Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions

INTRODUCTION

Although amorphous materials, such as grains, foams and glasses, are ubiquitous, their theoretical description remains rather rudimentary. Compared with ordered solids, which are central to any solid state textbook, the contrast could hardly be more glaring. The glass transition is an entirely different phenomenon than the first-order transition into a crystal. A glass is not the thermodynamic ground state, so the “glass transition” occurs as a liquid falls out of equilibrium and becomes rigid. The rigidity of glasses hence relies on altogether different principles than for crystals, and their materials properties bear that imprint.

Developing a theoretical description of amorphous materials is extremely challenging. Conventional paradigms for describing condensed matter rely on perturbative treatments around either the low-density, ideal gas limit (for moderately dense gases and liquids), or an ideal lattice (for crystals). For amorphous materials, however, both strategies fail badly. Because these materials interact strongly, low-density starting points are unreliable, and equilibrium particle positions are not forthcoming. A natural controlled reference system around which to make a small-parameter expansion is thus missing. Understanding the rich non-equilibrium behavior of these materials, including the glass and the jamming transitions, has instead been left to uncontrolled treatments, differently balancing rigor and guesswork. Constructing a first-principle theory of amorphous materials thus remains one of the major challenges of theoretical condensed matter physics.

Amorphous materials are not alone without a natural reference for building a perturbative description . A similar situation is encountered in liquids , strongly coupled electrons , atomic physics , and gauge field theory . For all these problems, dimensional expansion has been developed as an alternate line of attack. By solving the problem in infinite-dimensional space, $d\rightarrow\infty$, and treating $1/d$ as a small parameter, one may hope to recover–to a good approximation–the behavior of physical systems in $d=3$. Such a program for glasses was first proposed by Kirkpatrick, Thirumalai and Wolynes in the late 1980s , as part of a larger effort to construct the “Random First Order Transition” (RFOT) theory of the glass transition . Yet advances have been a long time coming, because the necessary tools to first solve the $d=\infty$ problem were not readily available. It is only a decade ago that some of us started attacking the problem head first for the simplest model glass former, hard spheres .

The large-$d$ approach may be a successful calculation device in various fields of physics, but for glasses it plays an additional–perhaps even more important–role. It helps clearly define entities that are otherwise only phenomenological. Consider, for instance, the “configurational entropy”, i.e., the “entropy of glassy states” . In order to define this quantity properly, one needs to count the logarithm of the number of metastable glassy states. However, it is not possible to have an exponential number of states that are absolutely stable in a finite-$d$ system . One thus defines complexity as the logarithm of the number of states with a lifetime larger than an (arbitrarily chosen) $t^{*}$. The large-$d$ approach, by contrast, allows the precise enumeration of states that are stable in the limit $d\rightarrow\infty$, thus providing a clear-cut definition. A related question is the definition of activated processes, which are notably responsible for blurring the dynamical glass transition. It seems mathematically clear and physically plausible that activated processes may be identified as those that take an exponentially long time, $t\sim e^{\mathcal{O}(d)}$, to complete. This definition gives a hint of the (nonperturbative) techniques needed to study such processes. It may even be that the entire RFOT scenario is but a metaphor based on entities that only truly exist in the limit of large dimensions.

The essential counterpart to the large $d$ computation is to assess the robustness of the physical phenomena under changing $d$. There would be but limited relevance to a $d=\infty$ solution for processes that acutely depend on spatial details. Consider, for instance, ordered packings of hard spheres in $\mathsf{R}^{d}$. Asymptotic packing bounds for $d\rightarrow\infty$ have little to say about the singularly dense triangular lattice in $d=2$, root lattice in $d=8$ , or Leech lattice in $d=24$ . Does something similar occur for amorphous materials? Computational answers to this question have been emerging from various directions over the last decade . It has now become clear that many features of these materials are, at least qualitatively and sometimes even quantitatively, remarkably independent of $d$.

The main purpose of this review is to describe the astounding theoretical and numerical advances triggered by these realizations. For the sake of concision and in order to be relatively self-contained, we here only consider hard-sphere glass formers. Note, however, that this choice is largely done without loss of generality, as we argue in the conclusion.

We consider the behavior of a system of $N$ hard spheres of diameter ${\cal D}$ within a box of volume ${\cal V}$, in $d$ spatial dimensions. We denote $x_{i}$ the $d$-dimensional vector that encodes the coordinates of sphere $i$, with $i=1,\ldots,N$, and $x_{i}^{\mu}$ its coordinates with $\mu=1\cdots d$. The singular nature of the hard-core interaction results in temperature playing but a trivial scaling role. Hence, the only equilibrium control parameter is the packing fraction, $\varphi$, i.e., the fraction of ${\cal V}$ occupied by spheres. For the number density of spheres $\rho=N/{\cal V}$, we thus have $\varphi=\rho{\cal V}_{\cal D}$, where ${\cal V}_{\cal D}$ is the $d$-dimensional volume of a sphere of diameter ${\cal D}$. Throughout the text we also rescaled version of physical quantities, $\widehat{\cdot}$, such that their numerical values remain of order unity in the $d\rightarrow\infty$ limit, e.g., $\widehat{\varphi}=2^{d}\varphi/d$.

In the following, Section 2 presents the hard sphere liquid dynamics and compares the exact solution with mode-coupling theory. Section 3 details what happens to an out-of-equilibrium amorphous solid upon a quasi-static compression, and the jamming endpoint to these compressions is separately discussed in Section 4. Section 5 presents the results of various out-of-equilibrium processes. Each section first describes the conceptual framework and the theoretical results from the exact $d\rightarrow\infty$ solution, and then compares these with experimental and numerical results for finite-$d$ systems. Note, however, that when the $d\to\infty$ solution straightforwardly reproduces well-known results from other approaches that have been extensively reviewed elsewhere, we limit ourselves to a minimal discussion of the low-$d$ results.

EQUILIBRIUM DYNAMICS AND MODE-COUPLING THEORY

Consider the following experiment: take at random a valid configuration of hard spheres at density $\varphi$, randomly assign each sphere a velocity from the Maxwell distribution, and evolve the Newtonian dynamics. At very low $\varphi$, dynamics is mostly ballistic, but the rare collisions eventually give rise to a diffusive regime. The system is then nearly ideal gas-like. Increasing $\varphi$ makes dynamics more liquid-like, as particles move a distance no further than ${\cal D}$ before their travel direction gets randomized by collisions and their diffusive behavior ensues. Further increasing $\varphi$ from this regime is the interest of this review.

If crystallization is successfully avoided–see for practical ways of achieving this, the liquid dynamics grows increasingly sluggish with $\varphi$. Sphere diffusivity decays and, correspondingly, the structural relaxation time, $\tau_{\alpha}$, grows so much that the system remains structurally similar to its initial configuration over experimentally accessible timescales. This loss of ergodicity defines the experimental glass transition. Why does diffusive dynamics become so slow and eventually freeze? In the $d\to\infty$ limit we can answer this question precisely because the equilibrium dynamics of hard spheres has been solved . In this section, we review this result and comment on its connection with other theoretical descriptions. We then discuss how it relates to the phenomenology of finite-$d$ systems.

In order to solve the liquid dynamics, it is convenient to substitute Newtonian by Langevin dynamics and to momentarily relax the hard-core constraint. We thus assume that spheres interact through a pair potential $V(r)=\overline{V}[d(r-{\cal D})]$, where $\overline{V}(h)$ remains finite for $d\to\infty$. The hard sphere limit corresponds to $\overline{V}(h)=0$ for $h>0$ and $\overline{V}(h)=\infty$ for $h<0$. The system dynamics is then obtained by solving

where $m$ is the mass of the spheres, $H=\sum_{i<j}V(|x_{i}-x_{j}|)$ is the system energy, $\zeta$ is the friction (or drag) coefficient, and $\xi_{i}$ is white noise with zero mean and variance $\langle\xi_{i}^{\mu}(t)\xi_{j}^{\mu^{\prime}}(t^{\prime})\rangle=2T\zeta\delta_{ij}\delta_{\mu\mu^{\prime}}\delta(t-t^{\prime})$ with temperature $T=1/\beta$. We are interested in understanding the behavior of the dynamical response and correlation functions,

Equilibrium dynamics is described by starting the Langevin process in an equilibrated configuration at density $\varphi$ and temperature $T$; in this case the solution of the dynamical equations satisfies both time translational invariance, i.e., $\Delta(t,t^{\prime})=\Delta(t-t^{\prime})$ and $R(t,t^{\prime})=R(t-t^{\prime})$, and the fluctuation-dissipation theorem, i.e., $R(t)=\beta\theta(t)\dot{\Delta}(t)$ .

For $d\rightarrow\infty$, and in the thermodynamic limit $N\to\infty$, $\Delta(t)$ obeys

where $\widehat{\zeta}={\cal D}^{2}\zeta/(2d^{2})$, $\widehat{m}={\cal D}^{2}m/(2d^{2})$, and the memory kernel $M(t)$ is the solution of the self-consistent equations

where $y(t=0)\equiv y_{0}$, $\langle\Xi(t)\Xi(t^{\prime})\rangle=2\widehat{\zeta}T\delta(t-t^{\prime})+M(t-t^{\prime})$, and $F(y)=-\overline{V}^{\prime}(y)$ is a scaled interaction force.

The complex problem of $N$ interacting particles in $d$ dimensions has thus been reduced to a simple one-dimensional problem: an effective degree of freedom moving in an effective potential $\overline{V}(h)-Ty$ in presence of a colored noise, whose memory kernel $M(t)$ is determined self-consistently as the average of a force-force correlation. Remarkably, these equations are akin to those from the generalized schematic mode-coupling theory (MCT) . The key difference is that within the MCT approximation, the memory kernel is a simple function of $\Delta(t)$ alone (essentially, $M\sim\Delta^{2}$), while the exact expression for $M(t)$ in $d\rightarrow\infty$ is given by Eq. (4) as an implicit functional of $\Delta(t)$. Because the critical scaling of the dynamical equations is insensitive to the precise form of $M(t)$, however, many qualitative features of standard MCT persist in that case as well.

Recall that the above result was derived for a soft potential $V(r)$ and for a Langevin dynamics with friction and noise, but one can safely take the hard sphere limit of $V(r)$ and the limit $\zeta\to 0$ at which the noise disappears and the Langevin dynamics becomes Newtonian. The latter limit, however, should be taken after taking the limits $N\to\infty$ and $d\to\infty$. Newtonian dynamics is then described by a self-consistent effective Langevin dynamics for which noise is generated by interactions.

2 Dynamical transition

The order parameter for the dynamical transition is the long time limit of the dynamical mean-square displacement, $\Delta(t)$. At low $\varphi$, the short-time ballistic behavior $\Delta(t)\sim t^{2}$ is promptly followed by a diffusive regime $\Delta(t)\sim\widehat{D}t$. The dynamical equation directly gives the (rescaled) diffusion coefficient (3), and a similar analysis gives the shear viscosity ,

hence the two transport coefficients obey

with $\Theta(x)=[1+\textrm{erf}(x)]/2$. We will see in Section 3 that these results can also be obtained using a completely static approach.

3 Dynamical critical exponents and susceptibility

with critical exponents that are related through a non-universal exponent parameter

At a standard phase transition, one commonly detects a critical point by considering the correlation function of the order parameter and its integral over $\mathcal{V}$. By analogy, in the case of the dynamical transition one can study the dynamical susceptibility

where the overline denotes averaging a quantity over both the system thermal history, $\langle\cdot\rangle$, and initial configurations $\{x_{i}(0)\}$. $\chi_{4}(t)$ encodes the fluctuations of dynamical correlators, here represented by the mean square displacement.

4 Liquid dynamics in finite dimensions

Because many physical features of the exact dynamical solution qualitatively coincide with those of standard MCT (the latter preceding the former by more than 30 years), substantial efforts have already been devoted to evaluating some of the theoretical predictions . The two-step behavior of $\Delta(t)$ (Fig. 1a), the scaling relations between the critical dynamical exponents, and the growing dynamical susceptibility that accompanies the lengthening plateau in $\Delta(t)$, have indeed been carefully documented in $d=3$ .

5 Exploring dynamics through equilibrium calculations

The dynamical equations that describe the equilibrium liquid in the limit $d\rightarrow\infty$ reveal that, upon approaching a critical density, the relaxation timescale diverges. Equilibration is impossible beyond that point. An equilibrium liquid that is “crunched” (quickly compressed) to such densities does not, however, immediately acquire an infinite relaxation time, but ages instead; as time passes, $\Delta$ relaxes ever more slowly. Hence, $\Delta(t,t_{w})$ becomes a function of both the (waiting) time $t_{w}$ elapsed since the crunch and of the time difference $t-t_{w}$ . Given that equilibrium is not achieved over reasonable times in this regime, equilibrium calculations may seem of limited interest. By modifying the set of configurations considered, i.e., the measure, however, one may nonetheless hope to extract dynamical information. And because they are often technically simpler–using the replica and cavity methods –than their dynamical counterpart, a panoply of methods have been developed. Results for various equilibrium methods are presented in the following sections, but before proceeding this brief overview aims to guide disoriented readers through some of the relevant spin-glass literature, where these ideas were first developed. Note that the precise dynamical justification, varies from scheme to scheme. In most cases, however, such justification is only known for a normal glass, and it remains unclear how to interpret equilibrium calculations in a marginal glass.

Counting metastable states: Bray and Moore first showed that the number of metastable states (as defined by the Thouless-Anderson-Palmer equations, assumed by Bray and Moore to be relevant for dynamics) is exponential. This computation can also be implemented from the dynamical equations .

Dynamics starting from various equilibrium measures: The dynamical equations can be modified to describe exactly the evolution of an equilibrium configuration at a different temperature or density . In a normal glass phase, the long time evolution can be described by computing the probability distribution of a configuration kept at fixed distance away from a reference one ; this construction – reviewed in Sec. 3 – accesses certain quantities associated with a full dynamical computation . Once can also extend it to an entire chain of copies, which is conjectured to mimic slow dynamics in its full generality, including in the marginal glass phase .

Threshold level and marginality: In a normal glass, computing the level at which states become marginally stable – as measured by their diverging four-point correlations–is justified from the dynamical solution. Without any special assumption, this analysis finds that aging takes place at free energies above which the landscape connects and below which it disconnects . This threshold level corresponds to the neighborhood of marginal states. We review this construction in Sec. 5.

Effective temperatures via Edwards-like assumptions: Another feature that appeared in the dynamical solution, and that validates a quasi-equilibrium approach, is that during aging the system samples states with a fixed energy uniformly. This assumption, which is equivalent to that proposed by Edwards for granular matter , is surprisingly exact within the dynamical solution . We review this construction in Sec. 5.

Parisi matrices as generating functions of (certain) dynamical diagrams: A recent application of replicas exploits the mathematical correspondence between the diagrams generated by Parisi matrices, which can be efficiently evaluated, and those used to compute some dynamic quantities , such as the exponents for time relaxation within and away from a state. This correspondence is purely mathematical – it implies no physical assumption.

FOLLOWING GLASSES UNDER SLOW COMPRESSION

Moreover, because $Y$ is sampled from the equilibrium distribution, the dynamics of $X(t)$ must be stationary, i.e.,

Next, consider first compressing (or decompressing) the initial configuration $X(0)$ to $\widehat{\varphi}\neq\widehat{\varphi}_{\rm g}$, and then evolving the system dynamics at this new density. If the diffusion constant remains zero, then we expect Eq. (11) to hold, and $X(0)$, $X(t)$ and $X(t+\tau)$ to always be close to one another. Although the overall dynamics for this protocol is not stationary in general, it nonetheless becomes so at long times, i.e.,

In finite $d$ the configuration $X$ is not expected to remain trapped around $Y$ for an infinite time. This residence time, however, can be made sufficiently long for a clear separation of timescales between sampling a glass state and leaving that state. The region initially visited by $X$ around $Y$ is thus only a metastable state.

2 The Franz-Parisi restricted free energy and the order parameters

3 Equilibrium Glasses

4 Gardner transition

5 Marginal glass phase

Beyond the Gardner transition the system enters a marginal glass phase. Recall that liquid dynamics is diffusive and ergodic (Fig. 3 first column), while in a normal glass there exists an exponential number of metastable states, and dynamically the system quickly and fully explores one basin but cannot visit other basins. The mean square displacement then has a finite long-time limit $\Delta(t)\to\Delta\equiv\Delta_{1}$ (Fig. 3 second column). By contrast, the typical relative mean-square distance between two amorphous lattices corresponding to two different basins is infinite.

At higher densities, however, the system enters the marginal glass phase (Fig. 3 third column). There, various amorphous lattices (glassy states) belong to a same metabasin and are organized hierarchically. For the sake of illustration, consider a $k$-step replica symmetry breaking ($k$RSB) approximation of phase space in that regime. At the very bottom of the hierarchy a state is made by vibrations around an amorphous lattice with an amplitude $\Delta_{k}$. Two amorphous lattices can be at only $k-1$ different mean square displacements $\{\Delta_{k-1},\ldots,\Delta_{1}\}$. For each triplet of amorphous lattices, at least two of the three distances must be equal, which makes the space ultrametric (see Fig. 3 top-right panel for $k=4$). The complete solution of hard spheres in $d\rightarrow\infty$ requires an infinite number of hierarchical levels and $k\to\infty$ (fullRSB) with infinitesimally close distances $\Delta_{i},\Delta_{i+1}$. The spheres vibrate around positions on an amorphous lattice followed by (infinitely) slowly changes of the amorphous lattice itself, giving rise to a sequence of infinitely close plateaus in the mean square displacement (Fig. 3). Moreover, the amplitude of vibrations fluctuates and is correlated over large regions. The system is thus much more heterogeneous than a normal glass.

6 Gardner transition and marginal glass in finite dimensions

JAMMING

The endpoint of the compression presented in Sec. 3, i.e., the end of the marginal glass phase, has a diverging pressure. At that point spheres are in direct mechanical contact with each other; the glass reaches its densest (or close) packing point. This point has properties similar to granular materials at jamming , which is a mechanical rigidity transition that describes the emergence of solidity in athermal systems, such as foams and grains. These systems jam when an applied pressure leads to slight deformations (elastic or not) of its components. The jamming transition then corresponds to the point where the applied pressure vanishes . Theoretical understanding of jamming was further enriched by including marginal stability in the description , as we describe in this section.

The exact solution for the vibrational modes around jamming presents a vast excess of low-frequency modes compared with the Debye model for solids (a so-called Boson peak) . The low-frequency limit of the density of states does not vanish as in the Debye model but tends to a constant . The structure of these modes is as extended as phonons, but their organization in no way resembles plane waves .

2 Scaling Relations

3 Jamming in finite dimensions

Many properties of jammed configurations are dimensionally robust and universal:

(ii) It has long been appreciated that the distribution of small interparticle voids in jammed hard sphere configurations displays an anomalous power-law scaling . Numerically, various measures of that exponent give $\gamma=0.40(4)$ . This result holds, irrespective of preparation protocol, for $d=2$ to 12, and agrees with the $d\rightarrow\infty$ prediction (Fig. 4b).

(iv) The small excitations associated with opening and breaking force contacts have been separated between extended and quasi-localized excitations . The former is qualitatively predicted from the exact solution, while the latter likely results from the presence of bucklers and of otherwise dimensionally anomalous local geometrical arrangements. A clear geometrical description of these localized excitations remains to be obtained, because these processes are beyond the scope of the $d\rightarrow\infty$ solution.

(v) The anomalous nature of $\kappa$ was suggested from mechanical marginality considerations , but extracting this quantity is riled with challenges. Unlike the previous exponents, it requires dynamical simulations, and is thus not purely a property of jammed configurations. At finite $p$ within the Gardner phase, all of the processes not captured by the exact solution can further hinder its measurement in low $d$. Recent numerical determination nonetheless give $\kappa=1.40(4)$ in $d=3-8$ , which is once again in excellent agreement with the $d\rightarrow\infty$ solution (Fig. 4d).

The triangular comparison between the $d\rightarrow\infty$ solution, mechanical marginality considerations, and finite-$d$ numerical results has thus been remarkably productive in clarifying the jamming phenomenology.

SAMPLING GLASSY STATES

In the previous sections we have considered what happens to a system of hard spheres that is slowly compressed in the liquid and in the equilibrium glass regimes. In this section we take a different approach. We construct a measure that gives equal weight to all existing glassy states at a given density $\widehat{\varphi}$ and pressure $p$ to determine where glasses can be found and in what number. This computation allows us to obtain information on strongly non-equilibrium dynamical protocols for preparing glasses, and is related to the Edwards ensemble.

We have seen above that at high densities, the equilibrium measure for hard spheres is supported on a large set of distinct metastable states, which we now label by an index $\alpha$. Our aim here is to sample these states using non-equilibrium weights. For finite systems, one can construct a partitioning of phase space, i.e., the unique association of each configuration $X$ to a state $\alpha$, denoted by $X\in\alpha$, by identifying each state with a potential energy minimum $\alpha$, and each configuration $X$ to the minimum reached by energy minimization via steepest descent . In the thermodynamic limit, and especially in analytic computations, this procedure cannot be used directly , but it remains informative – see, e.g., Appendix A of Ref. . We introduce the free energy of a state and a generalized partition function , respectively

An explicit calculation of $Z_{m}$ and of all the derived quantities, including $\Sigma(p,\widehat{\varphi})$, can be performed using the replica method . The resulting phase diagram in Fig. 5 indicates where $\Delta$ has a finite solution (white region). Outside of that region, the number of glassy states is negligible in the thermodynamic limit. In the following subsections, we describe various aspects of this phase diagram in more details.

2 Equilibrium sampling

The equilibrium partition function corresponds to $m=1$, where $\beta_{\rm eff}=\beta$. In this case, we recover the equilibrium line where, for each $\widehat{\varphi}$, the pressure of the dominant glass states is the liquid equilibrium pressure $p=p_{\rm liq}(\widehat{\varphi})$ (black line in Fig. 5b). We can thus compute the configurational entropy of equilibrium glassy states, $\Sigma_{\rm eq}(\widehat{\varphi})\equiv\Sigma[p_{\rm liq}(\widehat{\varphi}),\widehat{\varphi}]$, which is plotted in Fig. 5a. When $\widehat{\varphi}\geq\widehat{\varphi}_{K}\sim\log d$ is reached , the configurational entropy vanishes. The ensuing Kauzmann (or ideal glass) transition results from the population of glassy states becoming subextensive in system size. It therefore has a thermodynamic signature, and contrary to the dynamical transition it may remain a phase transition even in finite dimension. Whether that is the case, however, remains an open question. Note that this question is a logically separated issue from the existence of metastable states. As such, it is irrelevant for most of the previous discussion about out-of-equilibrium glasses. The relevance of the $d\to\infty$ solution to describe finite-$d$ glasses is not related in any way to the existence of a Kauzmann transition and of an ideal, thermodynamically stable glass.

3 The Gardner line

For sufficiently high $\widehat{\varphi}$, it is possible to find states with a pressure, different from that of the equilibrium liquid, $p\neq p_{\rm liq}$. At each state point in the white zone of Fig. 5b we can also compute the stability of the glassy states with respect to the marginal glass phase . The blue line separates stable from unstable states. In the latter regime, each glass becomes a metabasin of marginally stable glassy states. This Gardner transition has the same properties as that found in Sec. 3.4, but now distinct groups of glass states are sampled at each state point $(\widehat{\varphi},p)$: in other words, moving in the plane $(\widehat{\varphi},p)$ does not correspond to adiabatically following a given group of states.

4 Jamming line: the Edwards ensemble

Within the glassy regime, the limit $p\to\infty$ is particularly interesting. It corresponds to giving a uniform weight to all stable jammed states of hard spheres at a given density $\widehat{\varphi}$ , which is precisely the Edwards ensemble prescription. The effective temperature formalism thus represents a generalization of the Edwards measure to finite pressures. Figure 5b shows that there exists a line of jammed states along which the configurational entropy $\Sigma_{J}^{(E)}(\widehat{\varphi})\equiv\Sigma[p\to\infty,\widehat{\varphi}]$ can be computed (Fig. 5a), and that this line falls entirely within the marginal glass phase (Fig. 5b). Hence, all relevant glassy states within the Edwards measure are marginally stable. As discussed in Sec. 4 the marginality of glassy states at jamming is responsible for the critical behavior of jammed packings, and the critical exponents calculated in this ensemble are the same as the ones obtained by following states adiabatically . This correspondence is quite remarkable. Two very different ways of sampling jammed packings (uniformly à la Edwards, or by slowly annealing the liquid) give the same critical properties. This observation leads us to conjecture that all jammed states, however sampled, have the same universal critical properties.

This approach also reveals that there is at least one natural algorithm (adiabatic compression) that produces jammed packings sampled with weights that are distinct from the Edwards measure. To prove this point, Fig. 5 compares the configurational entropy of jammed states produced by adiabatic compression and of Edwards’ jammed states. Because for an adiabatic compression the jamming density is a unique function $\widehat{\varphi}_{\rm J}(\widehat{\varphi}_{\rm g})$ of the initial equilibrium density, the configurational entropy of the resulting packings is simply $\Sigma_{J}^{(SF)}(\widehat{\varphi}_{\rm J})=\Sigma_{\rm eq}[\widehat{\varphi}_{\rm g}(\widehat{\varphi}_{\rm J})]$. This quantity is systematically smaller than the Edwards configurational entropy (Fig. 5a), and that this line falls entirely within the marginal glass phase (Fig. 5b). Hence, adiabatic compression creates exponentially fewer packings than there exist. Moreover, the range of $\widehat{\varphi}_{\rm J}$ that can be constructed via adiabatic compression is smaller than the range of over which jammed packings exist.

5 The threshold and aging dynamics after a crunch

6 Out-of-equilibrium glasses in finite dimension

Studies of finite-$d$, out-of-equilibrium glasses have thus far focused on: (i) developing protocols for generating configurations at the jamming point, and (ii) assessing the validity of the Edwards measure. Although relatively few of the many predictions of the exact solution have been tested, (i) and (ii) establish a strong foundation for exploring these predictions. Because (i) and (ii) have already been extensively reviewed , we here only focus on the aspects most related to the exact solution.

Two main families of out-of-equilibrium protocols have been developed for generating hard-sphere configurations at jamming. The first obeys the volume exclusion of hard spheres throughout the preparation. It thus reaches the jamming transition from densities below it. Typical protocols include the Lubachevsky-Stillinger algorithm, which grows the diameter of hard particles at a fixed rate while running an event-driven molecular dynamics simulation , overdamped event-driven algorithm under external forcing , and a sequential linear programming algorithm , which iteratively approaches jamming by linearizing the optimization of the packing fraction. The second family of protocols allows particles to overlap during the preparation and steadily minimizes the system energy in order to systematically eliminate these overlaps, thus reaching the jamming transition from densities above it . Approaches typically vary in the choice of (purely repulsive) interaction potential and on the initialization condition .

As mentioned above the key structural properties of the resulting configurations are found to be invariant to the choice of preparation protocol. More subtle aspects, however, remain to be theoretically understood. Although the final density of the jammed configuration is expected to depend on the algorithmic details, systematic changes to the concentration of rattlers at jamming, for instance, are less easily rationalized .

Over the years, various efforts have probed the Edwards measure both in experiments and in simulations, but the results have often been challenging to interpret (see Ref. for a recent review). In particular, it has been difficult to obtain sufficient control over the preparation protocol to interpret discrepancies. A rather sophisticated (and computationally demanding) set of methodologies has recently been developed to avoid these pitfalls. For instance, a direct computation of the volume of the basins of attractions of jammed configurations, for a given minimization protocol, has been attempted . Yet even under these highly constrained conditions, the observed measure does not match that of Edwards . Although explanations for these discrepancies have been suggested , their inclusion within the framework of the exact solution remains to be achieved.

ONGOING AND FUTURE DIRECTIONS

The advances presented above hint at firm ground for completely solving the glass problem. In closing, we provide an overview actively explored questions.

Beyond the hard-sphere idealization, important avenues of research have yet to be fully explored. Most of the predictions discussed above for hard spheres are thought to apply to a broad selection of liquids, whatever the interaction type . The behavior of low-temperature glasses, however, may not be quite as universal. In particular, not all glasses may display a marginal phase. For instance, although the vibrational spectrum of soft spheres is remarkably robust with dimension, and marginality persists even when compressed far above the jamming transition, this regime is eventually extinguished .

Open theoretical questions also remain. First and foremost, the dimensional robustness of the jamming description is both remarkable and puzzling. The physical origin of this effect has no fully satisfying explanation. The universality of the phenomenon is also thought to apply far beyond structural glasses . Second, renormalization group analysis of these effects have yet to be fully developed . Whether perturbative or non-perturbative in nature, finite-$d$ analysis of the different transitions remains a work in progress. Third, the hard-sphere solution proposes tighter packing bounds than known, rigorous results (see for a recent discussion). Formalizing the results, as has been done for the solution of many problems with disorder over the last couple of decades, could thus open a new chapter in the venerable field of discrete geometry.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

The authors acknowledge funding from the Simons Foundation for the collaborative program “Cracking the Glass Problem”. PC acknowledges support from the National Science Foundation Grant no. NSF DMR-1055586. PU acknowledges support from NPRGGLASS.

References