal) eigenvalue of the Hessian. For example, when translational invariance holds in $d=3$ there are $3$ zero eigenvalues of the Hessian, corresponding to the three directions along which the system can be moved without changing the energy. Clearly, these directions correspond to trivial global movements of all degrees of freedom, and do not contribute to the relaxation of the system. To fix ideas, it is better to assume that all continuous symmetries are broken, and that the Hessian has no trivial zero modes..

A compact way to assess the eigenvalues of the Hessian is to define the spectrum, i.e. the fraction of the eigenvalues equal to a particular value $\lambda$,

The question now is: what is the typical spectrum $\rho(\lambda)$ of the minima trapping the dynamics for $T<T_{c}$?

What the topological transition has to do with the real dynamics of the system, and in particular with the dynamical transition at $T_{c}$? After all, there is nothing dynamic nor thermal in the energy landscape: it is the same at all temperatures and times. However, the properties of the landscape depend on the energy level $E$ at which we explore it, and this in turn depends on the temperature. We must only be careful to a point: when we consider a minimum with energy $E$, this is the energy density of the bottom of the well; but at finite temperature the system vibrates around the minimum, so that the average potential energy density of the system will be larger than $E$ by factor roughly equal to $1/2\;k_{B}T$. We will refer to $E$ as the bare energy of the system. At low temperature, the full average energy $\langle H\rangle$ is equal to the bare energy plus thermal fluctuations,

The bare energy $E$ is a better label than $\langle H\rangle$ because it is more directly connected to the topological properties of the stationary point. By $E(T)$ we indicate the bare energy of the typical minima trapping the system at temperature $T<T_{c}$. In the $p$-spin we can compute exactly the curve $E(T)$: as expected, the bare energy grows with the temperature, meaning that the system is trapped by higher energy minima the higher the temperature. If this is not particularly exciting, the following result is,

The bare energy of the minima visited by the system at the dynamical transition $T_{c}$ is exactly equal to the threshold energy. Moreover, it is possible to prove that for $T>T_{c}$ the closest stationary point to an equilibrium configuration is not a minimum, but a saddle; the fraction of unstable (negative) eigenvalues of this saddle goes to zero for $T\to T_{c}^{+}$ cgp-01.

Note that in the $p$-spin model for $T<T_{c}$ it is possible to sharply define the nonergodicity parameter $C^{\star}(T)$ as the infinite time limit of the correlation function once the system is thermalized within one state,

The discontinuous nature of the transition amounts to say that,

The $p$-spin model therefore provides in natural way a nonactivated mechanism of diffusion, whose slowing down is responsible for the appearance of the two steps relaxation in the correlation function. We recall that the exact dynamical equations of this model for $T>T_{c}$ are formally identical to those of MCT in the same temperature regime. Given that MCT describes supercooled liquids reasonably well for $T>T_{c}$, it is natural then to ask whether the nonactivated diffusion mechanism of the $p$-spin model above $T_{c}$ is valid also in supercooled liquids. Is the decreasing number of unstable directions of saddle points responsible for the two steps relaxation, and the corresponding slowing down, in the dynamics of supercooled liquids?

V.4 Topological origin of the glass transition

Saddles of the potential energy are present also in finite dimensional systems as real liquids, of course. However, after Goldtein’s paper in 1969 gold numerical investigations focused for a long time only on potential energy minima. In particular, given an equilibrium configuration of the system, it is possible by following the gradient of the potential energy, to associate a unique local minimum to the initial equilibrium configuration stillinger-weber. This local minimum is also called inherent structure in the literature. The partition of the phase space into basins of the local minima is well defined and unique, and it suites perfectly Goldstein’s scenario of an activated dynamics between different basins. This was the reason why local potential energy minima have largely dominated the analysis of supercooled liquids for quite a while. It has been indeed thanks to such numerical investigations that it has been possible to validate Goldstein’s picture sastry-00 and to understand many important properties of the dynamics of supercooled liquids at low temperatures angell-95; stillinger-95; sastry-98; wales-04; sciortino-05.

However, we know that for $T>T_{x}\sim T_{c}$ the description of the dynamics in terms of minima is not appropriate, since activation is no longer the main mechanism of diffusion in this regime. We also know that two steps relaxation is anyway present for $T>T_{c}$, and that MCT reproduces it. Finally, we have seen that the $p$-spin model gives a neat framework to explain nonactivated two steps relaxation: the cornerstone of this framework is the topological transition between minima and saddles taking place in the energy landscape. All these considerations suggest that in order to better understand the crossover between nonactivated to activated dynamics at $T_{x}$ we need to study the properties of saddles of the potential energy.

A first attempt to exploit unstable modes of the potential energy to study the diffusion properties of supercooled liquids was done by the Instantaneous Normal Mode (INM) approach inm. In this context, one would compute the fraction of negative eigenvalues of the Hessian averaged over all the instantaneous equilibrium configuration visited by the system. The extrapolated temperature where the fraction of negative modes goes to zero should give an estimate of the glass transition. Despite the great interest it stirred, the INM method is affected by a big problem: many of the negative modes visited by an instantaneous equilibrium configuration have nothing to do with diffusion, so that the INM negative modes do not correlate very well to the slowing down of the dynamics. What one should do is to compute the negative modes in correspondence of a saddle point, whereas the INM focuses on instantaneous equilibrium configurations that, due to thermal fluctuations, are quite far from any stationary point. For this very reason we considered the bare energy, $E$, rather than the average energy, $\langle H\rangle$, in the $p$-spin. Therefore, to go beyond both Goldstein’s picture and the INM approach, we must study the properties of unstable saddles in supercooled liquids. The first discussion of the role of saddles in supercooled liquids was done in cavagna-99, building on the study of saddles carried out for the $p$-spin model in cgp-98 and more in general on the topological approach to glassy dynamics discussed in laloux.

Finding unstable saddles numerically is less trivial than finding minima, but it can be done. The real problem, however, is that there is no unique way to associate to an equilibrium configuration a saddle. This is unlike what happens with minima, where the gradient descent method uniquely partition the phase space into basins. In the case of saddles, the mapping between equilibrium configuration and saddle depends on the particular algorithm one uses grigera-06, so that there is not a natural way to partition the phase space into basins of unstable stationary points. This is not small a problem, because it means that when we say sentences like, ‘the typical saddle visited by the system at temperature $T$’, we are in fact using an ill-defined concept.

There are two ways to face this problem. First, we can stick to one particular mapping (algorithm), and plot all saddle properties as a function of the temperature $T$ of the equilibrium configurations used to generate the saddles angelani-00. This first possibility is a natural generalization of the INM approach, with saddles in place of instantaneous configurations. The second possibility is to use a method that relies as little as possible on the mapping between equilibrium configurations and saddles broderix-00. The difference between the two methods is critically discussed in grigera-06. Here, we will explore the second method, which is the one that most directly connects to the $p$-spin scenario.

The interesting point is that the temperature at which we sample saddles (as well as the algorithm we use to do this), only changes the part of this backbone that we uncover, but not the backbone itself, which is a property of the landscape, independent of the temperature. Besides, this backbone is very weakly dependent on the sampling method: it has indeed been confirmed that different numerical algorithms used to sample saddles give rise to the same backbone in the $(E,k)$ plane grigera-06. The same, however, is not true if we use the generalization of the INM approach mentioned above: when we compute the average index $\bar{k}$ as a function of the temperature $T$ at which we are sampling saddles, we obtain a curve that strongly depends on the sampling algorithm grigera-06. This fact seems a good reason to prefer the ‘geometric’ $(E,k)$ approach, with respect to the ‘thermal’ $\bar{k}(T)$ approach.

From equation (78) we know that in the $p$-spin model the dynamical transition corresponds to the temperature where the bare equilibrium energy reaches the threshold. To test this conclusion in real liquids we must define in some way the bare potential energy. In accordance with equation (77), this can be done by subtracting from the average potential energy $\langle U\rangle$ at temperature $T$, the contribution of the thermal vibrations,

The scenario we have presented in these last sections gives a topological interpretation of the dynamic crossover from nonactivated to activated dynamics that takes place at $T_{x}\sim T_{c}$. This is the point where the systems stops visiting saddles, and thus runs out of unstable eigenvalues, which are equivalent to nonactivated diffusive modes. Below the crossover, the system spends most part of the time around minima, rather than around saddles, and an activated Goldstein-like diffusion mechanism kicks in.

Yet, we know that $T_{x}>T_{g}$ and we also know that in the temperature region $T_{g}<T<T_{x}$ fragile systems have a steep super-Arrhenius increase of the relaxation time. What can we say about this regime? Very little in this context. The topological approach is rooted in phase space, and knows nothing about the size of energy barriers, which is determined by the number of particles that really make the transition in real space. Numerical studies in fragile glass-formers broderix-00; grigera-02 show that at $T_{x}$, where the topological transition takes place, the size of the barriers is already quite large, typically,

VI Going deeply supercooled

One of the main conclusions of the former chapter is that at the Goldstein’s temperature $T_{x}$ there is a crossover from high-$T$ nonactivated to low-$T$ activated dynamics, where barrier crossing becomes the main mechanism of diffusion. This slows down dramatically the dynamics, so much that at $T_{g}$ the relaxation time exceeds the experimental time and the dynamic glass transition occurs. Clearly $T_{g}<T_{x}$, but whether or not the two temperatures are close to each other depends on how large the potential energy barriers are at $T_{x}$, when activation kicks in, and to what extent they increase with lowering the temperature.

At this stage of our study we do not have a clear idea about the precise nature of these potential energy barriers. Up to now we just assumed that they exist: there are minima in the potential energy landscape, and in order to visit different minima in the attempt of being ergodic the system will have to cross the barriers connecting these minima. Now this explanation is no longer sufficient. Once we have understood that activation is the central diffusion mechanism at low temperatures, it is essential to get a clearer understanding about barriers.

In order to do this we must first focus our attention on the empirical evidence in the low $T$ regime. Moreover, even though, by definition, we have no equilibrium data below $T_{g}$, the extrapolation down in the $T<T_{g}$ region is very interesting. Indeed, it was such an extrapolation that stimulated a rather famous observation by Kauzmann in 1948.

Let us quote the remarkable paper that Kauzmann wrote in 1948 kauzmann48: “The vitreous or glassy state of liquids evidently only exists because experiments performed by mortal beings must of necessity be of limited duration. It is interesting to speculate on the behaviour which liquids would show at very low temperatures if enough time could be allowed in the thermodynamic measurement to avoid vitrification.”

Basically, Kauzmann says that it seems restrictive not to investigate what happens at low temperatures solely because we do not live long enough to make an equilibrium measurement. What we can do, therefore, is to make an extrapolation of equilibrium data, and cross our fingers. This is what Kauzmann did, by performing a low-$T$ extrapolation of various thermodynamic quantities. In particular, Kauzmann was interested in the metastable nature of the liquid phase with respect to the crystal, and thus he plotted the liquid-crystal difference of quantities such as enthalpy, entropy, free energy and specific volume.

Let us focus our attention on the entropy. The liquid entropy decreases much more rapidly than the crystal one. This is obvious, since the derivative of the entropy is the specific heat,

and the specific heat of the liquid is larger than that of the crystal. We therefore conclude that the difference between liquid and crystal, also called excess entropy of the liquid,

decreases when the temperature is decreased. If we normalize this quantity by its melting point value, i.e. if we plot $\Delta S(T)/\Delta S(T_{m})$ vs. $T/T_{m}$ we can inspect many different liquids on the same diagram (Fig.16 is the original plot of Kauzmann’s paper). What immediately catches our attention is the fact that for some systems the extrapolated excess entropy seems to vanish at finite temperature. Abiding to the common convention, we will call this the Kauzmann temperature, and indicate it as $T_{k}$ (we shall see later that this is in fact a very poor conventional choice). Thus, if we trust the extrapolation, we conclude that there is a temperature $T_{k}$ such that,

As we have seen, extrapolations suggest that $T_{k}$ is finite for many substances. This is equivalent to say that the entropy of the supercooled liquid becomes lower than the entropy of the crystal for $T<T_{k}$. This phenomenon is known as entropy crisis or Kauzmann’s paradox. If we think about it, a metastable liquid with less entropy than its relative stable crystal is sort of counterintuitive. Even though there is no fundamental law of nature forbidding this fact, it certainly stirs our curiosity about the significance of $T_{k}$: is there something funny going on at this temperature?

The interesting point is that several other thermodynamic observables have an extrapolation qualitatively similar to that of the entropy: they all indicate a point where their liquid-crystal difference vanishes. However, the temperatures where this happens are not the same for different observables; for example, the point where the extrapolation of the entropy difference vanishes does not coincide with the point where the extrapolation of the enthalpy difference vanishes. Most importantly, Kauzmann noted that the liquid-crystal free-energy difference $\Delta F$ does not decrease when lowering the temperature, so that any (reasonable) extrapolation does not indicate that the free energies of the two phases converge to each other. From this observations, Kauzmann immediately ruled out the most naive interpretation of $T_{k}$, i.e. that it could be the locus of a continuous transition from the liquid to the crystal phase. This would require converging free-energies and similar values of $T_{k}$ for different observables, which is not the case.

An alternative way out of the conundrum posed by $T_{k}$ is to argue that the extrapolation is just wrong. In particular, the curve of $\Delta S$ vs. $T$ could become flat at some temperature below $T_{g}$ (note that the flattening of $\Delta S$ depicted by Kauzmann in Fig.16 - dotted lines - does not refer to this, but to the dynamic glass transition). If this were the case, $\Delta S$ would not vanish, and there would be no $T_{k}$. This is quite an effective way to cool down all the commotion about the Kauzmann temperature: the extrapolation is, indeed, just an extrapolation, so we cannot trust any result coming from it. Also this objection, however, is not completely reassuring. First, some empirical observations, both experimental and numerical, performed at rather lower temperatures, arrive quite close to $T_{k}$ without showing any evidence of an ‘elbow’ of the equilibrium $\Delta S(T)$ curve angell88. Second, such a flattening of $\Delta S$ would be for all practical purposes identical to what happens at the dynamic glass transition $T_{g}$: the entropy freezes at a certain level, and the specific heat drops to its crystalline value. Kauzmann rightly noted that this would be weird, since a dynamic glass transition certainly has nothing to do with the genuine thermodynamic behaviour we were after when we made the entropy extrapolation.

There is a second reason why Kauzmann’s way out of the problem is not fully satisfying, which is more profound. The sharp approach of the liquid entropy to the crystal value, seems an interesting fact by itself, something which cries for an explanation, irrespective of whether or not the metastable liquid phase actually survives down to $T_{k}$ or not. This same observation is also a reasonable reply to those who object that the entropy crisis is irrelevant because we will never be able to experimentally thermalize a liquid down to $T_{k}$. The incumbent entropy crisis at $T_{k}$ is important, even though we cannot equilibrate at $T_{k}$, because it may have consequences on the physics of the system even in the temperature range where we can equilibrate the system. To better appreciate this point, we must go deeper into the meaning of the excess entropy $\Delta S$ and the physical implications of its decrease. To do this we must go back to Goldstein’s description of the low-$T$ dynamics of a supercooled liquid. As we shall see, this deeper insight will also suggest a new interpretation of what happens at $T_{k}$.

VI.2 Configurational entropy

The configurational entropy $S_{c}$ is different from zero only if the number of minima is exponentially large in the size of the system. In fact, $S_{c}$ is defined by the relation,

We now make a crude, but rather sensible approximation: the vibrational entropy of an amorphous minimum is not too different from the vibrational entropy of the crystal. They are not identical, of course, since the shape of a typical amorphous minimum is in general different from that of the crystalline minimum, and so will be the vibrational frequencies, and thus the entropy sciortino-05. However, this difference is likely to be sub-dominant compared to the difference between vibrational and configurational entropies. We therefore assume,

This equation basically derives from an harmonic expansion of the potential around a minimum (local for the liquid, global for the crystal). Hence, such approximation does not make a bit of sense in all those cases, most notably hard sphere systems, where such an expansion cannot be done and where excluded volume effects are dominant over energetic considerations. In fact, many of the considerations in this section are ill-suited for hard-spheres and we suggest the reader to look at zamponi-08 for a review dedicated to such system.

From equation (91) we can rewrite the excess entropy as follows,

This is a very important equation in the physics of supercooled liquids: the excess entropy is roughly equal to the configurational entropy. Therefore, what Kauzmann actually discovered by means of his extrapolation was that the configurational entropy of the deeply supercooled liquid sharply decreases when the temperature is lowered. This means that the number of amorphous minima the liquid visits in its equilibrium wandering through the phase space becomes smaller and smaller when the temperature is lowered. Such result is rather intriguing, and it supports our suspicion that the decrease of the excess entropy is an interesting fact by itself, irrespective of its vanishing at $T_{k}$. The configurational entropy is a key concept in the physics of supercooled liquids.

It is crucial to understand that the decrease of the configurational entropy is not due to a lack of time to visit all available minima: we are talking about equilibrium here, the system is ergodic, it has enough time to visit whatever the Gibbs-Boltzmann distribution dictates and to jump all necessary barriers. Yet, the number of visited minima decreases with the temperature. Why is that? Let us answer this question in two steps.

First, the equilibrium potential energy decreases with the temperature. At low temperatures, where dynamics is dominated by vibrations around minima, this obvious fact implies that also the average energy of the minima visited by the liquid is lower the lower $T$ sdbs. This is rather reasonable. Second, the number of amorphous minima with given potential energy is smaller the smaller is the energy. This second fact is less obvious than the first, even though it is still quite intuitive. Let us make a very simple example. Imagine a system made up of $N$ non-interacting particles, each of them living into an asymmetric double well that has an energy difference $1$ between the lowest and highest minimum. Above the ground state (with conventional energy zero), this system has a trivial spectrum of excited states with discrete energies $1,2,3,\dots$. From purely combinatorial considerations, the number of minima $\cal N$ with energy equal to $K$ is given by,

If we consider excited states with extensive energy, $K=Nu$, we get for their configurational entropy (eq. (90)),

Therefore, the configurational increases with increasing energy, it is zero at $u=0$ and it reaches a maximum at $u=1/2$. This is a trivial example, of course. In real liquids the interaction plays a crucial role and the situation is much more complicated than this. In particular, we will show that the drop of the configurational entropy is intimately connected to the growth of a correlation length. For now, this simple example helps us to accept as plausible the general idea that the number of minima with a certain energy decreases with decreasing energy.

If we are prone to theoretical speculation, from this state of affairs we may argue that at $T_{k}$ there is a thermodynamic phase transition from the supercooled liquid to a new amorphous phase. Abiding to the convention, we will call this phase the thermodynamic glass or ideal glass phase. In fact, this is a possibility that Kauzmann himself mentioned in his 1948 paper kauzmann48. He acknowledged that the vanishing of the excess entropy at $T_{k}$ could suggest the existence of a new disordered phase of matter. However, he dismisses the hypothesis two lines after having mentioned it, proposing (as we have seen) the kinetic spinodal as a better resolution of the entropy crisis paradox, and dropped what seemed to him an unphysical hypothesis. Nevertheless, the hypothesis of a transition at $T_{k}$ enjoyed great success, mostly thanks to later theories that we will study in the next chapter.

The idea of a phase transition at $T_{k}$ is a fascinating one, especially within the physics of glass-formers, where all notable temperatures are invariably defined in a fishy way. Contrary to $T_{g}$, $T_{k}$ would be a well-defined phase transition, independent from any conventional time scale. As we have seen, also $T_{x}$ is independent from the experimental time scale; however, at $T_{x}$ there is a crossover between two physical mechanisms, rather than a sharp transition, and for this reason the position of $T_{x}$ is somewhat blurred. On the other hand, $T_{k}$ would be the underlying phase transition ruling the whole phenomenology of supercooled liquids at higher temperature. No surprise it is an appealing concept.

VI.3 The VFT law: just a fit?

It would be nice to have a theoretically motivated formula. Problem is that up to know the only theoretical scheme we have seen, able to say something about the dynamics of glassy systems, is Mode Coupling Theory (MCT), and we cannot use MCT to go below $T_{g}$. As the reader may remember, MCT predicts a power law divergence of the relaxation time at a temperature $T_{c}$ larger than $T_{g}$. We need something else.

It turns out that the best fit to the relaxation data was found a long time ago, quite independently from any theoretical framework. This is the celebrated Vogel-Fulcher-Tamman (VFT) law vogel; fulcher; tamman, which hit the physics journals almost one century ago. For the relaxation time, it reads,

If we decide to trust the VFT law even below $T_{g}$, we conclude that the relaxation time diverges at $T_{0}$. Even though the location of $T_{0}$ varies from system to system, it is fair to say that $T_{0}$ is below but close to $T_{g}$ in fragile systems, while it is far below $T_{g}$, and quite close to $0$ in strong systems. Thus, while at $T_{g}$ we observe a sharp slowing down of the dynamics, the VFT law tells us that at $T_{0}$ we eventually meet a real divergence of the relaxation time, that is a dynamical phase transition. All this would happen at equilibrium, thus this dynamical singularity cannot be simply due to the system remaining trapped within some metastable state (as it could happen in mean-field), and it must be associated to a true thermodynamic phase transition at $T_{0}$, accompanied by ergodicity breaking.

We now start to understand the link between the VFT empirical law and Kauzmann’s entropy crisis. This first predicts (on purely empirical basis) a divergence of the relaxation time, and thus a phase transition, at a temperature $T_{0}<T_{g}$. The second indicates a vanishing of the configurational entropy at a temperature $T_{k}<T_{g}$, which may be interpreted as a sign of a thermodynamic phase transition. It is important to stress that both these conclusions stretch to some degree the scenario they were developed within: the VFT law is meant to fit the data in the region where there are data, so that, in principle, $T_{0}$ is just a fitting parameter, without any particular relevance; the entropy crisis, as we have seen, can be interpreted in several other ways than a transition at $T_{k}$, including Kauzmann’s proposal that there is no $T_{k}$ at all. Yet, we cannot deny that the correspondence between $T_{k}$ and $T_{0}$ seems intriguing.

VI.4 A link between dynamics and thermodynamics

The most relevant question at this point is: is there a quantitative link between $T_{k}$ and $T_{0}$? Even though this rather crucial question is still debated, on balance we may say that the answer is: yes. Indeed, in most systems one can show that angell-97; angell-98,

Before diving into the implications of this relation, a few words of caution. In (97) we are comparing two temperatures, none of which is directly measured. One comes from a linear extrapolation of the data ($T_{k}$), the other is a parameter in a highly nonlinear fit ($T_{0}$), so that fixing them sharply is quite difficult. For what concern $T_{0}$, we must note that the VFT fit parameters depend on the temperature interval over which we fit the data, and in particular on the highest temperature included. As for $T_{k}$, the situation is perhaps even worse. First, Kauzmann’s extrapolation fixes the point $T_{k}$ where the excess entropy vanishes with a certain degree of uncertainty, which is the higher the larger the gap between $T_{g}$ and $T_{k}$. Secondly, we must remember that the configurational entropy is not exactly the same as the excess entropy, because of approximation (91), so that $T_{k}$ is not exactly the point where a thermodynamic transition is supposed to be. The bottom line is that equation (97) is accepted by those who mostly agree with its theoretical implications, whereas it is discarded by those who disagree. Personally, I believe the empirical evidence is in favour of (97), even though one should never forget the caveats above.

We therefore have this ‘coincidence’: the dynamic divergence predicted by the VFT fit is approximately equal to the point where the configurational entropy vanishes. This pseudo-fact seems indeed to establish a direct link between the dynamic and thermodynamic frameworks and it supports the scenario we discussed above of a true phase transition from the liquid to the thermodynamic glass at $T_{k}\sim T_{0}$. According to this scenario, the number of thermodynamic amorphous minima becomes sub-exponential at $T_{k}$, and the system remains trapped within one of these minima. For this reason ergodicity is broken and the relaxation time diverges, as in VFT.

As usual, we cannot avoid raising some conceptual concerns. The states that presumably trap the system, breaking the ergodicity at $T_{k}$, have nothing to do with the crystal, of course, but are rather the lowest lying amorphous minima of some free-energy functional. The crystal, on the other hand, will always be the absolute stable state of the system. Thus, below $T_{k}$ the thermodynamic glassy phase has a lower free energy than the liquid, and it is thus stable with respect to it, but it is metastable with respect to the crystal. This is a tricky point: how to define a thermodynamic phase transition between different metastable phases, excluding the crystal from the game, is far from clear.

The main problem is the following: a thermodynamic transition at $T_{k}$ requires the (free-energy) barriers around the lowest lying amorphous states to diverge. If this is true, however, not only the system would be unable to jump between the amorphous minima, but it could also never make the transition to the crystal. However, the crystal has a lower free-energy than the thermodynamic glass, and thus standard nucleation theory predicts that the amorphous states must collapse to the crystal. In which case, though, they cannot be surrounded by infinite free energy barriers. In other words, it is difficult (but perhaps not impossible) to imagine an ergodicity breaking mechanism that only affects the amorphous sector of the phase space, but that leaves open a transition channel to the crystal.

We can only see three ways out of this contradictory loop: first, there is no thermodynamic transition at $T_{k}$, relation (97) is just a coincidence, or perhaps it is wrong; second, elastic effects inhibits nucleation and stabilize lowest-lying amorphous minima, making them effectively ‘equilibrium’ and cutting the crystal out of the game; third, ‘equilibrium’ amorphous minima have the same free-energy as the crystal, or even lower than that, thus ruling out crystal nucleation.

Once we think about it, however, we realize that all these conceptual problems stem from the singularity at $T_{k}$. Imagine that the liquid has a kinetic spinodal at $T_{k}+\epsilon$. In this case, there would be no thermodynamic glassy phase and no conceptual problems. Yet, the relaxation time at the spinodal would still be enormous and the configurational entropy very small, and it would still be crucial to understand why the relaxation time becomes so large when the configurational entropy becomes so small. In other words, we believe that investigating whether or not there is a relationship between relaxation time and configurational entropy, and why it is so, is more important than speculating about the properties of the thermodynamic glass phase. The reason for this is that a relationship between these two concepts may have important consequences also on the supercooled phase that we can observe experimentally. As we already said, given that equilibrating the system at very low $T$ is, and always will be, an impossible task, the phenomenology associated to the incumbent transition is more relevant than the transition itself.

Therefore, our most urgent objective is to understand what is the relation, if any, between the growth of the relaxation time and the decrease of the configurational entropy. This must be done in a direct physical way, we can no longer invoke the existence of a transition where these two quantities are respectively infinite and zero, in order to link the two concepts.

VII The quest for a correlation length

In a finite dimensional system with short range interactions a rearrangement process must be local in space. This means that barriers arise because a certain (finite) number $n$ of particles have to locally rearrange in real space in order for the system to relax. Therefore, it is natural to think that when barriers increase, it is because the number $n$ of particle involved in the rearrangement increases, which is to say that the size $\xi$ of the region being rearranged becomes larger.

whereas in supercooled liquids we expect some sort of exponential law, due to the key role played by activation montanari-sem,

In this equation we have assumed that the barrier $\Delta$ scales like some power of $\xi$,

This view implies that super-Arrhenius relaxation in fragile supercooled liquids is due to the fact that larger and larger regions becomes correlated as the temperature is lowered, so that larger ensembles of particles have to be rearranged to relax the system. The aim of this chapter is to look for such increasing correlation length.

The problem in supercooled liquids is: how to detect $\xi$? In a standard statistical system (for example the Ising model) we would extract the correlation length from the decay of the correlation function of the order parameter parisi-book; binney. We already tried to do that in liquids, by using the standard correlation functions of density fluctuations. Unfortunately, it did not work: we did not find any clearly growing correlation length. As we already said, supercooled liquids are structurally unexciting and density fluctuations seem to do nothing extraordinary close to glassiness.

We have two paths, then. The first is to look for a growing lengthscale using nonstandard thermodynamic methods. However, what these methods may be is unclear at this level of our study. Thus, we will, in a way, invert the problem: we will first try to understand what is the nature of the static correlation length $\xi$ in low $T$ supercooled liquids, and then use this (hypothetical) knowledge to develop the tools needed to unveil $\xi$. This will be a long path, but it will allow us to see some interesting theoretical frameworks meanwhile, as the Adam-Gibbs-Di Marzio theory and the mosaic theory. The second path is to go dynamical. We have seen that dynamical correlation functions are exciting and that dynamically heterogeneous clusters become larger as the temperature decreases. Hence, so we can try to define a dynamic growing lengthscale $\xi_{d}$. These two paths are different, but not necessarily in contradiction with each other. We will illustrate both of them.

The first successful attempt to connect at the theoretical level the increase of the correlation length to the decrease of the configurational entropy is due to a remarkable series of papers published by Adam, Gibbs and Di Marzio between 1956 and 1965, even though one could argue that everything started with a calculation made by Flory in 1956, which had nothing to do with glass physics.

Before we proceed, it is essential to make a remark. Throughout the work of Flory’s and later the ones of Adam, Gibbs and Di Marzio, the word ‘configurations’ is used very much in the sense we now assign to the word ‘states’. In particular, these authors were not interested in the decomposition between vibrational and configurational entropy (see eq.(89)), they only focused on the configurational part, disregarding the (somewhat more trivial) vibrational part. In the case of the polymer on a lattice (Flory) the use of the word ‘configurations’ is quite approriate, whereas in the context of glass-forming liquids (Adam-Gibbs-Di Marzio) such word is somewhat misleading, because one expects a ‘state’ at finite $T$ to be actually made up of many ‘configurations’, contributing to the vibrational entropy of the ‘state’. Hence, unfaithful to the original papers, when discussing Adam-Gibbs-Di Marzio theory I will use as much as possible the word ‘states’. Unfortunately, I cannot accordingly change the wording for ‘configurational entropy’ (and promote it to ‘state entropy’ or similar), because this is far too much a well established notation in glass-forming liquids.

What Flory did in flory was to calculate the number of configurations available to a semi-flexible chain molecule, i.e. a polymer, on a lattice. The aim of the calculation was to compute the free energy of the disordered phase and compare it to that of the ordered one, thus finding the crystallization point. Flory’s analytic result demonstrated that below a certain temperature $T_{m}$, In fact, Flory considers the chain flexibility as the tuning parameter in his calculation, but then he connects flexibility to temperature by making the free energy stationary. Thus, one can equally well use temperature as the tuning parameter in his theory., the disordered phase has a larger free energy than the ordered one, and this was the main result he was after. However, while commenting his result, Flory noted en passant that at an even lower, but finite temperature, $T_{k}<T_{m}$, the number of disordered configurations available to the polymer becomes smaller than $1$, a fact that he deemed as physically unacceptable. He formally resolved this paradox by recalling that there is always at least one configuration, i.e. the perfect crystal, which goes undetected simply because the theory was designed to describe only disordered configurations, thus excluding the crystal. Moreover, Flory noted that it was not necessary to bother about this funny temperature $T_{k}$, because it was anyway below the crystallization point $T_{m}$, in a regime where the disordered phase is not stable, a regime he was not interested about.

We clearly see that, despite the different contexts and the different methods, the similarity with Kauzmann’s paper is striking. Kauzmann made an extrapolation of equilibrium empirical data in supercooled liquids, and found a temperature below which the excess entropy becomes negative. He resolved the paradox by arguing in favour of a kinetic spinodal: crystallization would avoid the paradox. Flory made an analytic calculation in the context of polymers, and found a temperature below which the number of accessible configurations in the disordered phase becomes smaller than $1$. He too, like Kauzmann, resolved the paradox arguing that crystallization would save the day.

The first to seriously bother about this ‘funny’ temperature $T_{k}$ and to make the crucial leap between the two approaches, was Gibbs agdm-1. In a very short Letter, written few months later than Flory’s work appeared, Gibbs noticed the similarity of the two paradoxes and saw the deep implications of the connection between Flory’s number of available configurations and Kauzmann’s excess entropy. Regarding Flory’s paradoxical temperature $T_{k}$ Gibbs asked the question agdm-1: “Could this be the glass transition?” .

Gibbs was not satisfied by Flory’s (and Kauzmann’s) resolution of the paradox, namely crystallization. He noted that in some amorphous systems a crystal is not even present, so that it could not be the conceptual way out of the problem. Gibbs was the first to tackle the paradox, instead of avoiding it, by arguing that a vanishing configurational entropy is the signal of a thermodynamic second order phase transition agdm-1. According to this hypothesis, the configurational entropy has a kink at the transition $T_{k}$, and it remains zero below this point, thus resolving Flory’s and Kauzmann’s paradox. Gibbs finally noted that the system’s dynamics had to become very sluggish when the configurational entropy becomes so small, even though in this first paper he did not elaborate about this rather crucial point.

Gibb’s arguments had to be developed in order to become a theory. This was done in 1958 in a paper by Gibbs and Di Marzio (GDM) agdm-2, where they apply Flory’s technique to tackle the problem of the glassy state. They showed that a second order phase transition has indeed to be associated to the vanishing of the configurational entropy at $T_{k}$ In GDM paper what we call here $T_{k}$ is indicated with $T_{2}$. Strictly speaking, this is necessary, since AGDM introduce a temperature where the analytical configurational entropy vanishes, whereas Kauzmann introduced a temperature where the empirical excess entropy vanishes. We know that excess entropy and configurational entropy are only connected by an approximation, rather than being equal, so this distinction is strictly speaking necessary. However, I prefer not to complicate the notation. Thus, I use the symbol $T_{k}$ also for the GDM transition.. According to GDM, at the transition the system remains trapped into one of the lowest-lying amorphous minima, whose number is sub-exponential in the size of the system, so that their configurational entropy is zero. Hence, below $T_{k}$ ergodicity is broken. This is, according to GDM, the thermodynamic glass, ”the fourth phase of matter” agdm-2.

The work of GDM is of purely thermodynamic nature, with no reference to relaxation times or energy barriers. They argue that the dynamic glass transition $T_{g}$ that we observe empirically is just a kinetic manifestation of the underlying transition at $T_{k}$. However, after reading their paper one is left with the same questions we had in the last chapter: why the relaxation time becomes very large when the number of accessible states becomes very small? What is the precise mechanism linking configurational entropy to barriers? From the theoretical point of view it is clear that a thermodynamic transition must have implications also at the dynamical level. Yet, this is a very nonstandard kind of transition, therefore having an intuition on how dynamics is practically linked to thermodynamics is essential.

GDM probably felt this urge as well and wrote a paragraph about this point, agdm-2: “Furthermore, the (free energy) barrier restricting flow of a system from one of these states (configurations) to another is very high in the neighbourhood of $T_{k}$ because, in this region … the few states that could conceivably occur are widely separated in phase space, and proceeding from one to another involves a considerable change in the topology of the molecular entanglements. Relaxation times … should become very long as $T_{k}$ is approached from above.”

First, we cannot help noting the rather smart use of the brackets, which allow GDM to skip all tricky explanations about two very nontrivial issues, namely the difference between states and configurations, and how to define free energy barriers. Apart from this, we have to admit that this clarification is far from satisfying. Where this ‘considerable change in the topology of the molecular entanglement’ comes from? And why should it be larger the fewer the states? An answer to these questions was provided seven years later by Adam and Gibbs agdm-3.

The key idea of Adam and Gibbs (AG) is that at low temperature relaxation proceeds through the rearrangement of larger and larger regions of correlated particles, which they called Cooperative Rearranging Regions (CRR). AG define the typical CRR as the smallest region that can be rearranged independently from its surrounding. This means that different portions of one CRR cannot choose their own configuration independently from each other, and thus cannot contribute to a combinatorial proliferation of the number of states available to the entire CRR. As a consequence of this fact, a typical CRR can be found in only a very small number $\Omega$ of locally stable states. According to AG, this number $\Omega$ is a constant: it does not depend on $T$, nor on the size of the CRR. The only requirement is that it must be larger than $1$, but even $\Omega=2$ would be a reasonable value agdm-3.

At this point AG ask: how many states $\cal N$ the global system can be found in? Given that, by definition, different CRRs are weakly interacting with each other, the answer to this question is very simple,

where $N$ is the total number of particles of the system, $n$ the typical number of particles in each CRR, and thus $N/n$ is the total number of independent CRRs. The configurational entropy $S_{c}$ is just the logarithmic density of the number of locally stable states, and thus we obtain,

Inverting and including the temperature dependence, we get,

Considered how little we had to work to get this relation, we must say that it is a remarkable result. As a consequence of the combinatorial product of the $\Omega$ states of each different CRR, the number of states of the entire system is exponential and the size $n$ of the CRR naturally increases when the configurational entropy $S_{c}$ decreases. Let us note that AG always talk about the size of CRRs in number of particles, $n$, rather than their linear size $\xi$. Of course, the two concepts are strictly related, since $n\sim\xi^{d}$. Thus, the AG mechanism explains why the correlation length increases when the configurational entropy decreases. This was the result we were after.

In order to use equation (103) to obtain the relaxation time, we need to work out a relation between size of the CRR and energy barrier $\Delta$ to be crossed in order rearrange the region. According to AG agdm-3, the barrier scales with the number $n$ of particles that must be rearranged cooperatively,

Given that $n\sim\xi^{d}$, AG result amounts to set $\psi=d$ in equation (100). Even though at the present stage this seems a reasonable result, we will see later on that $\psi=d$ is not necessarily the most natural choice. Anyway, at this point we can sum-up AG results and, by using Arrhenius formula, obtain,

where $B$ contains all constant factors. Equation (105) not only explains why the relaxation time increases when the configurational entropy decreases, but it also gives quite a neat interpretation of the strong vs. fragile behaviour of supercooled liquids. Let us see this in detail.

No one knows, of course, what is the precise behaviour of $S_{c}(T)$ at very low $T$. However, we can say something by using approximation (92), together with the thermodynamic relation between entropy and specific heat,

AG assume that the liquid-crystal difference of the specific heat $\Delta c_{p}$ is approximately independent of temperature. Within this approximation, and using $S_{c}(T_{k})=0$, we have,

and expanding the logarithm for $T\sim T_{k}$ we finally obtain,

which agrees with the linear behaviour of the excess entropy observed in the empirical data Of course, someone arguing that the entropy crisis paradox is resolved by a flattening of the configurational entropy at low, inaccessible temperatures, would say that this result is just unsupported by the real data.. Equation (109) tells us something interesting, although not surprising: the rate of change of the configurational entropy is proportional to the crystal-liquid difference of the specific heat $\Delta c_{p}$. We recall that at the dynamical glass transition $T_{g}$, when a liquid slips into the off-equilibrium glassy phase, its specific heat drops to a value comparable to that of the crystal (Fig.7). Therefore, $\Delta c_{p}$ is approximately equal to the jump of the specific heat at $T_{g}$ and equation (109) is telling us that the larger is this jump (which is an off-equilibrium feature), the sharper is the decrease of the configurational entropy (which is an equilibrium quantity).

Using the definition of fragility provided in equation (96), we see that the AG relaxation time tells us that the fragility of a system is proportional to the jump of the specific heat at the glass transition, $\Delta c_{p}$. Furthermore, if we are far from $T=0$ and closer to $T_{k}$, we can approximate (110) as,

where, as usual, we have wrapped up in $A$ all constant factors. Equation (111) is nothing less than the VFT law (95).

There is no doubt that Adam-Gibbs-DiMarzio (AGDM) theory gives several interesting results. First, it provides a neat resolution of Kauzmann entropy crisis in terms of a thermodynamic phase transition. Second, it provides a direct link between configurational entropy and size of the correlated regions (and thus size of the barriers), thus explaining the super-Arrhenius increase of the relaxation time in fragile systems. Third, it relates fragility to the decay rate of the configurational entropy, and thus to the jump of the specific heat at the glass transition. Fourth, it provides a formula for the relaxation time, which coincides with the most widely used fit in the empirical analysis of equilibrium data, the VFT law. Last, but not least, all these results were obtained at a very low price: the theory is simple and elegant, and yet with far reaching consequences. It is therefore no surprise that the AGDM theory has enjoyed such an enduring favour. Indeed, this theory has shaped much of the thermodynamic approach to glasses down to these days, and it has been a guide to some spin-glass physics too.

As usual, though, it is not all good news, so let us see the down sides. First of all, as we discussed in the previous chapter, the idea of a glass phase as the fourth equilibrium phase of matter, with sharp ergodicity breaking at $T_{k}$, has some interpretational problems. Secondly, in AGDM view, the dynamic glass transition at $T_{g}$ is nothing more than a mere precursor of the thermodynamic transition at $T_{k}$, which is, in this framework, the only relevant physical phenomenon. However, we have seen in the previous chapters that something quite interesting happens already at higher temperatures, with the emergence of two steps relaxation. Moreover, the crossover from nonactivated to activated dynamics at $T_{x}>T_{g}$ is quite important, and it is quite independent from the (possible) transition at $T_{k}$. Thus, in this respect, the claim of AGDM that all relevant glassy phenomenology is regulated by the transition at $T_{k}$ is a bit too extreme.

But there is a third, most tricky point in AGDM theory, and this regards the number of configurations $\Omega$ accessible to a typical cooperative rearranging region. Why it is constant? It seems awkward that the number of accessible configurations does not scale up with the size of the region that is actually rearranged. The issue is rather subtle, and it mainly concerns the difference between the number of configurations that are virtually accessible by a cooperative rearranging region and those that are actually visited by that region. It is not a purely semantic issue, even though it may seem so. In their argument in agdm-3, AG claim to be talking about the first number, while they are, in fact, considering the second. This aspect of the AGDM theory is somewhat misleading.

In the context of the mosaic theory this ambiguity about the number of configurations accessible to a CRR will be resolved. We will see that each region has an exponentially large number of accessible configurations, a number which does scale up with the region’s size. However, if the region is smaller than a certain critical size, the number of configurations actually visited by that region is in fact equal to $1$. We will show that this happens because of the constraints imposed on each cooperative region by the surrounding particles.

The mosaic theory was developed building on some key results obtained in the context of the $p$-spin model. It is thus essential first to go back briefly to this mean-field spin-glass model.

VII.2 The pp-spin model strikes back

One of the great advantages of the mean-field $p$-spin model is that metastable states can be rigourously defined. They are local minima of the mean-field free energy, which is function of the averaged local degrees of freedom. In spin-glasses this is the Thouless-Anderson-Palmer (TAP) free-energy, function of the local magnetizations thouless77. It is important to understand that, unlike the potential energy, the mean-field free energy depends on the temperature, so that the structure of the free energy landscape depends on $T$. Minima can appear and disappear on changing the temperature. Therefore, the definition of a certain state depends on the particular temperature at which we are working, at variance with the definition of potential energy minimum. In the $p$-spin, free energy barriers around these states are infinite due to the infinite range interaction. Thus, when the system is initially thermalized within one of these states it cannot escape burioni, and the lifetime of the metastable state is thus infinite. Because of this, the separation of the total entropy of the system in vibrational plus configurational part (equation (89)), is sharp. The configurational entropy (which is called complexity in the spin-glass context) is simply the logarithm of the number of metastable states divided by the size of the system $N$, and it is thus well-defined theoretically cavagna-review.

As a consequence, the phase space can be uniquely and unambiguously partitioned into basins of attraction of the various metastable states. If we label each metastable state by an index $\alpha$ running from $1$ to the total number of states $\cal N$, we can write the partition function as,

where $\sum_{\sigma\in\alpha}$ indicates a sum over all configurations $\sigma$ belonging to state $\alpha$. We can define,

as the free energy density of the metastable state $\alpha$, and write,

The sum under the integral is equal to the number of metastable states with free energy equal to $f$. We can define the $f$-dependent configurational entropy as,

This result is quite neat, but it must be reminded that it was only possible because we could define sharply metastable states and thus partition accordingly the phase space, which is hard to do out of mean-field. In fact, as we shall see later, partitioning the phase space into basins of the potential energy minima is not quite the same thing. Note also that at this point of the calculation the configurational entropy is a natural function of $f$, rather than $T$.

In the thermodynamic limit the integral in (116) can be solved by means of the saddle-point (or Laplace) method bender-orszag, i.e. by finding the stationary point (in fact, the minimum) of the expression between square brackets. The equilibrium free energy density of the system is thus,

where $f^{\star}(T)$ is the solution of the saddle-point equation,

The meaning of this result is the following: the global equilibrium state (equivalent to the ergodic liquid phase) consists in a superposition of an exponentially large number of metastable states, each one with free energy density $f^{\star}(T)$, whose value is fixed by equation (119). The interesting point is that from (118) we get,

The global equilibrium free energy density of the system is therefore smaller than the free energy density of the metastable states that dominate the partition function. This is due to the extra entropic term $TS_{c}(f^{\star})$ in (118). Individually taken, each metastable state has a free-energy $f^{\star}$ that is too large compared to the equilibrium one. This is why each metastable state is thermodynamically irrelevant when taken alone. It is only the collective contribution of all the exponentially many metastable states that gives rise to the correct free energy $F$.

The complexity $S_{c}(f)$ can be computed exactly in the $p$-spin model rieger92; crisanti95; monasson; recipes: it is a monotonously increasing function of $f$, with a negative second derivative. Equation (119) thus predicts that when $1/T$ increase, the solution $f^{\star}$ must decrease. This means that the lower the temperature, the lower the free-energy $f^{\star}(T)$ of the metastable states dominating the partition function at that temperature. We can define the equilibrium configurational entropy of the system as,

which is now an explicit function of the temperature. A measurement of the configurational entropy at equilibrium at temperature $T$, gives $S_{c}(T)$, which is therefore analogous to the configurational entropy we met in supercooled liquids. The decrease of $f^{\star}$ with $T$ implies that the equilibrium complexity decreases with decreasing temperature. The reader has certainly noted the similarity with the situation already described in supercooled liquids. In this case however, we have equations (118) and (119) telling us exactly why the configurational entropy decreases with $T$.

If one computes the spectrum of metastable states in the $p$-spin model, one finds a minimum free-energy density $f_{0}$, below which no metastable states are found. This implies that,

Below $f_{0}$ the complexity is negative, i.e. the number of states is exponentially small in the size of the system: there are no free-energy minima, neither stable nor metastable below $f_{0}$, which is thus the true ground state of the system (in the $p$-spin model there is no crystal). The crucial point is that the slope of $S_{c}$ at $f_{0}$ is finite, and therefore equation (119) implies that the temperature where $f^{\star}$ hits the lower band edge $f_{0}$ must be different from zero. This temperature is called $T_{s}$ in the $p$-spin, but here, for obvious reasons, we will call it $T_{k}$. We can define $T_{k}$ as,

or, equivalently, from the equilibrium configurational entropy,

At $T_{k}$ the configurational entropy vanishes and from (118) we conclude that at this temperature the two free energies, $F$ and $f^{\star}$, are the same: equilibrium is (at last!) given by the states with the lowest free-energy, they are no longer metastable and their number is sub-exponential. Below $T_{k}$, the system cannot dig lower than $f_{0}$, so it remains stuck in one of the lowest-lying states, just reducing gradually its vibrational entropy. Each one of these states has a low enough free energy to individually dominate the partition function. This means that below $T_{k}$ ergodicity is broken also at the thermodynamic level, just like it happens in the Ising models below the critical temperature. This is confirmed by all thermodynamic studies of the $p$-spin model kirk-2; kirk-3; rieger92; crisanti92: by using the replica method parisi79; parisi80, one finds a true thermodynamic transition at this same temperature $T_{k}$.

What we have just described is an exact realization of Kauzmann’s entropy crisis scenario at a temperature $T_{k}$ that is marked by a thermodynamic phase transition. Within the $p$-spin model we do not need to extrapolate anything, all comes from exact analytic calculations. The $p$-spin results provide a clearer conceptual framework for the interpretation of $T_{k}$ as the locus of a phase transition also in supercooled liquids. We recall that the $p$-spin is already quite similar to supercooled liquids at the dynamic level, i.e. close to $T_{g}$, where MCT equations are the same as the $p$-spin dynamical equations. It is therefore natural to try to extend this similarity down to low temperatures and to export the clear thermodynamic mechanism of the $p$-spin model to supercooled liquids. This program, though, is not straightforward.

All exact calculations in the $p$-spin model, including that of the configurational entropy and the replica calculation proving that there is a phase transition at $T_{k}$, are made possible by the mean-field nature of the model, which allows one to use the saddle point method in the limit $N\to\infty$. Doing the same is not possible out of mean-field, i.e. in finite dimensional systems as supercooled liquids are. Despite these difficulties, important steps have been done in the direction of a thermodynamic theory of the ideal glass transition mezard-96; cardenas-98; mezard-99; coluzzi-00. By means of nonstandard thermodynamic tools developed in the context of the replica method monasson; recipes and adopting an approximated scheme to treat the liquid (typically the hypernetted chain - HNC - approximation), one can compute analytically the configurational entropy and the position of the thermodynamic transition $T_{k}$. In the case of hard spheres, this approach has been recently reviewed in zamponi-08.

Even though results of such thermodynamic approach to the glassy phase are remarkable, there is a problem, namely metastable states. The approach of mezard-96; cardenas-98; mezard-99; coluzzi-00 basically assumes that we can define amorphous metastable states also in finite dimensions, and that the partition function can be decomposed as in equation (112). This is no trivial assumption. In fact, a critique that could be formulated is that by assuming the existence of metastable states one is effectively working within the mean-field approximation, while we have no guarantee that the mean-field results apply to finite dimensional systems. In other words, one may object that the nontrivial configurational entropy calculated within this thermodynamic approach, together with the thermodynamic transition at $T_{k}$, are just artefacts of an implicit mean-field approximation.

Such uneasiness about the role of metastable states in finite dimension is made particularly acute by one fact: the thermodynamic approach to the glass phase describes things purely in terms of phase space mechanisms, while it says little about what happens in real space (see, however, the discussion of Appendix A in zamponi-08). In fact, this is an expected limit of the $p$-spin inspired thermodynamic approach: due to its mean-field nature, the $p$-spin model says nothing of how its phenomenology would stage in real space, simply because there is no real space in it. However, in real liquids we need to reinterpret the $p$-spin thermodynamic mechanism in real space.

Summarizing, on one hand we have the AGDM picture, which is rooted in real space, but is somewhat unclear in the way it deals with the configurational entropy. On the other hand, we have the $p$-spin model, and the thermodynamic approach derived from it, where the thermodynamic role of the configurational entropy and the origin of the transition are clear, but where their real space role is absent. We still need to bridge this gap.

VII.3 Metastable states in finite dimension: just a delusion?

The great difficulty we have in exporting the $p$-spin mechanism to finite dimensional systems is how to define metastable states. We have long postponed this problem throughout this review, and often played a bit dirty on the ambiguity between local minima of the energy and metastable states, and also between energy and free energy. Even though we are definitely not alone in playing this game (see, for example, Gibbs-Di Marzio’s quotation in Section VII.A), it is worth to try and be more careful, just for once. In this section we will first illustrate the most naive definition of metastable states (that of local minima of the energy), highlight its problems, and finally discuss the need for metastable states to be defined locally both in time and in space. For a less pedestrian discussion of the general definition of metastable states in finite dimension see gaveau; for the specific case of glassy systems see kurchan-biroli and zamponi-08.

First of all, even though the supercooled liquid is of course metastable with respect to the crystal, that is not the metastability we are interested in. We assume that the crystal nucleation time is much larger than the liquid relaxation time, therefore the supercooled liquid is for all practical purposes our reference equilibrium phase. Its free energy is equivalent to what we called $F$ in the $p$-spin. The supercooled liquid is the ergodic phase of the system, and we are rather interested in the metastable sub-components of this ergodic phase, whose free energy is equivalent to $f_{\alpha}$ in the $p$-spin.

to be the free energy density of this local minimum of the energy, i.e. of this metastable state (as usual, this makes no sense for hard spheres). In this same way, one can associate a metastable state to each energy minimum and make a perfect matching between potential energy landscape and free energy landscape. In this framework, one can calculate the configurational entropy of metastable states just by counting the number of minima of the energy, rather than of the free energy, and get some interesting thermodynamic results based on this local energy minima approach sciortino-99; sciortino-00; sciortino-02.

Unfortunately, this program has got two serious shortcomings. First problem, the role of the temperature. Any reasonable free energy landscape depends on the temperature. In particular, not only the free energy $f$ of a state, but also the very existence of the state must depend on $T$. On the contrary, the energy landscape does not depend on the temperature, and a local (or a global) minimum is so irrespective of how large $T$ is biroli-00. Second, and far more serious, problem, the role of the size. As we have learnt when studying nucleation, activation time decreases when the system’s size increases, simply because there are more spots where we are tossing our coin (to form or not to form the nucleus). So, when we say that at low $T$ the system remains confined within a local energy minimum for a long time, we are very unclear, because by increasing the size we can actually make this time as short as we want. Hence, the notion of a global metastable state seems rather awkward. Let us discuss these two problems separately.

First, the temperature. The whole program of identifying energy minima with states rests on the rather casual sentence we wrote above: “when the temperature is low enough”, which, we must admit, is quite vague. At a certain temperature, minima surrounded by small barriers do not give rise to states, whereas those with large barriers may be good candidates. A certain minimum, or basin, with large barriers around can in turn contain a sub-structure of many minima separated by irrelevant barriers, so that only the larger basin may be identified with a state. Certainly, not all minima are relevant as metastable states. This view gave rise to the fruitful notion of metabasins heuer-1; heuer-2. When we change $T$ the structure of states may change more or less radically: minima that were too shallow to be identified with states at higher $T$, may become good candidates at lower $T$, and it may be hard to follow the structure of states in temperature.

We could say something safer if we had a precise notion of the size of barriers. For example, imagine there were only two scales of barriers (or a bimodal distribution of barrier size), a large barrier $\Delta_{1}$ and a small one $\Delta_{2}$, with $\Delta_{1}\gg\Delta_{2}$. Then, in the temperature regime $\Delta_{2}<k_{B}T<\Delta_{1}$, it would be somewhat safe to assume that all configurations connected by barriers at most of size $\Delta_{2}$ belong to the same ‘state’, whereas different ‘states’ would be separated by barriers $\Delta_{1}$. Unfortunately, in general we have no idea of what is the structure of barriers. Moreover, as we have seen, we have the strong feeling that barriers must depend on temperature, through the $T$-dependence of the correlation length. The situation is thus horribly complicated.

To make things a bit clearer, it is important to emphasize the crucial role of the time scale: if we fix a time scale $t$, it makes much more sense to say whether or not a certain region of the phase space is a metastable state: we check whether or not the time needed to get out (via activation) of that region is larger than $t$. Therefore, it is reasonable to talk about metastable states as regions of the phase space with finite lifetime kurchan-biroli; gaveau. Unless temperature is extremely low, however, any such phase space region does not consists of one single minimum: there are typically many relatively small rearrangements of the particles that contribute to the same state, simply because they can all be activated within the fixed time-scale $t$ at that particular temperature $T$. Of course, such view of metastable state is the safer the lower $T$, while we have to be very careful close to $T_{x}$, where activated dynamics is no longer the main mechanism of diffusion. Intuition tells us that the Goldstein’s crossover temperature $T_{x}$ (and thus the MCT transition $T_{c}$) should work as an upper stability limit of any sensible definition of metastable states.

The fact that the lifetime of any metastable state is finite in finite dimension, is actually quite useful when it comes to apply the $p$-spin mechanism out of mean field. In the $p$-spin, dynamics and thermodynamics are completely decoupled, and this is unpleasant. For $T_{k}<T<T_{c}$, the global equilibrium state of the system consists of a superposition of exponentially many metastable states. But how can the system visit all these states and be ergodic, if, due to the mean field nature of the model, barriers are infinite? Thermodynamics just sees one ergodic component, missing the fact that this component is in fact made up of various metastable states. On the other hand, dynamics tells us that a system thermalized within one of these metastable states remains stuck there forever. In the $p$-spin, thermodynamics is ergodic, but equilibrium dynamics is not. This is a typical artefact of mean-field. The finite (albeit long) lifetime of metastable states in real supercooled liquids, however, reconciles dynamics with thermodynamics. The exponentially many metastable states composing the liquid phase are surrounded by large, yet finite barriers, which can be crossed as long as the system is in equilibrium. At a given temperature $T$, only those states that optimize the balance between free energy and configurational entropy (equations (118) and (119)) dominate the partition function and the equilibrium dynamics. Their energy is $f^{\star}(T)$ and their configurational entropy is $S_{c}(T)\equiv S_{c}(f^{\star}(T))$. The lower the temperature, the lower $f^{\star}(T)$, and thus the lower the configurational entropy. As long as $T>T_{k}$, $S_{c}(T)>0$ and thus there are exponentially many metastable states the system actually visits. Their number becomes sub-exponential at $T_{k}$, where $S_{c}$ is zero and the entropy crisis occurs. In this picture, there is no difference between equilibrium dynamics and thermodynamics, and the configurational entropy is a measurable contribution to the total entropy of the system.

All is well, then? Not really. We still have to face the second, and definitely more serious problem in defining metastable states: their size. There is something badly missing in the notion of metastable states as described in this section, and in previous parts of these notes, and this is the role of real space. We have already briefly remarked this before, but it is now time to face directly this point, because it is really crucial. All along these notes we always talked about global metastable states, as if the entire system were at any given instant in just one state. However, if we go back to Goldstein’s remarks about the local nature of particle rearrangements in real space, we understand that this global view must be wrong. Goldstein himself noted that different parts, far from each other, of an infinitely large system, rearrange continuously, so that the whole system, represented by a single phase space point, is always on top of a barrier, rather than vibrating around the bottom of a global minimum. This means that, globally, the lifetime of a single minimum is zero, and all our nice picture above falls apart!

We already faced an identical problem when discussing crystallization: the nucleation rate is defined as the number of events per unit time per unit volume, so that the nucleation time becomes as small as we want when we increase the system’s size I hope the reader does not get confused by the fact that in this crystallization example the role of metastable state is played by the supercooled liquid phase.. Similarly, in a low temperature liquid, even though activated local rearrangements are rare, we can always increase the system size so much as to make the time between two consecutive rearrangements as small as we want: at any time, somewhere in the system a bunch of particles exits from its local state, thus bringing the whole system out of its global state. Clearly, the life-time of the state we are interested in is the local one, not the global one, which is trivially related to the occurrence of different independent rearrangements. Hence, the global landscape scenario must somehow ‘factorize’ when we consider local dynamics bu-bello.

Indeed, as we remarked when discussing Goldstein’s scenario, it is only locally that the intuitive notion of activation, and thus of metastable state, is recovered. If the observation time scale is short ‘enough’ and the region of the system we are observing is small ‘enough’, then it makes sense to say that this region is in a certain metastable state. Hence, metastable states must be defined locally not only in time, as we discussed above, but also in space. Of course, the key issue is how we define ‘enough’ in the sentence above. As for time, activation tells us that the time scale for rearrangement is approximately $\exp(\xi^{\psi}/T)$, where $\xi$ is the typical size of the rearranging region. So, any time smaller than this is fine for the existence of the metastable state. Regarding space, the crucial scale is $\xi$ itself: if the system under observation is much larger than $\xi$, different regions rearrange independently, and the whole system is most of the time on top of a barrier, as noted by Goldstein. If, on the other hand, we observe the system over a scale $\xi$, or smaller, then we really see a sound metastable state.

In conclusion, metastable states must go local. But how? We have understood that locality in both time and space depend on the typical size of the rearranging regions $\xi$. But what fixes $\xi$? In the next few sections we shall try to answer these questions and reconcile metastable states with real space structure.

VII.4 The mosaic theory

The mosaic theory (also known as Random First Order Transition - RFOT), was formulated by Kirkpatrick, Thirumalai and Wolynes in the late 80s. The original aim of their papers was to investigate the potential relationships between supercooled liquids and the $p$-spin class of mean-field spin-glasses, which was conjectured in kirk-0. After studying the dynamic and thermodynamic behaviour of the $p$-spin model kirk-1; kirk-2; kirk-3, it was introduced in mosaic a real space thermodynamic description of metastable states in supercooled liquids, partly inspired by the $p$-spin results. This was the mosaic theory.

Let us start with a rather crucial statement: if we accept that a finite dimensional system may have many states, then different physical portions of the system can be found in different states. When this happens, there must be some sort of interface separating the two states (let us call them $\alpha$ and $\gamma$) and it is reasonable to believe that an energy cost is associated to the creation of this interface, due to the mismatch of the two amorphous configurations along the interface.

This is the first key ingredient of the mosaic theory: in a finite dimensional system we can have interfaces between different states, and thus a surface tension. Note that the price we must pay in going from one state to another is the only sizeable consequence of the existence of different amorphous states; thus, assuming the existence of a surface tension is basically the same as assuming the existence of many amorphous states. Indeed, all metastable states we are talking about are amorphous, so that locally they all look the same, and it is not possible to distinguish them by using a standard local order parameter. Therefore, were it possible to crossover from state $\alpha$ to state $\gamma$ with no energy expense, it would be hardly reasonable to say that they are two different states.

Strictly speaking, surface tension is defined as the free energy cost per unit area to create an interface. This very definition assumes that the cost of an interface of linear size $L$ scales like $L^{d-1}$, i.e. $L^{2}$ in three dimensions. In supercooled liquids, however, we cannot be a priori sure about this point, and a safer assumption is to say that the free energy cost is,

In this case $Y$ is a ‘generalized’ surface tension, simply because if $\theta$ turns out to be strictly smaller than $d-1$, we cannot talk about free energy per unit area. There are at least two reasons why $\theta$ could be smaller than $d-1$: first, the free energy cost associated to an interface depends on the order parameter, for example it is different in the Ising model ($\theta=d-1$) and in the Heisemberg model ($\theta=d-2$). This exponent is typically smaller, the softer the order parameter shukla. We are not quite sure of what is the right order parameter for amorphous states in liquids, so we better be careful. Second, the very effect of disorder can be such as to reduce the value of $\theta$; this may happen when the interface can decrease its energy by passing through more favourable points and thus take a curly shape huse-85; henley-85. Also in this case, we have no idea how an interface between amorphous states in liquids looks like, so it would be a mistake being hasty about the value of $\theta$. The only assumption, thus, is that whenever two states are in contact one with another, there is a free energy cost ruled by a generalized surface tension $Y$.

Why is the surface tension important? Because in finite dimensional systems the surface tension is the main ingredient of the free energy barriers between states. Imagine that the system is in a state $\alpha$. Thanks to thermal fluctuations, there is a chance that a certain region (a droplet) of linear size $R$ rearranges. This is the idea of cooperatively rearranging region (CRR) in AGDM description. In a multi-state framework, it is reasonable to assume that the particles within the rearranging region do not rearrange to a random configuration, but rather make a transition to another of the many available states, say $\gamma$, so that the bulk free energy of the region in the new configuration will be similar as before. When this happens, there is a free energy price that has to be paid due to the $\alpha-\beta$ mismatch at the interface between the rearranging region and the rest of the system,

The crucial question we must answer now is: what fixes the typical size of the rearranging region? The AGDM theory fixes this size in a purely combinatorial way, completely disregarding the surface tension between different CRRs. The mosaic theory proceeds in a different way, getting its main inspiration from classic nucleation theory (CNT) kirk-1; mosaic. In CNT the size $R_{c}$ of the critical nucleus is fixed by the balance between the surface tension cost $\sigma R^{d-1}$ and the thermodynamic gain $\delta fR^{d}$, where $\delta f$ is the free energy difference between the background state and the nucleated one (see Section III.A). In liquids, we do have a surface tension term, albeit with a nonstandard exponent, $YR^{\theta}$. However, it is unclear what the thermodynamic gain should be: it cannot be $\delta f$, since the states dominating the partition function all have the same free energy density $f^{\star}$, fixed by equation (119). Indeed, the free energy of the system does not decrease by subsequent rearrangements, otherwise the system would not be at equilibrium. According to the mosaic theory kirk-1; mosaic, the thermodynamic drive to rearrange a droplet of radius $R$ is provided by the fact that such a region has an exponentially large number of available states. There is an entropic price that must be paid by the region to stay in just one of these many states, and this entropic price can be released if the rearrangement takes place. Therefore, the thermodynamic drive is the free energy contribution from the total configurational entropy available to that region, which is equal to,

where the minus sign emphasizes that this is a free energy gain, opposed to the free energy cost in (127). In (128) and in the following by $S_{c}(T)$ we actually mean $S_{c}(f^{\star}(T))$. Given that $\theta\leq d-1$ the balance of powers between surface tension cost and configurational entropy gain works very similarly to standard nucleation: for small $R$ the cost dominates and there is no net thermodynamic gain in the formation of a droplet. On the other hand, for large $R$ the entropic drive dominates; this happens for any size $R$ larger than a critical value $\xi$ fixed by the balance of (127) and (128),

where we have emphasized that both surface tension and configurational entropy depend on temperature. According to the mosaic theory, $\xi$ is the typical size of the rearranging regions. This size is inversely proportional to the configurational entropy, although with a nontrivial exponent, and in this respect we recover the most important result of the AGDM theory, that is the fact that the correlation length increases when the configurational entropy decreases. Note, moreover, that $\xi$ is also proportional to the surface tension, and this is an ingredient that is completely absent in AGDM. Expression (129) for the correlation length was first derived in kirk-1 in the specific case $\theta=d-1$.

The mosaic mechanism has deeper consequences than the growth of $\xi$ with decreasing $S_{c}$. Imagine we prepare a zero temperature system in a global local minimum of the potential energy. There is no contradiction in doing this and it can actually be done numerically. Now we heat up the system at our working temperature $T$, with the (naive) purpose to promote our global energy minimum to a global metastable state. We would soon find that this is not possible! What happens is that, due to thermal fluctuations, the global configurations is immediately unstable against the formation of droplets of size $R\sim\xi$ of other locally stable arrangements, so that after a while the system’s configuration would look like a ‘mosaic’ of many different local states, in a continuous process of rearrangement on a scale $\xi$ In fact, to us the system would look like nothing at all: local arrangements are all equally amorphous, so they would all look the same. We certainly would not ‘see’ the mosaic the way we can ‘see’ domains within a magnetic system.. We conclude that the very notion of a system globally living in a single metastable state does not make sense at all. The best we can do is to globally prepare it in a single energy minimum, which is a slightly different matter. This means that metastable states can only be defined over a scale $R<\xi$. The notion of metastable state is meaningless for portions of the system larger than $\xi$. This solves the real-space dilemma we faced in the last section. Note that this local definition of metastable state is very similar in spirit to Goldstein’s remark quoted in Section V.A: the notion of a system spending a long time vibrating in a local minimum of the energy makes sense only on a local scale.

Let us summarize the mosaic results up to now. First, compared to AGDM scenario, the role of the number of accessible states is much clearer: this number is of order $\exp(R^{d}S_{c})$ and it thus scales up with the size $R$ of the region. However, when $R<\xi$ the region is unable to explore all its available states, because the surface tension price it would pay in the rearrangement is larger than the entropic gain in getting out of the original state. Thus, the number of explored states is just $1$, when the region is not large enough. This clears up the AGDM ambiguity between the number of configurations that are virtually accessible, which is always exponentially large, and the number of configurations actually visited by a region, which is $1$ if the region is smaller than $\xi$, and exponentially large if he region is larger than $\xi$. Second, the mosaic introduces and deals with the key concept of generalized surface tension $Y$, thus confirming a rather reasonable expectation: the typical lengthscale, and thus the barrier, can be large not only as a result of a low configurational entropy, but also because there may be a large energy cost $Y$ in putting in contact two different states.

Hence, given that $\theta\leq d-1$, we conclude that the increase of the correlation length predicted by the mosaic theory is sharper than that in AGDM theory.

The position of the maximum is (up to a numerical factor) $R=\xi$, whereas the value of $\Delta F$ at the maximum, i.e. the barrier $\Delta$ is,

It is more interesting to note that the barrier $\Delta$ in (133) can be rewritten as,

This result is at variance with AGDM, according to which,

Given that $\theta\leq d-1$, the mosaic barriers grow slower than than the number of particles contained in the rearranging region. Conceptually, this makes a lot of sense: the barrier, according to the mosaic, is dominated by the surface tension term, because at the surface is where we actually pay energy. But the surface tension term is ruled by the exponent $\theta$, and so is the barrier. As anticipated before, the exponent $\psi$ linking the barrier to the linear size of the rearranging region (equation 100) is not necessarily equal in all frameworks: for AGDM $\psi=d$, while for the mosaic $\psi=\theta$. As we have seen, if $\theta=d/2$, as advocated in mosaic, the barrier scales with temperature in the same way in the mosaic and in the AGDM theory, $\Delta\sim(T-T_{k})^{-1}$.

Equation (133) shows that the surface tension critically enters into the barrier for rearrangement. Thus, barriers depend on the temperature also through the $T$ dependence of $Y$. Up to now we have not discussed how the surface tension can change in temperature. In fact, at very low $T$ it is reasonable to believe that $Y$ does not vary strongly with $T$, and that it levels to some finite value. Thus, the increase of the barrier $\Delta$ at low temperatures is predominantly triggered by the drop of the configurational entropy. At higher temperatures, on the other hand, the situation is different. From the discussion in the last section, we do not expect the multi-state scenario we are adopting here to hold also at high temperatures and we already suggested that states (whatever they are) become ill-defined above Goldstein’s crossover $T_{x}$. This requires the surface tension $Y$ to go to zero close to $T_{x}$. In the context of the mosaic theory, therefore, the Goldstein’s temperature $T_{x}$, and thus also the MCT transition $T_{c}$, acts like a thermodynamic spinodal point, where the surface tension vanishes, setting the limit of validity of the theory itself. We recall that $T_{x}$ can also be interpreted as the locus of a topological transition between stable minima and unstable saddles grigera-02. Saddles cannot sustain a finite surface tension, because the system can always rearrange by exploiting the negative-unstable modes. Therefore, the topological view of $T_{x}$ fits well with the idea of a spinodal fixed by the equation,

It is interesting to note that the interpretation of $T_{x}$ as a spinodal point is supported also by the $p$-spin model. As we have seen, $T_{x}$ must be identified with the MCT dynamical transition $T_{c}$, and this in turn is exactly the same as the dynamical transition $T_{c}$ of the $p$-spin model. In recipes it has been introduced an effective potential, given by the free energy cost one must pay to thermalize a system $B$ at fixed distance (in phase space) from an equilibrium system $A$. This effective potential (expressed as a function of the distance between the two systems), works similarly to the standard mean-field free energy in first-order phase transitions: above $T_{c}$ it has just one minimum, while at $T_{c}$ it develops a secondary, metastable minimum, due to the existence of metastable states in the system. Thus, in the first-order transition framework provided by this effective potential, the dynamical transition $T_{c}$ acts like a thermodynamic spinodal, i.e. the temperature above which the metastable minimum of the free energy disappears and the free energy barrier between states vanishes recipes.

Despite its many interesting aspects, there are some unclear points about the mosaic mechanism. First, in what precise sense the configurational entropy is the drive to rearrangement? How can the droplet know about the many states available to it? We would not ask the same in standard nucleation, because energy (and free energy) is a much clearer driving force. The same is not true for entropy: the new configuration would just be one of many others, exactly like the one configuration the droplet started from. Where is the gain? We need a pedestrian definition of entropic drive to rearrangement. Second, why rearranging droplets do not get larger than $\xi$? If standard nucleation is the inspiration of mosaic, then one would naively conclude that once $R>\xi$, the droplet is encouraged to grow indefinitely, just like a crystal nucleus does. Is it so? Third, at first sight the mosaic dynamics does not seem steady-state: the system starts in a homogeneous configuration, and then it breaks up in a multi-state pattern, just like a mosaic. What is this pattern? Is it a typical configuration of the system or not? Apparently, it is not, because there are many inter-states interfaces where surface tension energy is concentrated, which seem to be absent in the original homogeneous configuration. How do we solve this paradox? Let us see this three points one by one.

The expression ‘entropic drive to rearrangement’ is catchy, but in fact quite a misleading one. What actually drives the rearrangement, in the mosaic theory as in any other scenario, is simply thermal fluctuations. However, once the region has been pushed out from its original state by thermal fluctuations, it makes sense to ask whether it is more likely for the droplet to go back to the original state or to stay in the new, or any other, state. This is where entropic considerations about the number of available states come into play. If there are very few states, then the memory of the original state provided by the surface tension at the interface is sufficient to pull the droplet back. If, on the other hand, the droplet is large, there are exponentially many states available; once the transition has been done it is harder for the droplet to go back. Therefore, we see that it is somewhat easier to understand the mosaic mechanism if we think about an already rearranged region and talk about ‘surface tension drive to go back’, rather than the ‘entropic drive to get out’.

Standard nucleation theory seems a natural paradigm for the mosaic picture, but we must be in fact very careful about this comparison. In standard nucleation the growth of a nucleus beyond its critical size is guaranteed by the fact that this process decreases the total free energy of the system. In the mosaic scenario, nothing of this sort happens, because the newly formed droplet of a different state typically has the same free energy density as the rest of the system, so there is no gain in making it bigger. In fact, if we remember that the droplet formation is ruled by thermal activation and that the barrier will scale as a power of the size, it is reasonable to believe that the regions that rearrange have a typical size equal to the smallest thermodynamically stable droplet, and this is $\xi$. Smaller droplets are quick to form (small barrier), but are also thermodynamically unstable, because surface tension pulls them back; droplet larger than $\xi$, on the other hand, are thermodynamically stable, but need to cross an exponentially larger barrier and thus take a very long time to rearrange. We conclude that $\xi$ is indeed the typical size of the rearranging regions in the mosaic picture.

Finally, is the mosaic mechanism steady-state? Yes. Whenever a rearrangement takes place, the global mosaic pattern that is produced is in turn a typical configuration of the system, with the same physical and statistical properties as any other equilibrium configuration. Therefore, the misleading concept that must be rejected is the one of a homogeneous configuration, as opposed to a inhomogeneous mosaic configuration crossed by energy-leaden interfaces. To fix ideas, let us forget about states and consider a certain potential energy minimum $\alpha$. Now, imagine that a region of the system of size $\xi$ is rearranged, making a transition to a different configuration that locally would be part of a different minimum $\beta$. If we now re-minimize the energy, and relax the large stress that is produced at the interface, the system will be found again in a new energy minimum, which is very similar to $\alpha$ far from the rearranged droplet, and to $\beta$ within the droplet. Such new minimum $\gamma$ is statistically identical to $\alpha$ and $\beta$. In particular, the residual surface energy of $\gamma$ located close to the droplet interface is just a local etherogeneity of the system, as the typical stress and strains that are always found in disorder systems. The initial minimum $\alpha$ has exactly the same spatial etherogeneities in energy even before the formation of the new droplet $\beta$. These are the result of the continuous rearrangements taking place in the system. If we now switch from minima to states, we conclude, as already done before, that the very notion of global metastable state is in fact misleading. Only on scale $\xi$ it makes sense to talk about states, whereas for $R\gg\xi$ the system is unstable against the fragmentation in sub-regions of correlated particles. The concept of a homogeneous global state, crossed by no interfaces, does not correspond to anything real.

VII.5 A Gedankenexperiment

In gedanken it was proposed a reformulation of the mosaic theory that clarifies some of the points raised above and that puts on a firmer basis the theory. Let us make the hypothesis that a liquid, at low temperature, is trapped into a global metastable state $\alpha$. We want to prove that this hypothesis, i.e. the fact that the state is global, is thermodynamically unstable and show that only below a certain length scale we can talk about metastable states.

Within the system, let us focus on a sphere of radius $R$. We want to calculate the thermodynamics of such sphere, and in particular its probability to rearrange into a different state. To do this we assume that all particles outside the sphere are frozen (this is the ‘gedanken’ part of the experiment). This assumption is not that crazy: the particles far from the region do not interact with it, so that we can forget about them (if they rearrange, they do that independently from the considered region), whereas the non-rearranging particles in the vicinity of the rearranging region are quite immobile at low $T$, with only small vibrations around their local equilibrium positions. The main effect of these external particles is to produce a pinning field at the interface that tend to keep the sphere in the original state $\alpha$.

where we have used the definition of configurational entropy of the sphere,

The integral in (140) can be approximately calculated by using the saddle point method, provided that $R^{d}$ is large enough. Similarly to what we did for equation (116), this amounts to find the solution $f^{\star}$ of the saddle point equation,

State $\alpha$ is one of the many metastable states dominating the total partition function of the system, and thus also its free energy is equal to $f^{\star}$: as expected, the rearrangement of the sphere does not bring the system to a lower free-energy level (on average). Thus we have,

which is the mosaic correlation length (129). For small values of $R$, that is for $R<\xi$, the surface tension term dominates, and thanks to the sharp exponential form of the probabilities we have,