Can the glass transition be explained without a growing static length scale?

I Introduction

What is the physical mechanism explaining the dramatic slowing down of glass-forming liquids as the temperature is decreased? This question has motivated a vast amount of work since the late 50’s review1 ; review2 . The seminal intuition of Adam and Gibbs is that atoms must move more and more collectively in order to flow AG , leading to an increase of the activation barrier as the glass transition temperature $T_{g}$ is approached. The idea of an underlying ‘amorphous order’ that sets in over larger lengthscales has progressively been confirmed, as a result of intense theoretical biroli-bouchaud ; montanari , experimental Ladieu ; Albert and numerical efforts cavagna1 ; cavagna2 ; glen ; walter ; sho ; sho2 ; ceiling in the last 20 years. It is now well-accepted that a static lengthscale grows, albeit modestly, in supercooled liquids approaching the glass transition.

However, the physical relevance of these static correlations for the abrupt dynamical slowdown of supercooled liquids is still actively debated. The ‘elastic picture’ for instance proposes that the chief physical ingredient driving the glass transition is the growth of the plateau shear modulus, $G_{\text{pl}}$, which makes even local moves progressively more difficult dyre . The growth of the activation barrier to flow would then simply mirror the growth of $G_{\text{pl}}$, without having to invoke any growing lengthscale. This purely local point of view was developed by Dyre dyre ; dyre-review and further promoted by Wyart & Cates (WC) wyart-cates , who take stock of the recent numerical results on the influence of particle swaps on the dynamics of polydisperse mixtures hs2016 ; ninarello2017 ; ultrastable ; misakijamming ; daniele2018 ; 2d . The swap Monte Carlo algorithm (simply denoted here by SWAP) allows permutations of pairs of particles with different diameters swap1 ; swap2 ; swap3 ; cavity1 . SWAP can be thought of as the introduction of an additional fluctuating degree of freedom attached to each particle — its diameter ninarello2017 . The physical dynamics is recovered when diameters are no longer allowed to fluctuate and only displacements of the particles are permitted. For well-chosen models, such a change in the local dynamical rules leads to a spectacular acceleration of the equilibration, reducing the relaxation time by several orders of magnitude hs2016 ; ninarello2017 .

In a nutshell, the WC argument is that if local rules are so important for the dynamics, then collective effects, while undisputably present, can only play a minor role in the $10^{15}$-fold increase of the relaxation time occurring when a liquid freezes into a glass. WC further argue that the violation of the Stokes-Einstein (SE) relation between viscosity and diffusion ediger2000 can be used to bound from above the influence of collective effects in the slowing down of the dynamics. Since the violation factor at $T_{g}$ is about $10^{3}$ for fragile liquids — compared to the $10^{15}$ increase of the relaxation time — WC argue that 4/5 of the slowing down in log-scale should be attributed to local effects, thus disputing the experimental relevance of a static correlation length for glass formation.

Three more papers have successively appeared on the same topic. Firstly, Ikeda et al. Ikeda use a mean-field glass model introduced earlier by Mari and Kurchan mari-kurchan to propose a theoretical description of SWAP. In this model, SWAP acceleration can be explained by a downward shift, computed by means of a static replica calculation, of the critical temperature of the mode-coupling transition once swap moves are allowed. Secondly, Brito et al. brito2018 used computer simulations and heuristic stability arguments to suggest that SWAP delays the onset of activated dynamics to lower temperatures than in the physical dynamics. Thirdly, a dynamic mode-coupling calculation was performed szamel2018 , which shows that coupling diameter and density fluctuations can shift the location of the mode-coupling singularity to a lower temperature (or a higher density), while of course maintaining all static observables unaffected.

Overall, these investigations provide sound explanations for the dynamic acceleration due to SWAP, with which we basically agree. However, they differ on the mechanism responsible for the slowing down of the dynamics when the glass transition is approached in the absence of SWAP, which in the end is the real question we are interested in. Is this mechanism purely local, as originally advocated by Dyre dyre ? Is it ‘quasi-local’, i.e., related to vibrational properties rather than static glassy correlations, as envisaged by Brito et al. brito ; brito2018 ? Are instead collective rearrangements on the scale of the static correlations crucial to understand the physical nature of the glass transition?

The aim of this paper is to revisit these questions, arguments and scenarios from the point of view of thermodynamic theories of the glass transition, and more specifically the Random First-Order Transition (RFOT) theory rfot1 ; rfot2 ; rfot2b ; rfot3 . Our two basic claims directly conflict with the local scenario summarized above:

The dramatic speedup of the dynamics induced by local changes in the dynamical rules is actually consistent with the RFOT theory, and more generally with static, cooperative explanations of the glass transition AG ; FLDT . We describe this in terms of a ‘crumbling metastability’, by which we mean that a metastability of collective thermodynamic origin can be postponed to lower temperature and higher pressure by purely local dynamical rules that leave thermodynamic properties unchanged. We provide a concrete illustration of this phenomenon in the case of crystal nucleation.

The Stokes-Einstein decoupling can remain mild in the RFOT scenario, because local permutation processes are prohibitively more costly than collective relaxation. In this case, WC’s upper bound on the influence of collective relaxation is not apposite.

The paper is organised as follows. In Sec. II we provide a short recap of the RFOT theory of glass formation. In Sec. III we discuss metastability in finite dimensions and show that the change of local dynamical rules can allow a supercooled liquid to get around a barrier of thermodynamic origin and lose its metastability. In Sec. IV we discuss how to best interpret SWAP efficiency within the RFOT framework. In Sec. V we critically assess the ability of local mechanisms to account for the phenomenology of glassy liquids. We conclude our paper in Sec. VI. We also discuss kinetically constrained models and the Mari-Kurchan model in Appendix A, and present a simple bootstrap approach to emerging rigidity in Appendix B.

II RFOT theory: A short recap

The RFOT theory of glasses rfot1 ; rfot2 ; rfot2b ; rfot3 is inspired by mean-field spin-glass models in which the analogue of the liquid state becomes rigid in a two-step process as the temperature is lowered wolynes1 ; KT ; MP . At a first temperature $T^{*}$, some incipient local rigidity, absent for $T>T^{*}$, allows metastable states to appear and ‘trap’ the system for some amount of time. These metastable states are exponentially numerous, with an associated positive configurational entropy, $\Sigma(T)$. The mere existence of such a large number of metastable states allows the system to decorrelate with time: it is still a liquid, albeit one with some transient rigidity described by a nonzero shear modulus $G_{\text{pl}}$. At a lower temperature $T_{K}<T^{*}$, the configurational entropy vanishes, and the system undergoes a phase transition to an ideal glass phase. These statements can be made sharp in mean-field situations. However, their interpretation for realistic finite-dimensional systems is delicate and, although enticing, is still under construction.

The mode-coupling crossover at $T^{*}$ plays a crucial role in the understanding of how SWAP may speedup the thermalization of supercooled liquids. In fact, all recent explanations put forward in the literature Ikeda ; brito2018 ; szamel2018 , including WC’s original one, rely on a shift of $T^{*}$ to a lower temperature $T^{*}_{\text{swap}}$ due to SWAP dynamics. Within mean-field theory the temperature $T^{*}$ can be detected by analyzing a replicated system in the formalism of Franz and Parisi FP . Interestingly, it is numerically found that the emergence of a nontrivial Franz-Parisi potential in three-dimensional glass-formers takes place close to the MCT transition simuFP1 ; simuFP2 ; ceiling . Physically, this also signals the emergence of nontrivial static properties, which in mean-field theory corresponds to the appearance of metastable states FP ; charlotte .

III Crumbling metastability

All of the above theoretical construct relies on a liberally-used but rather elusive concept, that of ‘metastable states’. In fact, the main theoretical difficulty posed by the glass transition precisely lies in correctly handling this concept in non-mean-field situations.

Metastability is intrinsically a dynamical property. The fact that a collection of micro-states forms a bona fide metastable state depends both on the dynamical rules and on a timescale. This timescale should allow the system to evolve among the given set of micro-states (which define the metastable state) and yet be short enough for not allowing escape from the latter birolikurchan ; schulman ; bovier . Such a separation of timescales can hold for one set of dynamical rules and not for another.

In many physical systems, the mechanism leading to the existence of metastable states is of thermodynamic origin, as, e.g., for supercooled liquids which are metastable with respect to the crystal, superheated liquids which are metastable with respect to the gas, etc. debenedetti ; cavagna09 . In the case of glass-forming liquids the situation is more intricate since the role of thermodynamics is still hotly debated. Purely kinetic effects (i.e., dynamical rules) may play a dominant role in inducing metastability, even when the thermodynamic landscape is trivial. This is the idea put forward by theories of the glass transition based on kinetically constrained models reviewKCM ; garrahan-chandler ; steve . One then expects that if changing the local dynamical rules removes some of the constraints, this type of metastability will be destroyed (see also Appendix A.1). This is somehow the core of the argument used by WC (and also Ikeda et al.) to explain the acceleration generated by the SWAP, in which the strict enforcement of fixed particle diameters in a polydisperse mixture is waived.

The main point we want to make is that even in cases where metastability is of thermodynamic origin, the temperature (or density) at which metastable states emerge and govern the physics of the system may depend on the local dynamics. Thus, by changing the local dynamical rules, the onset of activated glassy dynamics can be postponed to lower temperature or higher pressure.

III.2 Affecting thermodynamic barriers by changing local dynamics

While it is clear that a change in the local dynamical rules can allow the system to navigate in configuration space by avoiding some barriers, the question of whether such a change lowers or circumvents thermodynamic barriers is more subtle. By thermodynamic barriers, we mean free-energy barriers resulting from a thermodynamic drive involving collective behavior over a typical lengthscale that is definable through static observables only; and the considered change of dynamics is ‘local’ in that the allowed elementary moves always involve a limited number of atoms that does not grow as one changes the control parameter(s), unlike the static length.

WC’s main point is that thermodynamic barriers cannot be altered by a change in local dynamics and, hence, observing such a change necessarily invalidates to a large extent a thermodynamic description. While this is likely true for asymptotically large barriers, such a general statement has limited scope in practice, as we now show in the well-understood case of crystal nucleation.

Below the melting temperature a supercooled liquid is metastable with respect to the crystal. It is known that the possibility to avoid crystallization and subsequently to form a glass involves kinetic effects, as illustrated by the form of the so-called TTT (Time Temperature Transformation) diagrams debenedetti . However, in this case the kinetic part is usually associated with the growth process and is a priori unrelated to the thermodynamic component of the time for crystallization given by the rate of nucleation $r_{\rm nucl}$. One can express the time to crystallization as the product

It is well established that: (1) $\tau_{\rm kin}$ is slaved to the typical relaxation time in the liquidIn practice, one may argue that it is rather the diffusion time that should be considered but the difference is immaterial to our argument, as we will study a moderately supercooled liquid in which the decoupling between diffusion and $\alpha$-relaxation is not significantly large.; (2) the nucleation rate, $r_{\rm nucl}$, is of thermodynamic origin and goes as $\exp(-\Delta F/k_{B}T)$, with $\Delta F$ the free-energy of the critical nucleation droplet; $\Delta F$ remains finite in the thermodynamic limit debenedetti .

To demonstrate that a change of the local dynamical rules can in some cases alter the thermodynamic barrier $\Delta F$ to crystallization, and not only the kinetic contribution, we have studied crystal formation in one of the three-dimensional supercooled polydisperse mixtures for which SWAP has proven to be very efficient ceiling ; ninarello2017 . We compare results obtained with the standard Monte Carlo dynamics and with that in which SWAP moves are allowed. Our first observation is that SWAP is often so efficient that many putative polydisperse glass-forming liquid models crystallize even before being able to thermalize in the metastable supercooled liquid phase ninarello2017 . This, in itself, is already an indication that metastability and nucleation phenomena can be circumvented by a change of the local dynamics.

In order to study this effect in more detail, we have chosen an $11\%$ polydisperse mixture of hard spheres in the moderately supercooled (or rather supercompressed) phase, at pressures that are just above the melting one, and followed the time evolution of several samples by constant pressure Monte Carlo simulation with and without swap moves. The complex crystalline structures formed by a system very similar to the present one were recently analysed thanks to SWAP truskett .

The most interesting outcome of our computer study is displayed in Fig. 1 where we show the packing fraction after a quench as a function of time measured relative to the $\alpha$-relaxation time of the liquid, as obtained from the appropriate dynamics with or without swap. We observe that over the time span that we have been able to cover (which is of the order of $450\tau_{\alpha}$ at $P=16.2$), all of the samples when evolved with the ordinary dynamics have remained in the metastable liquid phase, whereas the same samples evolved with SWAP have all crystallized extremely fast. Since according to Eq. (2) the crystallization time divided by $\tau_{\alpha}$ is directly related to the thermodynamic barrier encountered by the system, our results show that the finite nucleation barrier thwarting crystallization in the ordinary dynamics is reduced or bypassed by introducing swap moves, which leads to a greatly accelerated nucleation process (on top of the expected effect on the kinetic term $\tau_{\rm kin}$)A related effect was reported in an earlier work sanz2007 where it had been shown that introducing swap moves could change the path to crystallization in binary mixtures when distinct forms of crystals can form during nucleation..

In fact it is even difficult to get an accurate measure of $\tau_{\alpha}$ using SWAP, since the system crystallizes over a timescale comparable to the equilibration time (i.e., before the correlation function approaches zero). By contrast, we have not been able to crystallize the system without SWAP at any pressure, even when using many independent samples that were run over extremely large times (in units of $\tau_{\alpha}$). This directly shows that a well-established metastability for crystal nucleation — which is of thermodynamic origin — can totally ‘crumble’ as a result of purely local changes in the dynamics. The observed speedup might be due to a reduction of the standard nucleation barrier with a critical nucleus similar to the non-SWAP case, or to the opening of a completely different channel bypassing that barrier. This is in itself an interesting question that requires further investigations, both in the crystallisation case and in the more complex glassy relaxation case.

The phenomenon in which metastable states are destabilized by a change in the local dynamics can therefore take place whether barriers are of thermodynamic origin or not (provided they are not too large, see next subsection). We expect this crumbling metastability to be rather dramatic when considering the complex free-energy landscape of glass-forming liquids. Before building on this aspect to discuss SWAP efficiency within the context of the RFOT theory in Sec. IV, it is useful to discuss in more detail different crossover temperatures relevant for nucleation, which have counterparts in the case of glassy dynamics as well – although a strict analogy with nucleation can be misleading in that case.

III.3 Two characteristic temperatures

For the sake of concreteness, we henceforth consider temperature as the control parameter, but depending on the system and physical conditions, the control parameter may instead be pressure, applied magnetic field or chemical potential, density, etc. Although our discussion applies to all cases in which metastability is of thermodynamic origin, it is convenient to keep the example of nucleation associated with a conventional first-order transition, as the above crystallization case or the Ising model in dimension $d\geq 2$.

Metastability requires the existence of different thermodynamic states that can be envisaged as coexisting under some conditions. In finite-dimensional systems, nucleation is a mechanism through which, when changing the control parameter(s), the less stable states disappear and transform into the most stable one, and this activated process is driven by a thermodynamic force. All of this can only exist below a certain temperature $T_{\rm onset}$, which in the Ising model or the liquid-gas transition of a fluid is the critical temperature $T_{c}$. However, the temperature $T^{*}$ below which one can indeed observe metastability is generically lower than this upper limit of metastability. In the example of the Ising model, the metastability of a negatively magnetized state in the presence of a positive magnetic field $H$ appears at a temperature $T^{*}$ strictly below $T_{\rm onset}=T_{c}$, and it depends on the magnetic field: $T^{*}(H)<T_{\rm onset}$. Whereas the temperature $T_{\rm onset}$ is only determined by the thermodynamics of the system, $T^{*}(H)$, which signals a crossover at which the effect of metastability can be observed (i.e. when the rate $r_{\rm nucl}$ becomes small), depends on the local dynamics and on the observation timescale.

Purely kinetic effects can only alter thermodynamic metastability under some conditions, and one expects that one such condition is that the lengthscale characterizing the nucleation process and the resulting escape from a metastable state are not too large. As a matter of fact, in the case of the Ising model, it can be proven that for small magnetic fields, when the nucleation size is very large, the nucleation barrier is independent of local dynamics and given by the thermodynamic nucleation argument martinelli . Yet, in cases where characteristic lengthscales are not large, a change in the local kinetic rules can have a dramatic effect and destroy metastability, as we have illustrated above in the case of crystallization.

IV SWAP efficiency within RFOT theory

As already pointed out, $T_{\rm onset}$ is independent of the microscopic dynamics whereas $T^{*}$ a priori depends on it. We will keep the notation $T^{*}$ for ordinary dynamics and call $T^{*}_{\text{swap}}$ the corresponding temperature for SWAP, and use similar definitions for the characterisitic densities $\varphi_{\rm onset},\varphi^{*}$, and $\varphi^{*}_{\text{swap}}$. As we have shown in the previous section, SWAP allows a much faster equilibration in the example of the nucleation of the crystal phase in a metastable supercooled liquid, where barriers are of thermodynamic origin. Similarly, in glass-forming liquids around and below $T^{*}$ (or $\varphi^{*}$), SWAP is expected to wash out the metastability associated with the incipient amorphous order and to have dramatic consequences on the relaxation time in the range $T^{*}_{\text{swap}}<T<T^{*}$ (resp., $\varphi^{*}_{\text{swap}}>\varphi>\varphi^{*}$), as also suggested in Refs. Ikeda ; brito2018 ; szamel2018 . We illustrate this point below in the case of polydisperse hard-sphere systems.

IV.2 SWAP and effective landscape

In equilibrium, the typical configurations visited by the system do not depend on the dynamical rules, provided detailed balance is satisfied. However, the ‘effective landscape’ seen by the system, or more precisely by its representative point in configurational space, depends on the local dynamical rules: some channels allowing it to go from one configuration to another may be open for one type of dynamics and closed for another type. This is clearly true when one allows swap moves between particles of different diameters in a Monte Carlo algorithm. These moves correspond to displacements in real space that are at least of the order of one particle diameter. (They also correspond to large displacements in the configurational space associated with fixed-diameter particles.) Introducing swaps is equivalent to providing an extra dimension in which exchange of particles is much easier, while still fulfilling detailed balance. The landscape may also depend on the probability $p$ controlling the frequency of the swap moves in the Monte Carlo simulation. One can alternatively think of these moves as allowing each particle to change its diameter, which then adds one degree of freedom to each particle, with the constraint that the distribution of diameters is conserved at each stepThis strict constraint can be softened by considering as in Refs. brito2018 ; szamel2018 a grand-canonical version in which diameters are allowed to fluctuate, subject to a diameter-dependent chemical potential that enforces a constraint on the average size distribution only.. This additional degree of freedom can be discrete or continuous depending on the exact nature of the polydisperse mixture.

Opening new dynamical paths, as SWAP does, can only increase the number of unstable modes around stationary points and potentially destabilize states that would be metastable with the conventional dynamics. The effective landscape of the SWAP, which is a representation of configurations grouped into basins considered as metastable states over a given timescale and of the set of paths that connect one to another, must therefore be different from that in the absence of swap movesThe zero-temperature version of these free-energy landscapes are discussed in Ref. brito2018 in the context of the jamming transition..

As a case in point, we show in Fig. 2 the time dependence of the self-intermediate scattering function for the same three-dimensional hard-sphere model as studied in Refs. hs2016 ; ceiling ; ninarello2017 . (The polydispersity here is 23 % and is thus larger than the one used in Fig. 1). We choose a volume fraction that is intermediate between $\varphi^{*}$ and $\varphi^{*}_{\rm swap}$. Strikingly, whereas the familiar two-step relaxation process is observed in the standard case, there is no longer any hint of an inflexion point in the swap case. Configurations that were metastable in the former case are completely unstable in the latter case: Locally rigid systems are fluidized by SWAP, i.e., glassy metastability has crumbled. A similar point was made in Ref. brito2018 and is further discussed in Appendix B.

The effective landscape being of dynamical nature, it depends in principle on the probability $p$ controlling the frequency of the swap moves compared to the translational ones in the simulation. This raises the possibility to explore a continuous range of effective landscapes, obtained by a continuous change of the probability $p$. We illustrate in Fig. 2 the effect of changing $p$ on the slowdown of relaxation of the same three-dimensional hard-sphere model as above. For this model, the empirically determined mode-coupling crossover occurs at a packing fraction $\varphi^{*}\approx 0.6$ and one can barely go beyond it for $p=0$. SWAP is most efficient with $p\approx 0.2$ hs2016 ; ceiling ; ninarello2017 , because the dynamical speedup obtained by increasing $p$ further is not enough to compensate the increasing computational cost of the swap moves. This, then, corresponds to the optimal combined annealing of the diameter changes and particle displacements.

For very small values of $p$, one observes a crossover from the normal dynamics at moderate $\varphi$ to the optimal swap dynamics at large $\varphi$, suggesting that all these dynamics in fact smoothly interpolate between only two extreme cases. This implies that as soon as $p>0$, the system will ultimately explore the effective landscape where particle diameters are allowed to fluctuate. It is therefore convenient to focus only on the two extreme dynamics: the conventional one with $p=0$ and the one with the optimal choice of $p$ corresponding to the best joint annealing of all degrees of freedom. (Note that throughout this paper the term SWAP refers to the latter.)

IV.3 Self-consistent description of activated dynamics

It is easy to see graphically that this equation has an activated solution when $A$ is small enough, as we envisage for the normal dynamics: see Fig. 3. The relaxation time $\tau_{\alpha}$ decreases when $A$ increases, i.e., as the local dynamical fluctuations are increased, as in the case of SWAP. The activated solution then abruptly disappears for some value of $A$. This is our crumbling metastability scenario: Faster motion of the surroundings prevents the freezing of the inside of the cavity. Conversely, for a given $A$, an activated solution always appears at low enough temperature.

Now, since the temperature dependence of the relaxation time with SWAP is much weaker down to $T^{*}_{\rm swap}$ (and its accessible vicinity) and since the relaxation times for SWAP and ordinary dynamics must diverge in the same manner close to $T_{K}$, our picture suggests an extremely steep increase of the logarithm of the SWAP relaxation time in a narrower range of temperature $T_{K}<T<T^{*}_{\rm swap}$ than for the normal dynamics. This would correspond to an unusually ‘fragile’ behavior. It is however likely that this takes place in a range which is difficult to equilibrate for SWAP itself, and that the convergence of the SWAP and non-SWAP relaxation times only occurs at astronomically long times. Still, the available data such as the one shown in Fig. 2 is not incompatible with this view. Indeed the density dependence of the normal Monte Carlo dynamics appears less sharp than the one of the swap dynamics with a large frequency of swap moves.

Although the interpretation put forward above uses similar ideas as in Wyart & Cates wyart-cates and Brito et al. brito2018 concerning the different effective landscapes for SWAP and normal dynamics, it differs completely on the conclusions one should infer about the physical dynamics. We strongly disagree with the claim that the SWAP efficiency disproves the relevance of activated processes over free-energy barriers controlled by thermodynamic properties for the normal dynamics near $T_{g}$, a claim that we find unsubstantiated.

The disagreement might be related to the following subtle (and possibly confusing) considerations:

V Local versus collective scenario for glass formation

Core to the argument of WC that collective effects play a minor role in the slowing down of relaxation leading to glass formation is the value of the Stokes-Einstein product between the self-diffusion constant and the viscosity estimated at $T_{g}$: $\mathcal{S}_{g}\equiv D(T_{g})\eta(T_{g})$. It is of order $\mathcal{S}_{g}\sim 10^{3}$ in fragile liquids. In their paper, WC suppose that the temperature-dependent barriers to relaxation in supercooled liquids add up as

from which the amplitude of the decoupling is deduced:

WC’s conclusion is therefore that $B_{\text{coll}}(T_{g})/k_{B}T_{g}$ is at most $\log 10^{3}$, when the total barrier to relaxation at $T_{g}$ is, by definition, $B(T_{g})/k_{B}T_{g}\approx\log 10^{15}$. Hence, according to WC, the collective barrier only explains about $1/5$ of the slowing down (in number of decades) from high temperature to the glass transition, namely, 3 decades out of the 15 decades observed for the viscosity increase. Taking into account dynamical heterogeneities would reduce even further the part of the decoupling due to collective barriers.

There are several experimental and numerical facts that show that this scenario is moot. Very costly, local permutations must indeed become the dominant channel to single-particle dynamics very close to $T_{K}$ (because $B_{\text{coll}}$ then diverges whereas $E_{\text{loc}}$ remains finite), in which case we indeed expect a huge SE violation factor. However, both single-particle dynamics and collective-density relaxation use the very same collective mechanism when $T\gtrsim T_{g}$, and this also applies to diffusion. Indeed, several studies of the SE factor have shown that the decoupling between diffusion and relaxation can be explained in terms of a spatial heterogeneity of the relaxation. More precisely, viscosity and diffusion probe different moments of the distribution of local relaxation times $\tau$ ediger2000 ; book , i.e., $\eta\propto\langle\tau\rangle$ and $D\propto\langle\tau^{-1}\rangle$, which behave differently as temperature decreases, even when both these quantities are controlled by the same collective effects. Several direct indications that this picture is correct are:

The amount of SE violation is empirically correlated with the exponent $\beta$ characterizing the stretching of the time-dependent relaxation function ediger2000 , thus confirming the important role played by the broadness of the distribution of the local relaxation times;

The same amount of decoupling between diffusion and structural relaxation is observed when considering exclusively single-particle dynamics (but over a range of wavevectors) PRE2004 ; in the WC scenario this should not happen since decoupling is a direct consequence of comparing a single-particle observable with a collective one;

The dynamical susceptibilities (four-point correlation functions) for collective and self density relaxation both show similar peaks for the same $\alpha$-relaxation time of the system glotzer , thus suggesting that the very same collective phenomenon is responsible for both.

Several phenomenological approaches have shown that such a description is able to reproduce the observed values of the SE factor $\mathcal{S}_{g}$ as well as other dynamical features of supercooled liquids viot2000 ; xia2001 . We therefore claim that the physical relaxation channel leading to diffusion and that leading to viscosity are in fact the same — at least close to $T_{g}$ — and has to be collective, i.e., has to involve a substantial number of particles.

V.2 Emerging rigidity: shortcomings of a ‘quasi-local’ scenario

Invoking a purely local explanation of the glass transition appears to us to be a step back to the pre-1995 state of affairs, before the flurry of activity around spatial heterogeneities in glass-forming liquids ediger2000 ; book . Below the crossover temperature $T^{*}$ where most theories envisage a change of nature of the relaxation in fragile supercooled liquids, a purely local scenario is clearly at odds with many experimental and numerical data. In particular it cannot be reconciled with the continuous growth of spatial correlations in the dynamics of supercooled liquids that is detected by multi-point space-time correlation functions science ; quedalle and nonlinear responses Albert ; Ladieu .

WC argue that the local scenario should rather be interpreted as ‘quasi-local’ in the sense that local moves involve finite collections of particles wyart-cates . The associated lengthscale is surmised to be the dynamical correlation length $\xi_{\rm dyn}$, whose growth is aborted when $T\approx T^{*}$ (hence the quasi-local character below $T^{*}$). Following elastic models of viscous liquids such as the shoving model put forward by Dyre dyre ; dyre-review , WC implicitly assume that

where $V_{c}\sim\xi_{\rm dyn}^{3}$ is constant below $T^{*}$ and $G_{\text{pl}}(T)$ is the plateau shear modulus.

However, WC wyart-cates , as well as Brito et al. brito2018 , remain vague about the relaxation mechanisms around and below $T^{*}$ within their scenario, which nonetheless does not easily account for the well-established monotonous growth of multi-point correlation functions below $T^{*}$. In their view, spatial correlations in the dynamics should only be present near $T^{*}$, i.e., at the onset of rigidity where the soft modes associated to marginal stability become delocalized brito . Below $T^{*}$ the system departs from marginality and becomes increasingly more stable. This increased stability should naturally imply that the lengthscale related to the relaxation process [i.e., $V_{c}$ in Eq. (8)] decreases, in contradiction with experimental and numerical results which show instead that this length continues to grow quedalle . The possibility, invoked by WC and Brito et al., that the dynamics below $T^{*}$ is thermally activated along some soft dynamical modes spanning a length $\xi_{\rm dyn}$ that does not decrease is intriguing but lacks at present a substantial explanation, precise quantitative calculations and numerical support.

Another necessary input in a scenario based on Eq. (8) is the emergence of a nonzero plateau shear modulus $G_{\text{pl}}(T)$ and its sizable increase of as one cools the system. Even if it is not precisely described in wyart-cates ; brito2018 , the emergence of rigidity associated with the disappearance of soft marginal modes near $T^{*}$ appears akin to the MCT/RFOT scenario. As recalled in Appendix B, a nonzero plateau shear modulus $G_{\text{pl}}(T)$ indeed appears rather abruptly and grows as $T$ is decreased below $T^{*}$, with a concave temperature dependence yoshino2012 . Therefore, even with a nondecreasing $V_{c}$, Eq. (8) would predict a concave (downward) temperature dependence of the activation barrier. This qualitatively disagrees with the empirical observation of an convex (upward) growth as temperature decreases (see Tarjus2 ; rossler ), except if the experimental $G_{\text{pl}}(T)$ has itself a convex behavior, which would be at odds with most studies of the Debye-Waller factor (proportional to $T/G_{\text{pl}}$, see Appendix B) in the vicinity of $T^{*}$.

Finally, some additional numerical and experimental results appear to favor the collective scenario over the WC scenario. For example, Larini et al. Larini have shown that for many glassy systems, one can find a master curve that collapses the dependence of the logarithm of the relaxation time $\log\tau$ as a function of the ratio $\langle u^{2}\rangle_{g}/\langle u^{2}\rangle$, where $\langle u^{2}\rangle$ is the Debye-Waller amplitude of displacements in the plateau regime. According to Dyre dyre-review and WC, this relation should be linear, while the master curve of Larini et al. reveals a large nonlinear component. Taking their functional fit seriously suggests that the nonlinear contribution to the increase of $\log\tau$ is 6 times larger than that of the linear contribution. This is qualitatively similar to the results of Buchenau et al. Buchenau , who conclude that the collective barrier contribution to fragility at $T_{g}$ is between 1 and 6 times that of the local barrier contribution, with a ratio that increases with fragility itself. These results again suggest that collective effects play a large role in the increase of the effective energy barrier $B_{\text{tot}}$ as the temperature is lowered.

V.3 Insights from amorphous confinement

This dynamical phenomenon, which becomes more prominent as the point-to-set length increases, emerges below $T^{*}$ and above $T^{*}_{\rm swap}$. In this temperature regime, the bulk normal dynamics is deep into the (inaccessible) super-Arrhenius activated regime and the bulk SWAP dynamics is still very fast and characterized by a small dynamic correlation length $\xi_{\rm dyn,swap}$.

VI Conclusion

We have argued that the observations recently put forward to dismiss the role of collective effects in the dynamical slowing down of glass-forming liquids — namely, the efficiency of the SWAP Monte Carlo algorithm to thermalize polydisperse glass-formers below $T_{g}$ and the Stokes-Einstein decoupling at $T_{g}$ — can actually be accounted for within theories in which these collective effects are central. We contended that alternative local and quasi-local explanations of SWAP acceleration have trouble coping with several experimental and numerical results on supercooled liquids.

Our main point is that introducing swap moves can wash out metastability and allow the system to bypass free-energy barriers, even of thermodynamic origin. This crumbling metastability postpones the temperature at which collective activated processes are required and therefore dramatically accelerate the dynamics of the system — at least when the cooperatively rearranging regions are not too large. We have shown numerically that this is the case for standard nucleation.

Our general conclusion is that the success of SWAP, while generating a genuinely useful debate about the validity of the RFOT approach, does not directly favor one view of the glass transition nor disproves any. However, since it allows the exploration of new and unexplored territory in glass physics scalliet ; gardner ; deposited ; yielding ; lowTmodes , we might expect some progress towards discriminating the relevance of various approaches, when their predictions are pushed and tested over an unprecedently wide range of temperature and studied as a function of dimension $d$ as wellGiven that by lowering the dimension $d$, fluctuations are expected to decrease thermodynamic barriers for a fixed value of the point-to-set length, we expect that SWAP should allow the exploration of a larger range of point-to-set lengths in two dimensions 2d , and a smaller range in dimensions larger than three.. We hope that progress along these lines will be available soon.

Appendix A Other scenarios for SWAP efficiency

The dynamic facilitation approach to the glass transition garrahan-chandler mostly relies on the idea that dynamics slows down because of kinetic constraints which forbid some pathways steve . Barriers thus have a purely kinetic origin, and are totally unrelated to the thermodynamic properties of the system. The approach is illustrated by simple noninteracting spin models known as kinetically constrained models (KCM) reviewKCM . Although swap moves per se may not be relevant for such models, the success of a change of local dynamical rules seems easily explained conceptually, since the kinetic constraints are purely local and can thus, in principle, be extremely sensitive to the microscopic dynamics (see however below). In a sense, it is an implicit prediction of this approach (although it was never truly put forward or explored) that certain types of local algorithms should be able to considerably accelerate the thermalization of these systems. Indeed in KCMs, thermalization of initial conditions is not a difficult issue at all, and only the dynamical relaxation processes are of physical interest.

On the other hand, this issue might be more subtle than it seems at first sight. In fact, in the only microscopic models where kinetic constraints were derived from first principles, the so-called plaquette models juanpe , any local change of the spin-flip dynamics would not be able to fully accelerate the dynamics. In these systems, dynamics is glassy at low temperature and there is a diverging static correlation length with a thermodynamic singularity at zero temperature. The origin of glassiness is the existence of dilute defects at low temperature. Local changes in the dynamics can only speed up the dynamics of defects by increasing their diffusion coefficient but not reducing their dilution.

Ideally, one would like to know better how kinetic constraints emerge in the dynamics of realistic supercooled liquids, in order to invent smart Monte Carlo moves that at once preserve the equilibrium properties and bypass the kinetic constraints. For now, it is unclear why SWAP would achieve precisely this nontrivial goal. Similarly, it appears difficult to predict, in this view, what is the temperature dependence of the SWAP dynamics and what are its characteristics. Such an understanding would be useful, as it could allow the potential determination of even better algorithms to achieve thermalization at lower temperatures.

A.2 The Mari-Kurchan model

In their analysis of the Mari-Kurchan (MK) model mari-kurchan , Ikeda et al. Ikeda discuss the emergence of two distinct mode-coupling temperatures. Their main conclusion is that the normal dynamics essentially freezes at some temperature $T^{*}$, whereas the SWAP dynamics does at a lower temperature, $T^{*}_{\rm swap}<T^{*}$. Although this is along similar lines to ours and WC’s discussion, the interpretation of some of these results and their meaning for finite-dimensional systems is quite different, as we now explain.

Let us first address a crucial point. The MK model is a ‘mean-field’ description of a finite-dimensional system but a very special one, where neither the MCT dynamical transition nor the thermodynamic glass transition are truly realized. Indeed, the genuine thermodynamic equilibrium of the model is attained when including all local moves, which comprise changes of particle diameters (the equivalent of the swaps) and individual particle hopping. In consequence, the emergence of a ‘high’ transition temperature $T^{*}$ when arbitrarily preventing diameter changes results from a purely kinetic constraint. If instead one allows by fiat changes of diameters without changing the positions of the particles, freezing and lack of diffusion take place at a lower temperature, identified as $T^{*}_{\rm swap}$. If one allows all local moves, including diameter change and particle hopping, then no freezing takes place at any finite temperature.

This situation is very different from generic finite-connectivity mean-field models. Indeed, for mean-field models on Bethe lattices where locality has a well-defined meaning, it is conjectured and to a large extent proven montanari that any local change of the dynamics cannot generically alter the value of $T^{*}$. This statement is related to the general idea that infinite thermodynamic barriers, such as those emerging at $T^{*}$ in mean-field models, cannot be destroyed by any local change of the dynamics. For example, consider a finite temperature lattice-glass binary mixture on a Bethe lattice bethe . Such a model has a unique MCT transition, irrespective of the local dynamics (as long as it is an irreducible Markov chain), at which the Boltzmann measure breaks up into many thermodynamic states separated by infinite barriers. Certainly, the grand-canonical dynamics, which resembles the SWAP one, is faster than the canonical one but they both have the same $T^{*}$. This is the usual mean-field scenario on which the RFOT theory is based. In the zero-temperature limit, where the Markov chain becomes reducible since hard kinetic effects intervene, an MCT transition which is similar to those of KCM’s on Bethe lattices bethe can preempt the thermodynamic MCT transition. It is the counterpart of the former kind of transition that has been investigated by Ikeda et al. and that, arguably, can take place in other mean-field models as well, due to kinetic constraints in the dynamicsThe infinite dimensional limit certainly plays an important role in inducing the kinetic transition found in Ikeda ..

In short, the work of Ikeda et al. introduces a smart approximation to take into account kinetic effects within a static replica computation. We agree that it provides a possible explanation of the difference between $T^{*}$ and $T^{*}_{\rm swap}$, which is of kinetic origin. Overall, it reiterates from a mean-field perspective the dynamic facilitation view that kinetic constraints are the main cause of slow dynamics in supercooled liquids. Although this is certainly possible, and realized in some models endowed with specific dynamical rules, we think that experimental and simulation results for supercooled liquids with (standard) dynamics rather point toward an explanation in which a growing static length plays a key role.

In summary, our disagreement stems from the interpretation of the MK results and their implications to finite-dimensional supercooled liquids, not from the analytical results of Ikeda et al. which, we believe, are correct.

Appendix B A bootstrap theory for emerging rigidity

Clearly, the appearance of local rigidity must itself be a collective, bootstrap phenomenon. This idea permeates many approaches to the glass transition, from the “cage” picture advocated in the context of MCT to the isostatic theories of jamming. For a particle not to move easily, its neighbours must themselves be blocked, and so on. One can formalize this scenario by using the recent replica theory of Yoshino and Mézard yoshinomezard ; yoshino2012 that allows one to estimate the plateau shear modulus $G_{\rm pl}$ from first principles. The physical idea is to write $G_{\rm pl}$ as a difference of two contributions

where $G_{\rm born}$ is the so-called Born term coming from the contribution of affine displacements under shear, and $G_{\rm fluc}$ is a contribution induced by thermal fluctuations. In an ergodic, liquid, phase translation invariance imposes that $G_{\rm fluc}=G_{\rm born}$, and, thus, that the shear modulus is zero, as expected. At smaller temperatures, $G_{\rm fluc}$ decreases, allowing for the possibility of rigidity, at least on intermediate timescales. The detailed replica calculations are quite intricate yoshinomezard ; yoshino2012 , but one can grasp the correct scenario by approximating $G_{\rm fluc}$ as:

where $b$ is a coefficient that depends on the microscopic details of the model. This equation states that local fluctuations in a solid are proportional to temperature and inversely proportional to the shear modulus (which assumes that the latter is small compared to the bulk modulus). Inserting Eq. (10) back into Eq. (9) leads to a second-order equation for $G_{\rm pl}$, whose solution is

This simple description predicts that the plateau modulus jumps from to a finite value at $T=T^{*}$, with a singular square-root contribution when $T<T^{*}$, exactly as predicted by standard MCT. Intuitively, MCT captures the self-consistent appearance of local rigidity, that can only exist if the rest of the system is itself rigid enough. By the same token, if the local dynamical rules (like SWAP) lead to an increase of local fluctuations (i.e., an increase of $a$), one expects $T^{*}$ to decrease. Note that one can rephrase the above argument in terms of bootstrap percolation with the effect of swaps modelled as an increased number of neighbors required to stabilize each particle FranzSellitto ; BiroliIkedaMiyazaki . This latter formulation is conceptually very close to the framework proposed by Brito et al. brito2018 .

In real three-dimensional systems, the above rigidity transition is replaced by a continuous crossover, with a plateau shear modulus that appears smoothly below $T^{*}$, but with concave, square-root like dependence on temperature, qualitatively similar to that predicted by MCT. This was actually considered as one of the early success of MCT mct1 ; mct2 .

References