Perspective: Configurational entropy of glass-forming liquidsLudovic Berthier,1 Misaki Ozawa,1 and Camille Scalliet1

Laboratoire Charles Coulomb (L2C), Universite de Montpellier, CNRS, Montpellier,France

(Dated: 7 February 2019)

The configurational entropy is one of the most important thermodynamic quantities characterizing super-cooled liquids approaching the glass transition. Despite decades of experimental, theoretical, and computa-tional investigation, a widely accepted definition of the configurational entropy is missing, its quantitativecharacterization remains fraud with difficulties, misconceptions and paradoxes, and its physical relevance isvividly debated. Motivated by recent computational progress, we offer a pedagogical perspective on the con-figurational entropy in glass-forming liquids. We first explain why the configurational entropy has becomea key quantity to describe glassy materials, from early empirical observations to modern theoretical treat-ments. We explain why practical measurements necessarily require approximations that make its physicalinterpretation delicate. We then demonstrate that computer simulations have become an invaluable toolto obtain precise, non-ambiguous, and experimentally-relevant measurements of the configurational entropy.We describe a panel of available computational tools, offering for each method a critical discussion. Thisperspective should be useful to both experimentalists and theoreticians interested in glassy materials andcomplex systems.

I. CONFIGURATIONAL ENTROPY AND GLASSFORMATION

A. The glass transition

When a liquid is cooled, it can either form a crystalor avoid crystallization and become a supercooled liquid.In the latter case, the liquid remains structurally disor-dered, but its relaxation time increases so fast that thereexists a temperature, called the glass temperature Tg,below which structural relaxation takes such a long timethat it becomes impossible to observe. The liquid is thentrapped virtually forever in one of many possible struc-turally disordered states: this is the basic phenomenologyof the glass transition.1–4 Clearly, Tg depends on the mea-surement timescale and shifts to lower temperatures forlonger observation times. The experimental glass transi-tion is thus not a genuine phase transition, as it is notdefined independently of the observer.

The rich phenomenology characterizing the approachto the glass transition has given rise to a thick litera-ture. It is not our goal to review it, and we refer in-stead to some reviews.1–9 There are convincing indica-tions that the dynamic slowing down of supercooled liq-uids is accompanied by an increasingly collective relax-ation dynamics. This is seen directly by the measure-ment of growing lengthscales for these dynamic hetero-geneities,10–12 or more indirectly by the growth of the ap-parent activation energy for structural relaxation, as seenin its non-Arrhenius temperature dependence. These ob-servations suggest an interpretation of the experimentalglass transition in terms of a generic, collective mecha-nism possibly controlled by a sharp phase transition.13

‘Solving the glass problem’ thus amounts to identifyingand obtaining direct experimental signatures about thefundamental nature and the mathematical description ofthis mechanism.

Why is this endeavor so difficult as compared to otherphase transformations encountered in condensed mat-ter?14,15 The core problem is illustrated in Fig. 1 by twoparticle configurations taken from a recent computer sim-ulation.16 The left panel shows an equilibrium configura-tion of a two-dimensional liquid with a relaxation time oforder 10−10 s, using experimental units appropriate for amolecular system. The right panel shows another equi-librium configuration now produced close to Tg with anestimated relaxation timescale of order 100 s. The systemon the right flows 1012 times slower than the one on theleft, but to the naked eye both configurations look quitesimilar. In conventional phase transitions,14,15 a struc-tural change takes place and some form of (crystalline,nematic, ferromagnetic, etc.) order appears. Glass for-mation is not accompanied by such an obvious structuralchange. Therefore, the key to unlock the glass problemis to first identify the correct physical observables to dis-

FIG. 1. Two equilibrium configurations of a two-dimensionalglass-forming model characterized by relaxation times thatdiffer by a factor 1012. The two density profiles appear sim-ilarly disordered and featureless. These two states in factdiffer by the number of available equilibrium states, ratherthan their aspect. The configurational entropy quantifies thisdifference.

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tinguish between the two snapshots in Fig. 1.Several theories, scenarios and models have been de-

veloped in this context.5,17–27 Some directly focus onthe rich dynamical behavior approaching the glass tran-sition,24 while others advocate some underlying phasetransitions of various kinds,17,19,20 possibly involvingsome ‘hidden’ or amorphous order.

In this perspective, we explore one such research line,in which configurational entropy associated with a grow-ing amorphous order plays the central role.19,20,28 Weargue that recent developments in computational tech-niques offer exciting perspectives for future work, allow-ing the determination of complex observables that arenot easily accessible in experiments, as well as the explo-ration of temperature regimes relevant to experiments.

B. Why the configurational entropy?

The fate of equilibrium supercooled liquids followed be-low Tg with inaccessibly long observation times was dis-cussed 70 years ago by Kauzmann in a seminal work.29

Since the supercooled liquid is metastable with respect tothe crystal, Kauzmann compiled data for the excess en-tropy, Sexc ≡ Sliq − Sxtal, where Sliq(T ) and Sxtal(T ) arethe liquid and crystal entropies, respectively. Kauzmannobserved that Sexc(T ) decreases quite sharply with de-creasing the temperature of the equilibrium supercooledliquid.

An extrapolation of the temperature evolution of Sexc

from equilibrium data to lower temperatures suggeststhat Sexc will become negative at a finite temperature,which led Kauzmann to comment:29 ‘Certainly it is un-thinkable that the entropy of the liquid can ever be verymuch less than that of the solid.’ To avoid this para-doxical situation, referred to as the Kauzmann paradoxor entropy crisis, he mentioned the possibility of a ther-modynamic glass transition occurring well below Tg, ata temperature now called the Kauzmann temperature,TK . Although Kauzmann suggested that crystallizationeventually prevents the occurrence of an entropy crisis,Kauzmann’s intuition remains very influential, for goodreasons.

Gibbs and DiMarzio were the first to give theoreticalinsights to the temperature evolution of Sexc, by analogywith a lattice polymer model whose entropy is purelyconfigurational.30,31 Hence the conventional name, ‘con-figurational entropy’ and notation Sconf , widely used inthe experimental literature.32 We show below that thereis no, and that there cannot be any, unique definitionof Sconf . We nevertheless use the same symbol for allestimates, in particular Sconf ≈ Sexc.

We compile state-of-the-art experimental32,34,35 andnumerical16,33 data of Sconf , and their extrapolation tolow temperatures in Fig. 2. We employ a representationclose to Kauzmann’s original analysis,29 rescaling Sconf

by its value at some high temperature (we choose themode-coupling temperature Tmct,

36 for convenience).

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

T/Tmct

2MTHF OTP Ethylbenzen Propanol Salol Toluene SS, d=3(Isochoric ) SS, d=3(Isobaric ) HS, d=3 SS, d=2(Isochoric ) KA, d=3(Isochoric )

Vapor deposited

S conf(T)/S conf(T

mct)FIG. 2. Experimental and numerical determinations of theequilibrium configurational entropy in various models16,33 andmaterials.32,34,35 Data points extracted from vapor deposi-tion experiments35 are indicated by the ellipse. Both axisare rescaled using the mode-coupling crossover as a referencetemperature at which the relaxation time is about 10−7s. Forhard spheres, the inverse of the reduced pressure, 1/p, playsthe role of the temperature. Extrapolation to low tempera-tures suggests the possibility of an entropy crisis at a finiteTK in d = 3, whereas TK = 0 in d = 2.

In calorimetric experiments, the configurational en-tropy becomes constant below Tg upon entering the non-equilibrium glass regime, defining a residual entropy.29,34

The glass residual entropy is a non-equilibrium effect thathas been extensively discussed.37–41 Here, we focus onlyon equilibrium supercooled liquids and do not discussfurther the glass residual entropy and do not show non-equilibrium measurements in Fig. 2.

The data for ethylbenzene and toluene are extendedby combining conventional calorimetric measurements todata indirectly estimated from ultrastable glasses pro-duced using vapor deposition.35,42 Various computationalmodels using hard,43 soft,44 and Lennard-Jones poten-tials,45 along isochoric and isobaric paths, in spatial di-mensions d = 216 and 333 are included along with exper-iments.32,34,35 This representative data set demonstratesthat all glass formers in dimension d = 3 display a sharpdecrease of Sconf , even down to a temperature regime un-available to Kauzmann. These results reinforce the ideathat Sconf can vanish at a finite temperature, TK > 0.Simulation data in d = 2 suggest instead that Sconf van-ishes only at TK = 0, suggesting that a finite TK entropycrisis can only occur for d > 2.16

Of course, the data in Fig. 2 do not rule out theexistence, at some yet inaccessible temperature, of acrossover in the behavior of Sconf that makes it smoothlyvanish at T = 0 in all these systems,46,47 or remain fi-nite with an equilibrium residual entropy in classical sys-tems,48–52 or a discontinuous jump due to an unavoidablecrystallization,29,53,54 or a liquid-liquid transition,22 or a

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conventional (kinetic) glass transition.55 These alterna-tive possibilities are not supported by data any betterthan the entropy crisis they often mean to avoid. It isimpossible to comment on the many articles supportingthe absence of a Kauzmann transition46,50,56–59, but weclarify below that none of them resists careful exami-nation. The existence of a thermodynamic glass transi-tion remains an experimentally and theoretically valid,but unproven, hypothesis. Thus, extending configura-tional entropy measurements to even lower temperaturesremains an important research goal.60

As emphasized repeatedly, a negative Sexc is not pro-hibited by thermodynamic laws.56 This is also not ‘un-thinkable’ since entropy is not a general measure of dis-order. As a counterexample, think of hard spheres forwhich the crystal entropy is larger than that of the fluidabove the melting density. A stronger reason to ‘resolve’the Kauzmann paradox is that if Sliq continues to de-crease further below Sxtal, the third law of thermody-namics could be violated.61 However, the third law is con-ventionally interpreted as a consequence of the quantumnature of the system.62 This implies that the Kauzmannparadox is not really problematic if considered within therealm of classical physics. In summary, there is no theo-retical need to avoid the entropy crisis.

However, theoretical treatments rooted in Gibbs andDiMarzio’s theory30,31 relate the configurational entropyto the (logarithm of the) number of distinct amorphousstates available to the system at a given temperature. Aproper enumeration of those states must therefore resultin a non-negative configurational entropy. In this inter-pretation, Fig. 2 suggests that a fundamental change inthe properties of the free-energy landscape must underlieglass formation.

A strong decrease of the configurational entropy an-swers the question raised by the structural similarity ofthe snapshots in Fig 1. Conventional phase transitionsdeal with the ‘structure’ of a single configuration,14,15 forinstance the periodic order of the density profile for crys-tallization, see Fig. 3(a). By contrast, it is not the natureof the density profile that changes across the glass tran-sition, but rather the number of distinct available pro-files.1 There are many distinct states available to the liq-uid, leading to a finite configurational entropy, but onlya subextensive number in the putative thermodynamicglass phase, where Sconf = 0. ‘Glass order’ can thus onlybe revealed by the enumeration of equilibrium accessiblestates, see Fig. 3(b).

A final general question is: how can a purely thermody-namic quantity be useful to understand slow dynamics?After all, the above phenomenological description of theglass transition relies on dynamics, and a connection toconfigurational entropy is not obvious. The first quanti-tative connection arose in 1965, when Adam and Gibbsproposed that the timescale for structural relaxation in-creases exponentially with 1/(TSconf).

17 Quantitatively,the modest decrease of Sconf(T ) in Fig. 2 is sufficient toaccount for the modest increase in the apparent activa-

FIG. 3. (a) Crystallization at the melting temperature Tm

corresponds to the emergence of periodic order in the densityprofile of a single configuration. (b) The glass transition atTK is detected by enumerating equilibrium configurations inconfiguration space C. Glass order is revealed by comparingthe degree of similarity (in practice, the overlap in Eq. (7)) ofamorphous density profiles.

tion energy, and for the large increase of relaxation times.Testing the Adam-Gibbs relation has become a straw

man for a deeper issue:32,58,63,64 how can one (dis)provethe existence of a causal link between the rarefaction ofequilibrium states and slow dynamics? In essence, thephysical idea to be tested is that the driving force behindstructural relaxation for T > TK is the configurationalentropy gained by the system exploring distinct disor-dered states. Slower dynamics then arises when fewerstates are available at lower T , since the system hardlyfinds new places to go. In this view, the two configura-tions in Fig. 1 relax at a much different rate not becausethey are structurally different, but because much fewerequilibrium configurations are accessible to the configu-ration on the right. This is indeed hard to recognize bythe naked eye.

C. Mean-field theory of the glass transition

Despite the diversity of theoretical work related toglass formation, the configurational entropy plays a cen-tral role. This is natural for theories rooted in thermo-

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dynamics and describe an entropy crisis,17,18,65 but the-ories based on a different mechanism must also explainthe observed behavior of Sconf , and the role played by a(possibly avoided) entropy crisis.21,22,46 Finally, theoriesbased on dynamics must explain why a rapidly changingSconf is an irrelevant factor.24,66–68 This makes the con-cept of configurational entropy, a careful understandingof its physical content, and the development of precisenumerical measurements important research goals.

The first theory ‘predicting’ an entropy crisis appearedabout a decade after Kauzmann’s work.30,31 Inspired bylattice polymer studies,69 Gibbs and DiMarzio identifiedthe decrease of Sexc presented by Kauzmann with thereduction of the entropy computed within a set of mean-field approximations. In their lattice model, ‘states’ wereidentified with microscopic configurations, with no needto subtract any vibrational contribution, Sconf ≈ Stot.An approximate statistical mechanics treatment of theirmodel yields Stot → 0 at a finite temperature.

Revisions and extensions of the Gibbs-DiMarzio workabound.48,70,71 Modern studies offer more detailed treat-ments of the polymer chain and refined approxima-tions.72 The entropy may or may not vanish, dependingon the approximations used and the ingredients enter-ing the model.52,73 An entropy crisis is thus not alwayspresent within the Gibbs-DiMarzio line of thought, butone cannot draw general conclusions about the existenceof an entropy crisis in supercooled liquids. Moreover, thedistinction between individual configurations and free-energy minima is generally not considered, which maybe problematic.74 Finally, these works rely heavily onthe polymeric nature of the molecules to make predic-tions whose validity for simpler particle models or molec-ular systems is not guaranteed. These works neverthelesssuggest that the presence of a Kauzmann transition couldwell be system-dependent. This is demonstrated by somespecific colloidal models in which the entropy crisis is in-deed avoided with a finite configurational entropy at zerotemperature.49,51

A coherent mean-field theory of the glass transitionwas recently derived for classical, off-lattice, point parti-cle systems interacting with generic isotropic pair inter-actions.75–79 The ‘mean-field’ nature of the theory stemsfrom the fact that it becomes mathematically exact inthe limit of d → ∞, whereas it amounts to an approxi-mate analytic treatment for physical dimensions d <∞.The nature of the glass transition found in this mean-fieldlimit agrees with results obtained in the past in a varietyof approximate treatments, starting with density func-tional theory of hard spheres,80 replica calculations offully-connected spin glass models,18,81–85 and others.86–88

The fact that distinct models and treatments yieldsimilar results reflects a universal evolution of the free-energy landscape in glassy systems, with results rediscov-ered in a variety of contexts.28,89 The theory reveals theexistence of sharp temperature scales where the topog-raphy of the free-energy landscape changes qualitatively.There exists a first temperature scale, Tonset, above which

0.0 0.2 0.4 0.6 0.80.00

0.01

0.02

0.03

0.04

0.05

0.55 0.60 0.65 0.700.00

0.02

0.04

0.06

s conf

Glass

TK

TmctV(Q)

Q

Tonset

Liquid TsconfTTK Tonset

FIG. 4. Schematic plot of the Franz-Parisi free energy inmean-field theory. Inset: Temperature evolution of the con-figurational entropy.

a single global free energy minimum exists, and belowwhich a large number, N , of free-energy minima appear.This number scales exponentially with the system size,which allows for the definition of an entropy, Σ = lnN ,90

also called complexity. At a second temperature scale,Tmct < Tonset, the partition function becomes dominatedby those multiple free-energy minima. This transitionshares many features with the dynamic transition firstdiscovered in the context of mode-coupling theory.36 Thethird critical temperature is TK < Tmct, below which thenumber of free-energy minima becomes subextensive, re-sulting in a vanishing complexity, Σ(T → TK)→ 0.

An entropy crisis is thus an analytic result in mean-field theory, which provides a clear physical interpreta-tion of the configurational entropy as the logarithm ofthe number of free-energy minima, Sconf ≈ Σ = lnN . AKauzmann transition is exactly realized, and is referredto as a random first order transition (RFOT).

The idea that the existence, number, and organizationof distinct free-energy minima control the glass transi-tion was elegantly captured by an approach developedby Franz and Parisi.91,92 As in Landau theory, they ex-pressed the free-energy, or effective potential V (Q), as afunction of a global order parameter Q. As illustrated inFig. 3(b), the distinction between liquid and glass phasesstems from the degree of similarity of particle configu-rations drawn from the Boltzmann distribution. Let usdefine an overlap, Q, as the degree of similarity of thedensity profiles of two equilibrium configurations, suchthat Q ≈ 0 for uncorrelated profiles (liquid phase), andQ ≈ 1 for similar profiles (glass phase); see Eq. (7) below.

The free-energy V (Q) can be computed analytically formean-field glass models, as shown in Fig. 4. As expected,the global minimum of V (Q) is near Q ≈ 0 for T >TK as there exists so many distinct available states thattwo equilibrium configurations chosen at random haveno similarity. All critical temperatures mentioned above

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have a simple interpretation in this representation. Thefree-energy V (Q) has non-convexity when T < Tonset, itdevelops a secondary minimum when T < Tmct, and thislocal minimum becomes the global one when T reachesTK . In this description, mean-field glass theory sharessimilarities with ordinary first-order transitions.

In the interesting regime, TK < T < Tmct, the glassphase at high Q is metastable with respect to the liquidphase at low Q. The free-energy difference between theliquid and glass phases results from confining the sys-tem within a restricted part of the configuration space.Preventing the system to explore the multiplicity of avail-able free-energy minima entails an entropic loss, preciselygiven by the complexity, TΣ(T ). The temperature evo-lution of the configurational entropy Sconf is thus read-ily visualised and quantified from the Franz-Parisi free-energy as shown in Fig. 4. The inset of Fig. 4 shows thata finite configurational entropy emerges discontinuouslyat Tonset, and vanishes continuously at TK .

The entropy crisis captured by the random first-ordertransition universality class is now validated by exactcalculations performed in the large dimensional limit,d → ∞.79 This confers to RFOT a status similar to vander Waals theory for the liquid-gas transition. With itswell-defined microscopic starting point, mean-field the-ory confirms that the configurational entropy is centralto the understanding of supercooled liquids, and the rig-orous treatment it offers puts phenomenological and ap-proximate ideas introduced earlier by Kauzmann, Gibbs,DiMarzio, Adam, and others on a solid basis. This nowserves as a stepping stone to describe finite dimensionaleffects.93–98

D. Conceptual and technical problems

Physically, the configurational entropy quantifies theexistence of many distinct glassy states that the systemcan access in equilibrium conditions. There are two mainroutes to measure Sconf .

First, one can subtract from the total entropy a con-tribution that comes from small thermal vibrations per-formed in the neighborhood of a given reference config-uration: Sconf(T ) ≈ Stot(T ) − Sglass(T ). In this view,Sglass(T ) should be the entropy of an equilibrium systemthat does not explore distinct states at temperature T .This quantity can be measured straightforwardly in equi-librium for T < TK , whereas some approximations are byconstruction needed to measure Sglass for T > TK .

Experimentally, it is often assumed that Sglass ≈ Sxtal,because it is possible to measure Sxtal in equilibrium us-ing reversible thermal histories.32 This represents a well-defined and physically plausible proxy. It has been testedfor some systems,99–104 and its validity seems to be non-universal.104 We shall introduce in Sec. III E a computa-tional method to determine Sglass that makes no referenceto the crystal.105–107

The second general route to Sconf is to directly enumer-

ate the number of distinct glassy states available to thesystem in equilibrium, N , and use Sconf = lnN . Here,mean-field theory provides a rigorous definition of glassystates as free-energy minima. However, just as for or-dinary phase transitions (e.g. van der Waals theory)local free-energy minima are no longer infinitely long-lived when physical dimension is finite, and states canno longer be defined precisely. Thus, strictly speaking,the complexity that vanishes at TK in mean-field the-ory is not defined in finite dimensional systems. Again,approximations must be performed to measure a phys-ical analog. Two such methods based on the Franz-Parisi free energy91,92 and glassy correlation length,65

are now available, as discussed in Secs. IV and V. Theexistence of an entropy crisis in finite dimension is notdirectly challenged by the approximate nature of theseestimates. To determine whether a Kauzmann transitioncan occur in finite d, one should rather study the effectof finite-dimensional fluctuations within a d-dimensionalfield theory using the Franz-Parisi free-energy as a start-ing point.93–98 There exists no ‘proof’ that the Kauz-mann transition should be destroyed in finite dimensionsas divergent conclusions were obtained using distinct ap-proximate field-theoretical treatments. This is a difficult,but pressing, theoretical question for future work.

A popular alternative is the enumeration of poten-tial energy minima using the potential energy landscape(PEL), which was actually proposed long before the de-velopment of mean-field theory, first by Goldstein,108 andfurther formalized by Stillinger and Weber.109,110 ThePEL approach assumes that an equilibrium supercooledliquid resides very close to a minimum of the potentialenergy, also named inherent structure. Assuming fur-ther that each inherent structure corresponds to a dis-tinct glassy state, the number of inherent structures,NIS, provides a proxy for the configurational entropy,Sconf ≈ lnNIS. This assumption offers precise and simplecomputational methods to estimate the configurationalentropy,46,111,112 discussed below in Sec. III.

The confusion between inherent structures and thefree-energy minima entering the mean-field theory shouldnot be ignored, and explicit examples were proposedto show that it is generally incorrect.74,113 Physically,it is believed that free-energy minima may contain alarge number of inherent structures. The concept of‘metabasins’114 has been empirically introduced to cap-ture this idea, but there is no available method to enu-merate the number of metabasins to obtain a configu-rational entropy. The hard sphere model is a strikingexample of the difference between energy and free energyminima. In large dimensions, hard spheres undergo anentropy crisis, but it does not correspond to a decreaseof the number of inherent structures, which are not evendefined for this potential.

Using the PEL approach, several arguments were givento question the existence of a Kauzmann transition insupercooled liquids. By considering localized excitationsabove inherent structures, Stillinger provided a physical

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argument showing that the PEL approximation to theconfigurational entropy can not vanish at a finite tem-perature.46 The effect of such excitations on the free-energy landscape has not been studied, therefore thisargument does not straightforwardly apply to the ran-dom first order transition itself. In the same vein, Donevet al. directly constructed dense hard disk packingsof a binary mixture model to suggest that NIS cannotyield a vanishing configurational entropy. This againdoes not question the Kauzmann transition of that sys-tem, since it should be demonstrated that the equilib-rium free-energy landscape is sensitive to these inherentstates, whose relevance to the equilibrium supercooledfluid is not established.115 Finally, the ambiguous natureof inherent structures becomes obvious when consideringcolloidal systems composed of a continuous distributionof particle sizes. Starting from a given inherent struc-ture, each permutation of the particle identity providesa different energy minimum and a naive enumeration ofthe configurational entropy116 would contain a divergentmixing entropy contribution, again incorrectly suggestingthe absence of a Kauzmann transition.113 A similar argu-ment was proposed for a binary mixture.117 The problemof the mixing entropy in the PEL approach is consideredfurther, and solved in Sec. III F.

II. COMPUTER SIMULATIONS OF GLASS-FORMINGLIQUIDS

A. Why doing computer simulations to measure theconfigurational entropy?

Let us start with some major steps in computer simula-tions of supercooled liquids, referring to broader reviewsfor a more extensive perspective.118,119 Early computa-tional studies date back to the mid-1980s,120–124 followedby intensive works strongly coupled to the development ofmode-coupling theory during the 1990s.45 The nonequi-librium aging dynamics of glasses,125 along with conceptsof effective temperatures,126–128 rheology129,130 and dy-namical heterogeneities10,131–133 were in the spotlight atthe end of the 20th century. The search for a growingstatic lengthscale,134 linked to a Kauzmann transitionand configurational entropy,111,112 have continuously an-imated the field until today. Over this period spanningabout 3 decades, the numerically-accessible time windowincreased about as many orders of magnitude, mainly dueto improvements in computer hardware. Until 2016, com-puter studies lagged well behind experiments in terms ofequilibrium configurational entropy measurements, butrecent developments in computer algorithms have sud-denly closed the gap with experiments.44 Even better,temperatures below the experimental glass transition arenow numerically accessible in equilibrium conditions,making computer simulations an essential tool for con-figurational entropy studies in supercooled liquids.16,135

As illustrated in Fig. 1, theories for the glass transition

need to make predictions for complex observables that re-flect nontrivial changes in the supercooled liquid, suchas multi-point time correlation functions,136 point-to-set correlations,134,137 non-linear susceptibilities,138,139

as well as properties of the potential and free-energylandscapes.140–142 Most of these quantities are extremelychallenging, or sometimes even impossible, to measurein experiments. Computer simulations are particularlysuitable because they generate equilibrium density pro-files from which any observable can be computed. Ob-taining the same information in experiments is possibleto some extent in colloidal glasses, but still a challengein atomistic or molecular glasses.

Computer simulations take place under perfectly con-trolled conditions, and are therefore easier to interpretthan experiments. All settings are well-defined: micro-scopic model, algorithm for the dynamics, statistical en-semble (isobaric or isochoric conditions), external param-eters, etc. Computer simulations are also very flexible.Since the mean-field theory for the glass transition pro-vides exact predictions for the configurational entropy ininfinite dimensions, it is crucial to understand how finite-dimensional fluctuations affect them. Along with cur-rent efforts that strive to develop renormalization groupapproaches to this problem, numerical simulations giveprecious insights into the effect of dimensionality on thephysics of glass formation. Numerical simulations canbe performed in any physical dimensions, and the ranged = 1 − 12 was explored in that context.143–146 Eventhe space topology can be varied.147,148 One can studythe effect of freezing a subset of particles with arbitrarygeometries by means of computer simulations.137,149–152

The size of the system under study can be tuned andfinite-size scaling analysis can reveal important length-scales for the glass problem.153,154

B. Simple models for supercooled liquids

The features associated with the glass transition, suchas a dramatic dynamical slowdown and dynamic hetero-geneities, are observed in a wide variety of glassy materi-als composed of atoms, molecules, metallic compounds,colloids and polymers. It may be useful to focus on simplemodels exhibiting glassy behavior to understand univer-sal features of the glass transition. We consider classicalpoint-like particles with no internal degrees of freedomthat interact via isotropic pair potentials. These modelsmay not capture all detailed aspects of glass formation,e.g. β-relaxations due to slow intramolecular motion inmolecules, but their configurational entropy can never-theless be measured. The numerical study of simple mod-els is especially relevant in the context of configurationalentropy, since mean-field theory was precisely derived forsuch simple models, which allow direct comparison be-tween theory and simulations.79

We illustrate this paper using results obtained for threesimple glass-formers. The Lennard-Jones (LJ) potential

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was first introduced to model the interaction betweenneutral atoms and molecules. The interaction potentialbetween two particles separated by a distance r reads

vLJ(r) = 4ε

[(σr

)12

−(σr

)6], (1)

where ε and σ set the energy and length scales. Thestiffness of the repulsion in the soft-sphere (SS) po-tential vSS(r) = ε(σ/r)ν can be tuned with ν.44 Thehard-sphere (HS) potential, defined as vHS(r) = ∞ ifr < σ, and vHS(r) = 0 otherwise, models hard-core re-pulsion between particles of diameter σ. This highly-idealized model efficiently captures the glass transitionphenomenology.155,156

The homogeneous supercooled liquid is metastablewith respect to the crystal in the temperature regimewhere the configurational entropy is measured. Design-ing glass-forming models in which crystallization is frus-trated, and defining strict protocols to detect crystalliza-tion is crucial.44 Mixtures of different species are goodexperimental glass-formers: colloidal glasses are madeof polydisperse suspensions,157 and metallic glasses arealloys of atoms with different sizes.158 Inspired by ex-periments, numerical models use particles of differentspecies which differ by their size σ or interaction ε. TheKob-Andersen (KA) model is a bidisperse mixture with80% larger particles and 20% smaller particles, inter-acting via the LJ potential with adjusted parameters σand ε to describe amorphous Ni80P20 metallic alloys.45

Many numerical models with good glass-forming abilityhave been developed,44,45,122,124,159,160 although develop-ment in computational power now leads to crystallizationfor some of those models.44,161–163 Thus, developing newmodels robust against crystallization is an important re-search goal.

While the situation may seem satisfactory to theorists,numerical glass-formers are probably too simplistic formany experimentalists. Future developments should aimat designing minimal models for more complex particlesand molecules, in order to also close this conceptual gap.

C. Molecular dynamics simulations

The two main classical methods used to simulate theabove models are Monte Carlo (MC) and Molecular Dy-namics (MD) simulations.164,165 Quantum effects, par-tially included in ab initio simulations, are irrelevant inthe present context.

The course of a numerical simulation is very similarto an experiment. A sample consisting of N particles isprepared and equilibrated (using either MC or MD dy-namics) at the desired state point, until its properties nolonger change with time. After equilibration is achieved,the measurement run is performed. Common problemsare just as in experiments: the sample is not equilibratedcorrectly, the measurement is too short, the sample un-

0.5 1.0 1.5 2.0 2.5 3.0 3.510-1

101

103

105

107

109

1011 Ethanol Propylene carbonate Propylene glycol MD MD + SWAP

/ 0

T0/T

Tg

Extra

polatio

n

FIG. 5. Isobaric relaxation time of supercooled liquids as afunction of the inverse temperature for ethanol,168 propylenecarbonate,169 and propylene glycol,170 as well as the standardmolecular dynamics (open squares) and its combination withthe swap Monte-Carlo algorithm (open circles)171 for three-dimensional polydisperse soft spheres44. We renormalize axisusing the onset of glassy dynamics, (τ0 = 10−10s in experi-ments), and the corresponding T0. We fit MD results witha parabolic fit, which provides a reasonable estimate of Tg

for this system (vertical dashed line). The SWAP algorithm(open circles) can equilibrate the numerical model well belowTg.

dergoes an irreversible change during the measurement,etc.

A noticeable difference between computer and experi-mental supercooled liquid samples is their size. Numer-ical studies of the configurational entropy are limited toaround 104 particles, to be compared to around 1023

atoms or molecules in experimental samples. Periodicboundary conditions are applied to the simulation box inorder to avoid important boundary effects, and simulatethe bulk behavior of ‘infinitely’ large samples. Length-scales larger than the system size are numerically inac-cessible. Up to now, this limit was not really problematicsince all important lengthscales associated with the glasstransition are not growing to impossibly large values.

The difference between MD and MC is the way thesystem explores phase space. The molecular dynamicsmethod simulates the physical motion of N interactingparticles. As an input, one defines a density profile rN0 ,particle velocities vN0 , and an interaction potential be-tween particles. The method solves the classical equa-tions of motion step by step using a finite differenceapproach. As an output, one obtains physical particletrajectories (rN (t),vN (t)) from which thermodynamicquantities can be computed, see Sec. II E. By construc-tion, the trajectories sample the microcanonical ensem-ble. Other ensembles can be simulated by adding degreesof freedom which simulate baths which generate equilib-rium fluctuations in any statistical ensemble.164,166,167

Molecular dynamics mimics the physical motion of par-

8

ticles, very much as it takes place in experiments, butcomputers are much less efficient than Nature. Long MDsimulations of a simple glass model (about a month) canonly track the first 4-5 orders of magnitude of dynamicalslowdown in supercooled liquids approaching the glasstransition, to be compared to 12-13 orders of magni-tude in real molecular liquids. In Fig. 5, we show re-laxation time τα of some molecular liquids of variousfragilities,168–170,172 and MD simulations of polydispersesoft spheres under isobaric condition. The temperaturerange accessible with MD simulations is far from the ex-perimentally relevant regime, and stops well before Tg isreached (estimated from a parabolic fit173).

D. Beating the timescale problem: Monte Carlosimulations

Monte Carlo simulations aim at efficiently sam-pling the configurational space with Boltzmannstatistics.174,175 A stochastic Markov process is gener-ated in which a given configuration rN is visited witha probability proportional to the Boltzmann factor

e−βU(rN ), where β = 1/T and U are the inverse of thetemperature and the potential energy, respectively. Themethod only considers configurational, and not kinetic,degrees of freedom, and is suitable for configurationalentropy measurements.

A Markov process is defined by the transition proba-

bility T (rN → r′N

) to go from configurations rN to r′N

.To sample configurations with a probability P (rN ) givenby the Boltzmann factor, the global balance conditionshould be verified∑

r′N

P (rN )T (rN → r′N

) =∑r′N

P (r′N

)T (r′N → rN ) .

(2)We consider a stronger condition and impose the equality

in Eq. (2) to be valid for each new state r′N

. This detailedbalance condition reads

T (rN → r′N

)

T (r′N → rN )=P (r′

N)

P (rN )= exp

[−β(U(r′

N)− U(rN )

)].

(3)

In practice, T (rN → r′N

) = α(rN → r′N

) × acc(rN →r′N

), where α and acc are the probabilities to proposea trial move and to accept it, respectively. We con-sider a symmetric matrix α for trials such that thematrix acc obeys the same equation as T in Eq. (3).If trial moves are accepted with probability acc(rN →r′N

) = min

1, exp[−β(U(r′

N)− U(rN ))

](Metropo-

lis criterion),174 the configurations are drawn from thecanonical distribution at equilibrium at the desired tem-perature.

Contrary to MD simulations, dynamics in a MonteCarlo simulation is not physical, since it results froma random exploration of configurational space. This is

actually good news, since there is a considerable free-dom in the choice of trial moves, opening the possibilityto beat the numerical timescale problem illustrated inFig. 5. The choice of trial move depends on the pur-pose of the numerical simulation. A standard trial moveconsists in selecting a particle at random and slightlydisplacing it. For small steps, the dynamics obviouslyresembles the (very physical) Brownian dynamics.176

If instead efficient equilibration is targeted, more ef-ficient but less physical trial moves should be preferred.In the SWAP algorithm,44,160,177–181 trial moves combinestandard displacement moves and attempts to swap thediameters of two randomly chosen particles. Since thetrial moves satisfy detailed balance in Eq. (3), the SWAPalgorithm by construction generates equilibrium config-urations from the canonical distribution.

Using continuously polydisperse samples, this algo-rithm outperforms standard MC or MD, as equilibriumliquids can be generated at temperatures below the ex-perimental glass transition.44 In Fig. 5, we show the equi-librium relaxation time τα of an Hybrid scheme of MDand SWAP MC developed recently in Ref. 171. The re-laxation dynamics with this scheme is significantly fasterthan with standard MD, which makes equilibration ofthe system possible even below the estimated experimen-tal glass transition temperature Tg. Accessing numeri-cally these low temperatures is crucial to compare simu-lations and experiments. From a theoretical perspective,the concept of metastable state applies far better at lowtemperatures. In particular, numerical estimates for theconfigurational entropy become more meaningful in theseextreme temperature conditions.

To conclude, Monte Carlo simulations are very relevantin the present context, because their flexibility allows usto compute and compare different estimates for the con-figurational entropy of supercooled liquids.33,135 Thesemeasurements are done under perfectly controlled condi-tions, in a temperature regime relevant to experimentalworks, and even at lower temperatures.16

E. From microscopic configurations to observables

The output of a numerical simulation consists in a se-ries of equilibrium configurations. To measure an observ-able numerically, one must first express it as a functionof the positions of the particles.

Static quantities describing the structure of the liquidare easily computed.182 In particular, the density field isgiven by

ρ(r) =

N∑i=1

δ(r− ri) . (4)

Two-point static density correlation functions such as

9

the pair correlation function

g(r) =1

ρN

⟨∑i 6=j

δ(r + ri − rj)

⟩, (5)

where ρ = N/V is the number density and the bracketindicate an ensemble average at thermal equilibrium, andthe structure factor

S(k) =1

N〈ρkρ−k〉 , (6)

are evaluated, where ρk =∑Ni=1 e

ik·ri is the Fouriertransform of the density field. Even if these quantitiesare not relevant to describe the dynamical slowdown ofthe supercooled liquid (see Fig. 1), they are convenient todetect instabilities of the homogeneous fluid (crystalliza-tion, fractionation). Thermodynamic quantities (such asenergy, pressure), and their fluctuations (e.g. specificheats, compressibility), related to macroscopic responsefunctions, can be computed directly from the two-pointstructure of the liquid.

As presented in Sec. I C, the relevant order parameterfor the glass transition is the overlap Q that quantifiesthe similarity of equilibrium density profiles. This quan-tity compares the coarse-grained density profiles of twoconfigurations, to remove the effect of short-time thermalvibrations. Numerically, the following definition is veryefficient

Q =1

N

∑i,j

θ(a− |r1i − r2j |) , (7)

where r1 and r2 are the positions of particles in distinctconfigurations, and θ(x) is the Heaviside step function.The parameter a is usually a small fraction (typicall 0.2−0.3) of the particle diameter. The overlap is by definitionequal to 1 for two identical configurations, and becomesclose to zero, or more precisely 4πa3ρ/3, for uncorrelatedliquids.

III. CONFIGURATIONAL ENTROPY BY ESTIMATINGA ‘GLASS’ ENTROPY

A. General strategy

The configurational entropy enumerates the number ofdistinct glassy states. One possible strategy to achievethis enumeration is to first estimate the total number ofconfigurations, or phase space volume, Ntot. If one canthen measure the number of configurations belonging tothe same glass state, Nglass, the number of glass statesNconf can be deduced, Nconf = Ntot/Nglass. Taking thelogarithm of Nconf yields the configurational entropy

Sconf = Stot − Sglass. (8)

Whereas the measurement of the total entropy Stot isstraightforward, the art of measuring the configurational

entropy lies in the quality of the unavoidable approxima-tion made to determine Sglass. Recall that experimental-ists typically use Sglass ≈ Sxtal. This is not a practicalmethod for simulations, because numerical models whichcan crystallize are generally very poor glass-formers. Inthis section, we describe several possible strategies tomeasure Sglass which do not rely on the knowledge ofthe crystal state, and present their limitations.

Let us now introduce our notations for the entropycalculations. We consider a M -component system in thecanonical ensemble in d spatial dimensions, with N , V ,and T = 1/β the number of particles, volume, and tem-perature, respectively. We fix the Boltzmann constantto unity. We take M = N to treat continuously polydis-perse systems. The concentration of the m-th species isXm = Nm/N , where Nm is the number of particles of

the m-th species (N =∑Mm=1Nm). A point in position

space is denoted as rN = (r1, r2, · · · , rN ). For simplic-ity, we consider equal masses, irrespective of the species,which we set to unity.

For this system, the following partition function in thecanonical ensemble is conventionally used183

Z =Λ−Nd

ΠMm=1Nm!

∫V

drNe−βU(rN ), (9)

where Λ and U(rN ) are respectively the de Broglie ther-mal wavelength and the potential energy. The only fluc-tuating variables are the configurational degrees of free-dom rN , since momenta are already traced out in Eq. (9).

B. Total entropy Stot

An absolute estimate of the total entropy at a givenstate point can be obtained by performing a thermody-namic integration from a reference point where the en-tropy is exactly known,107,111,184,185 typically the idealgas at ρ → 0 or β → 0. This approach works for allstate points which can be studied in equilibrium condi-tions, and are connected to the reference point by a seriesof equilibrium state points. This is usually doable alsoin most experiments. However, this constraint preventsa direct thermodynamic study of vapor-deposited ultra-stable glasses produced directly at very low temperature.In practice, to perform the thermodynamic integrationand access Stot, we need to distinguish between contin-uous ‘soft’ interaction potentials, such as the Lennard-Jones potential, and discontinuous ‘hard’ potentials, asin the hard sphere model:

Stot = Sid + βEpot(β)−∫ β

0

dβ′Epot(β′) (soft),(10)

Stot = Sid −N∫ ρ

0

dρ′(p(ρ′)− 1)

ρ′(hard), (11)

where Sid, Epot and p = P/(ρT ) are the ideal gas entropy,the averaged potential energy, and the reduced pressure,

10

respectively. The ideal gas entropy Sid can be written as

Sid = N(d+ 2)

2−N ln ρ−N ln Λd + S

(M)mix , (12)

where S(M)mix is the mixing entropy of the ideal gas,

S(M)mix = ln

(N !

ΠMm=1Nm!

). (13)

When M is finite and Nm 1, Stirling’s approximationcan be used, lnNm! ' Nm lnNm − Nm, and Eq. (13)

reduces to S(M)mix /N = −

∑Mm=1Xm lnXm.

As a representative example, Fig. 6(a) shows the tem-perature dependence of the numerically measured totalentropy in the Kob-Andersen model.45 It decreases mono-tonically with decreasing the temperature.

C. Inherent structures as glass states

The first strategy that we describe to identify glassstates and estimate Sglass is based on the potential en-ergy landscape (PEL).108–110,114,142 The central idea isto assume that each configuration can be decomposed as

rN = rNIS + ∆rN , (14)

where rNIS is the position of the ‘inherent structure’, i.e.the potential energy minimum closest to the original con-figuration. This trivial decomposition becomes meaning-ful if one makes the central assumption that different in-herent structures represent distinct glass states. It followsimmediately that the glass entropy, Sglass, then quantifiesthe size of the basin of attraction of inherent structures.

Assuming that temperature is low, ∆rN can be treatedin the harmonic approximation. Expanding the potentialenergy U(rN ) around the inherent structure rNIS, one gets

Uharm(rN ) ' U(rNIS) +1

2

∑i,j

∂2U(rNIS)

∂ri∂rj∆ri∆rj . (15)

Injecting this expansion in the partition function, Eq. (9),gives

Zharm = e−βU(rNIS)ΠNda=1(β~ωa)−1, (16)

where ω2a are the eigenvalues of the Hessian matrix. We

also considered that each inherent structure is realizedΠMm=1Nm! times in the phase space volume, as permut-

ing the particles within each specie leaves the config-uration unchanged (see related argument in mean-fieldtheory)186–188. This factorial term cancels out with thedenominator in Eq. (9).

We have implicitely assumed that exchanging two dis-tinct particles produces a different inherent structure,116

which is consistent with the identification of energy min-ima as glass states. Physically, this implies that thereis no mixing entropy associated with inherent structures.

0.3 0.4 0.5 0.6 0.7 0.8-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Kob-Andersen

Entro

py

T

Stot /N

Sharmglass /N

Sanhglass /N

SFLglass /N

(a )

0.0 0.2 0.4 0.6 0.8 1.0-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

E anh/NT

Simulation data Fitting

(b )

Kob-Andersen

10-3 10-2 10-1 100 101 102 103 104 105 106 10710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Kob-Andersen

T

T = 0.4003 T = 0.4238 T = 0.4374 T = 0.4500 T = 0.4700 T = 0.5000

min

Einstein solid

(c )

Debye-Waller factor

FIG. 6. (a) Total entropy Stot and various estimates of theglass entropy Sglass: harmonic Sharm

glass , with anharmonic cor-

rection Sanhglass, and the Frenkel-Ladd entropy SFL

glass. (b) An-harmonic energy Eanh from simulations (black points) andpolynomial fitting to third order (red line). (c) Constrainedmean-squared displacement in the Frenkel-Ladd method. Thedashed horizontal line correspond to the Debye-Waller factorindependently measured in the bulk dynamics at the lowesttemperature. The vertical arrow indicates αmin.

11

As realized recently,113 this assumption produces unphys-ical results for systems with continuous polydispersity.

Averaging over independent inherent structures (de-noted by 〈(· · · )〉IS), the free energy of the harmonic solidis obtained,

−βFharm = 〈lnZharm〉IS (17)

= −β⟨U(rNIS)

⟩IS−

⟨Nd∑a=1

ln(β~ωa)

⟩IS

.

The internal energy of the harmonic solid is

Eharm =Nd

2T +

⟨U(rNIS)

⟩IS

+Nd

2T, (18)

where the first and last terms are the kinetic and har-monic potential energies. Using Eqs. (17) and (18), wefinally obtain the glass entropy in the harmonic approx-imation:

Sharmglass = β (Eharm − Fharm) ,

=

⟨Nd∑a=1

1− ln(β~ωa)

⟩IS

. (19)

In practice, this method requires the production ofa large number of independent inherent structures, ob-tained by performing energy minimizations from equilib-rium configurations using widespread algorithms such asthe steepest decent or conjugate gradient methods,189 orFIRE.190 The energy U(rIS) is measured, and the Hes-sian matrix is diagonalized to get the eigenvalues ω2

a. Us-ing Eq. (19), these measurements then provide the glassentropy Sharm

glass . The numerical results for Sharmglass (T ) in

the Kob-Andersen model are shown in Fig. 6(a). Thedifference Stot − Sharm

glass is a widely used practical defini-tion of the configurational entropy in computer simula-tions.111,112,152,184,191,192

D. Anharmonicity

Although presumably not the biggest issue, it is possi-ble to relax the harmonic assumption in the above proce-dure.142 First, the anharmonic energy, Eanh, is obtainedby subtracting the harmonic energy in Eq. (18) from thetotal one,

Eanh = Epot −⟨U(rNIS)

⟩IS− Nd

2T. (20)

The anharmonic contribution to the entropy can then beestimated as

Sanh =

∫ T

0

dT ′

T ′∂Eanh(T ′)

∂T ′, (21)

which requires a low-temperature extrapolation of themeasured Eanh(T ). This can be done using an empiri-cal polynomial fitting, Eanh(T ) =

∑k≥2 akT

k, where the

sum starts at k = 2 to ensure a vanishing anharmonicspecific heat at T = 0. By substituting this expansion inEq. (21), we obtain

Sanh(T ) =∑k≥2

k

k − 1akT

k−1. (22)

We show the numerically measured Eanh for theKob-Andersen model, along with its polynomial fit inFig. 6(b). The non-trivial behavior of Eanh suggests thatthe harmonic description overestimates phase space atlow T , but underestimates it at high T , a trend widelyobserved across other fragile glass-formers.142,193 The re-sulting Sanh using Eq. (22) is thus negative and is of theorder of Sanh/N ≈ −0.1, which is a small but measur-able correction to Sconf . As a result, the improved glassentropy Sanh

glass = Sharmglass +Sanh is slightly smaller than the

harmonic estimate, as shown in Fig. 6(a).

E. Glass entropy without inherent structures

The identification of inherent structures with glassystates is a strong assumption which can be explicitlyproven wrong in some model systems.74,113,194 Moreover,inherent structures cannot be defined in the hard spheremodel, which is an important theoretical model to studythe glass transition.

A more direct route to a glass entropy which automat-ically includes all anharmonic contributions and can beused for hard spheres is obtained by using the followingdecomposition, 105–107,135,184,194–196

rN = rNref + δrN , (23)

where rNref is a reference equilibrium configuration. Thefirst difference with Eq. (14) is that inherent structuresdo not appear, since deviations are now measured froma given equilibrium configuration.

The second difference is the strategy to estimate thesize of the basin surrounding rNref , which makes use of aconstrained thermodynamics integration about the fluc-tuating variables δrN . The potential energy of the sys-tem is βU(rN ), is augmented by a harmonic potential toconstrain δrN to remain small, leading to

βUα(rN , rNref) = βU(rN ) + α

N∑i=1

|ri − rref,i|2. (24)

We consider the statistical mechanics of a given basin,specified by rNref , under the harmonic constraint. Thepartition function and the corresponding statistical aver-age are

Zα = Λ−Nd∫V

drNe−βUα(rN ,rNref ), (25)

〈(· · · )〉Tα =

∫V

drN (· · · )e−βUα(rN ,rNref )∫V

drNe−βUα(rN ,rNref ). (26)

12

Note that the factorial term ΠMm=1Nm! in Eq. (25) is

treated as in Eq. (16) within the PEL approach. Weconsider the entropy of a constrained system as Sα =

β(Eα − Fα), where βEα = Nd2 + β

⟨Uα(rN , rNref)

⟩Tα

andβFα = − lnZα are the internal energy and free energy,respectively.

In the putative glass phase, the system remains closeto the reference configuration for any value of α, includ-ing α = 0. For the liquid, this is true only for timessmaller than the structural relaxation time. For α smallbut finite, however, the system must remain close to thereference configuration and explore the basin whose sizewe wish to estimate. We therefore define the glass en-tropy in the Frenkel-Ladd method as105

SFLglass = lim

αmin→0Sαmin

, (27)

where (· · · ) represents an average over the reference con-figuration. The limit in Eq. (27) is a central approx-imation in this method, which is directly related toconceptual problems summarised in Sec. I D. Becausemetastable glassy states are not infinitely long-lived infinite dimensions, a finite value of α is needed to preventan ergodic exploration of the configuration space, and thelimit in Eq. (27) is difficult to take in practice. The choiceof αmin amounts to defining ‘by hand’ the glassy state asthe configurations that can be reached at equilibrium fora spring constant αmin.

The practical details are as follows. At very large α (=αmax), the entropy is known exactly because the secondterm in the right hand side of Eq. (24) is dominant. Theentropy of the system is described by the Einstein solid,

Sαmax=Nd

2−N ln Λd − Nd

2ln(αmax

π

). (28)

By performing a thermodynamic integration from αmax,one gets Sαmin

, and thus SFLglass from Eq. (27)

SFLglass = Sαmax

+N limαmin→0

∫ αmax

αmin

dα∆Tα , (29)

where ∆Tα is defined by

∆Tα =

1

N

⟨N∑i=1

|ri − rref,i|2⟩T

α

. (30)

To perform the integration and take the limit αmin → 0in Eq. (27), we write:

limαmin→0

∫ αmax

αmin

dα∆Tα ' αmin∆T

αmin+

∫ αmax

αmin

dα∆Tα . (31)

The practical choice for αmax is simple, as it is sufficientthat it lies deep inside the Einstein solid regime where∆Tα = d/(2α) is satisfied. For αmin, a more careful look

at the simulation results is needed.In Fig. 6(c), we show ∆T

α for the Kob-Andersen modelat low temperature. The Einstein solid limit is satisfied

FIG. 7. Mixing entropy problem in continuously polydisperseglasses. Should one treat these 3 configurations as three dis-tinct glass states, or only two by grouping (a) and (b) to-gether? In Sec. III G, a computational measurement is de-scribed that measures the correct answer, instead of guessingit.

for large α, and we can fix αmax = 107. When α de-creases, deviations from Einstein solid behavior are ob-served, and a plateau emerges. Decreasing α further,the harmonic constraint for ∆T

α becomes too weak andthe glass metastability is not sufficient to prevent thesystem from diffusing away from the reference configu-ration, which translates into an upturn of ∆T

α at smallα. It is instructive to compare the plateau level withthe Debye-Waller factor measured from the bulk dynam-ics,163 indicated by a dashed line. This comparison showsthat αmin ≈ 2 is a good compromise: it is in the middleof the plateau, and corresponds to vibrations compara-ble to the ones observed in the bulk. Using this value forαmin, we obtain the Frenkel-Ladd glass entropy shown inFig. 6(a). We observe that SFL

glass is smaller than Sharmglass ,

and becomes comparable to the anharmonic estimate us-ing inherent structures, Sanh

glass, as temperature decreases,confirming that anharmonicities are automatically cap-tured by the Frenkel-Ladd method.194

We show the resulting Sconf = Stot − SFLglass in Fig. 2.

Comparing with experimental data, the temperaturerange where Sconf can be measured is limited since theSWAP algorithm is not efficient for binary mixtures suchas the Kob-Andersen model.191 Nevertheless an extrap-olation to lower temperature suggests that Sconf/N mayvanish at a finite TK.33

F. Mixing entropy in the glass state

Using multi-component mixtures is essential to studysupercooled liquids and glasses for spherical particle sys-tems, as exemplified by metallic158 and colloidal157,197

glasses. This is also true for most computer simulations,since monocomponent systems crystallize too easily, ex-cept for large spatial dimensions146 or exotic mean-fieldlike model systems.198 For such multi-components sys-tems, a mixing entropy term appears in the total en-tropy, see Eq. (12), with no analog in the glass entropy,see Eqs. (19) and (29). Physically, this is because we de-cided to treat two configurations where distinct particles

13

had been exchanged as two distinct glass states.For typical binary mixtures studied in computer sim-

ulations, the mixing entropy is about as large as theconfigurational entropy itself over the range accessibleto molecular dynamics simulations.107,111 Therefore, ne-glecting the mixing entropy can change the configura-tional entropy by about 100%, which in turn producesa similar uncertainty on the estimate of the Kauzmanntemperature. Properly dealing with the mixing entropyis thus mandatory.113

For discrete mixtures, such as binary and ternarymixtures, with large size asymmetries, the abovetreatment produces an accurate determination ofSconf .

111,112,152,184,191,192 However, for systems with acontinuous distribution of particle sizes, such as col-loidal particles and several computer models, this leadsto unphysical results. In the liquid, the mixing en-tropy is formally divergent, since for M = N it becomes

S(M=N)mix /N = (lnN !)/N ' lnN −1→∞.199,200 Because

the glass entropy remains finite in conventional treat-ments, the configurational entropy also diverges, leadingto the conclusion that no entropy crisis can take place insystems with continuous polydispersity.113,201 A similarargument was proposed by Donev et al to suggest thatan entropy crisis does not exist in binary mixtures.117

In fact, the above treatments do not accurately quan-tify the mixing entropy contribution in the glass en-tropy. This can be easily seen by considering a con-tinuously polydisperse material with a very narrow sizedistribution, which should physically behave as a mono-component system, but has a mathematically divergentmixing entropy. In addtion to this trivial example,the fundamental problem is illustrated in Fig. 7, whichsketches three configurations which differ by the ex-change of a single pair of particles. The inherent struc-ture and the standard Frenkel-Ladd methods treat thosethree configurations as distinct. Physically, configura-tions (a) and (b) should instead be considered as the sameglass state, since they differ by the exchange of two par-ticles with nearly identical diameters. The glass entropyshould contain some amount of mixing entropy, takinginto account those particle permutations that leave theglass state unaffected.113

Recently, two methods were proposed to estimate theglass mixing entropy. The first method provides a simpleapproximation to the glass mixing entropy using infor-mation about the potential energy landscape.113 We de-scribe the second one in the next subsection, which leadsto a direct determination of the glass mixing entropy us-ing a generalized Frenkel-Ladd approach.33

G. Generalized Frenkel-Ladd method to measure the glassmixing entropy

A proper resolution to the problematic glass mixingentropy is to directly measure the amount of particlepermutations allowed by thermal fluctuations, instead of

making an arbitrary decision.33 Technically, one needsto include particle permutations in the statistical me-chanics treatment of the system. In addition to the posi-tions, we introduce the particle diameters, represented asΣN = σ1, σ2, · · · , σN. Let π denote a permutation ofΣN , and ΣNπ represents the resulting sequence. There ex-ist N ! such permutations. We define a reference sequenceΣNπ∗ = (σ1, σ2, σ3, · · · , σN ). The potential energy now de-pends on both positions and diameters, U(ΣNπ , r

N ). Forsimplicity, we write U(rN ) = U(ΣNπ∗ , r

N ) for the refer-ence ΣNπ∗ .

Including particle diameters as additional degrees offreedoms, the partition function reads

Z =1

N !

∑π

Λ−Nd

ΠMm=1Nm!

∫V

drNe−βU(ΣNπ ,rN ). (32)

This partition function is the correct starting point tocompute the configurational entropy in multi-componentsystems, including continuous polydispersity. The re-sulting method is a straightforward generalization of theFrenkel-Ladd method.105

As before, we introduce a reference configuration anda harmonic constraint,

βUα(ΣNπ , rN , rNref) = βU(ΣNπ , r

N ) + α

N∑i=1

|ri − rref,i|2,

(33)where rNref is a reference equilibrium configuration.

For the unconstrained system with α = 0, the par-tition function in Eq. (32) reduces to the conventionalpartition function in Eq. (9), because diameter permuta-tions can be compensated by the configurational integral.Therefore, the computation of Stot is not altered by theintroduction of the permutations. For the glass statewith α > 0, the partition function in Eq. (32) and thecorresponding statistical average become

Zα =1

N !

∑π

N !Λ−Nd

ΠMm=1Nm!

∫V

drNe−βUα(ΣNπ ,rN ,rNref ).(34)

〈(· · · )〉T,Sα =

∑π

∫V

drN (· · · )e−βUα(ΣNπ ,rN ,rNref )∑

π

∫V

drNe−βUα(ΣNπ ,rN ,rNref )

, (35)

We add a factor N ! in the numerator of Eq. (34), becausethere exist N ! configurations defined by the permutationsof the particle identities of the reference configurationrNref . More crucially, due to the presence of rNref , the par-tition function in Eq. (34) is not identical to the one inEq. (25).

Following the same steps as before we get the glassentropy by a generalized Frenkel-Ladd method, definedas

SGFLglass = Sαmax

+N limαmin→0

∫ αmax

αmin

dα∆T,Sα

+S(M)mix − Smix(rNref , β), (36)

14

with

∆T,Sα =

1

N

⟨N∑i=1

|ri − rref,i|2⟩T,S

α

, (37)

and Smix(rNref , β) is obtained as

Smix(rNref , β) = − ln

(1

N !

∑π

e−β(U(ΣNπ ,rNref )−U(rNref ))

).

(38)Note that in Eq. (36), the mean-squared displacement∆T,Sα is evaluated by simulations where both positions

and diameters fluctuate, and we expect ∆T,Sα ≥ ∆T

α .Practically, ∆T,S

α is computed by Monte Carlo simula-tions including standard translational displacements anddiameter swaps. In addition to this, SGFL

glass in Eq. (36)

contains another non-trivial contribution, S(M)mix − Smix,

which requires Monte Carlo simulations of the diameterswaps for a fixed rNref . In practice the entropy in Eq. (38)is evaluated by a thermodynamic integration.33

For mixtures with large size asymmetry such as theKob-Andersen model, particle permutations of un-likeparticles rarely happen,191 and the generalized Frenkel-

Ladd method yields Smix = S(M)mix and ∆T,S

α = ∆Tα , so

that Eq. (36) reduces to the conventional Frenkel-Laddmethod in Eq. (29). On the other hand, for continu-ously polydisperse systems or mixtures with small size

asymmetry, we expect Smix/N < S(M)mix /N → ∞ and

∆T,Sα > ∆T

α . In the limit case of a very narrow con-tinuous distribution, we would have Smix/N = 0 and∆T,Sα = ∆T

α , and we automatically get back to the treat-ment of a mono-component material.

We finally obtain the configurational entropy asSconf = Stot − SGFL

glass, which finally resolves the paradoxraised by the mixing entropy in conventional schemes.For polydisperse systems, both the total entropy and theglass entropy in Eq. (36) contain the diverging mixing en-tropy term, which thus cancel each from the final expres-sion of the configurational entropy. Instead, the physical

mixing entropy contribution is quantified by Smix(rNref , β)in Eq. (38), which is finite, and whose value depends onthe detailed particle size distribution of the system.

In Fig. 8, we show the measured Smix(rNref , β) forthree representative glass-forming models. For the Kob-Andersen binary mixture, the combinatorial mixing en-

tropy, S(M=2)mix /N ≈ 0.5,111,184 is found, whereas for con-

tinuously polydisperse soft44 and hard spheres43 withpolydispersity≈ 23%, a finite value of the mixing entropyis measured, with a non-trivial temperature dependence.The data also directly confirms that the mixing entropycannot be used to disprove the existence of a Kauzmanntransition.117

Figure 2 shows the final result, Sconf = Stot−SGFLglass, for

polydisperse hard and soft spheres along isochoric33 andisobaric paths (in preparation), in d = 216 and 3. Forthe hard sphere model, we use the inverse of the reduced

0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

SSHSKA

S mix/N

T/TmctFIG. 8. Mixing entropy Smix/N as a function of the normal-ized temperature T/Tmct for polydisperse soft spheres (SS),hard spheres (HS), and the Kob-Andersen model (KA). Thedashed-line corresponds to the combinatorial mixing entropyfor the KA mixture.

pressure, 1/p = ρT/P , as the analog of the tempera-ture. Thanks to the massive performance of the SWAPalgorithm for these models, we can measure a reductionof the configurational entropy comparable to experimen-tal molecular liquids, and even access values measuredin vapor deposited ultrastable glasses.202 Therefore, thesimulation results presented here, together with experi-mental ones, offer the most complete and persuasive dataset for existence of the Kauzmann transition at a finitetemperature in d = 3 and at zero temperature in d = 2.

IV. CONFIGURATIONAL ENTROPY FROM FREEENERGY LANDSCAPE

A. Franz-Parisi Landau free energy

The mean-field theory of the glass transition intro-duced in Sec. I C suggests a well-defined route to theconfigurational entropy,203 based on free energy measure-ments of a Landau free energy V (Q), expressed as a func-tion of the overlap Q between pairs of randomly chosenequilibrium configurations.91,92 A practical definition ofthe overlap was given in Eq. (7). The introduction of theappropriate global order parameter to detect the glasstransition driven by an entropy crisis is the first key step.

The second key point is the assumption that V (Q) con-tains, for finite dimensional systems, the relevant infor-mation about the configurational entropy. As illustratedin Fig. 4, mean-field theory suggests that the glass phaseat large Q, for TK < T < Tmct, is metastable with re-spect to the equilibrium liquid phase at small Q, with afree-energy difference between the two phases that is con-trolled by the configurational entropy. To measure thisconfigurational entropy, one should first demonstrate the

15

existence of the glass metastability, and use it to inferSconf as a free energy difference between liquid and glassphases.

The computational tools to study V (Q) and metasta-bility are not specific to the glass problem, but canbe drawn from computer studies of ordinary first-orderphase transitions.179 To analyze the overlap and its fluc-tuations, we introduce a reference equilibrium configura-tion rNref . We then define the overlap Qref = Q(rN , rNref)between the studied system rN and the reference configu-ration, and introduce a field, ε, conjugate to the overlap,

Uε(rN , rNref) = U(rN )− εNQ(rN , rNref), (39)

where U is the potential energy of the unconstrained bulksystem (ε = 0). The corresponding statistical mechanicsand average become

Zε = Λ−Nd∫V

drNe−βUε(rN ,rNref ), (40)

〈(· · · )〉ε =

∫V

drN (· · · )e−βUε(rN ,rNref )∫V

drNe−βUε(rN ,rNref )

, (41)

and the related Helmholtz free energy is obtained as

− βFε = lnZε, (42)

where the overline denotes an average over independentreference configurations. All thermodynamic quantitiescan then be deduced from Fε, such as the average overlap〈Q〉ε = −(1/N)∂Fε/∂ε.

Following the spirit of the Landau free energy,15 weexpress the free energy as a function of the order param-eter Q, instead of ε. The Franz-Parisi free energy V (Q)is obtained by a Legendre transform of Fε,

V (Q) =1

N

(minεFε + εNQ − F0

), (43)

where F0 = −β−1 lnZ0 is the free energy of the uncon-strained system, which simply ensures that V (Q) = 0for the equilibrium liquid phase at small Q. Followingstandard computational approaches for free-energy cal-culations,179 V (Q) is directly obtained by probing theequilibrium fluctuations of the overlap,

V (Q) = − TN

lnΛ−Nd

Z0

∫V

drNe−βU(rN )δ(Q−Qref),

= − TN

lnP (Q), (44)

where P (Q) = 〈δ(Q − Qref)〉 is the probability distribu-tion of the overlap function for the unconstrained bulksystem.

This method naturally solves issues about the mixingentropy.113 As captured in Eq. (43), this constructiontreats free energy differences, with no need to define ab-solute values for the entropy. The combinatorial terms inEq. (41) are therefore not included since they eventuallycancel out. Additionally, the constraint applied to the

system acts only on the value of the overlap Q. Sinceparticle permutations do not affect the value of the over-lap, see Eq. (7), particle permutations can occur both inthe liquid, near Qliq, and in the glass, near Qglass, witha probablity controlled by thermal fluctuations.

In finite dimensions, the secondary minimum in V (Q)obtained in the mean-field limit (see Fig. 4) cannot ex-ist, as the free energy must be convex, for stabilityreasons.204 At best, V (Q) should develop a small non-convexity for finite system sizes, and a linear part forlarger systems, as for any first-order phase transition. Inthe presence of a finite field ε, a genuine first-order liquid-to-glass transition is predicted,91,92 where 〈Q〉ε jumpsdiscontinuously to a large value as ε is increased. Thisphase transition exists in the mean-field limit, and canin principle survive finite dimensional fluctuations.

The existence of this constrained phase transition in-duced by a field ε in finite dimensional systems is neededto identify the Franz-Parisi free-energy with the config-urational entropy in the unconstrained bulk system. Ifa metastable glass phase is detected in some tempera-ture regime T > TK , then it is possible to measure thefree-energy difference between the equilibrium liquid andthe metastable glass, namely V (Qglass). This quantityrepresents the entropic cost of localizing the system in asingle metastable state: this is indeed the configurationalentropy.

B. Computational measurement

The free-energy V (Q) in Eq. (44) is the central physicalquantity to measure in computer simulations.135,203,205

It follows from the measurement of rare fluctuations ofthe overlap, since P (Q) ∼ e−βNV (Q). Measuring suchrare fluctuations (indeed, exponentially small in the sys-tem size) in equilibrium systems is a well-known prob-lem that has received considerable attention and power-ful solutions in the context of equilibrium phase transi-tions,179 such as umbrella sampling. Physically, the ideais to perform simulations in an auxiliary statistical en-semble where the Boltzmann weight is biased by a knownamount, and from which the unbiased canonical distribu-tion is reconstructed afterwards.206,207 Combining thistechnique to the swap Monte Carlo44 and parallel tem-pering methods208 to sample more efficiently the relevantfluctuations makes possible the numerical measurementof V (Q) over a broad range of physical conditions.

The same numerical techniques can also be used toprobe the existence and physical properties of the phasetransition induced by a field ε. The ε-transition has givenrise to number of theoretical and computational analysis,which conclude that the transition is present in spatialdimensions d > 2.203,205,207,209–211 The phase transitionemerges for temperatures lower than a critical temper-ature T ∗, which is the analog of Tonset defined in themean-field theory. For T < T ∗, a first-order phase tran-sition appears at a finite value ε∗(T ) of the field, where

16

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 Qglass

Qliq

V(Q)

Q

= 0.559 = 0.573 = 0.588 = 0.603 = 0.612 = 0.619 = 0.628 = 0.635 = 0.645 Tsconf

FIG. 9. Franz-Parisi free energy in three dimensional polydis-perse hard spheres, using a combination of swap Monte Carlo,parallel tempering, and umbrella sampling techniques.135

V (Qglass) decreases progressively with increasing the volumefraction φ. The vertical arrow indicates the estimate of Sconf

as the free energy difference between the low-overlap (Qliq)and large-overlap (Qglass) phases.

the overlap jumps discontinuously to a value Qglass(T ).The existence of the transition allows the quantitative

determination of the configurational entropy, namely

Sconf =N

TV (Q = Qglass). (45)

A nearly equivalent determination can be obtained di-rectly from the properties of the constrained phase tran-sition, since ε∗ represents the field needed to tilt theFranz-Parisi free energy and make the local minimumat large Q become the global one.135,203 Taking into ac-count the small but positive Qliq > 0, we can estimatethe configurational entropy as

Sconf 'N

Tε∗(Qglass −Qliq). (46)

In Fig. 9, we show the evolution of the Franz-Parisi freeenergy V (Q) for a system of continuously polydispersehard spheres in three dimensions, with N = 300 particles.The value of Qglass is identified by a separate study of theε-transition, and is indicated as a vertical dashed line.For each value of the volume fraction, V (Qglass) providesan estimate of the configurational entropy using Eq. (45),as shown by the vertical arrow.

More broadly, the data in Fig. 9 suggests that Kauz-mann’s intuition of an underlying thermodynamic phasetransition connected to the rapid decrease of Sconf is re-alized in deeply supercooled liquid. The evolution ofthe Franz-Parisi free energy shows that the glass stateat large Q is metastable as T > TK , but its stabilityincreases rapidly as T decreases, controlled by the de-crease of the configurational entropy. It is still not knownwhether a finite temperature entropy crisis truly takes

place as a thermodynamic phase transition, but the keyidea that ideal glass formation is accompanied by the de-crease of the associated free energy difference (and hencethe configurational entropy) is no longer a hypothesis,but an established fact.

Finally, the evolution of the free-energy V (Q) with su-percooling is quite dramatic. This large change quantita-tively answers the question raised by the surprising sim-ilarity of the two particle configurations shown in Fig. 1.The density profiles of those two state points are not verydifferent, but their free energy profiles V (Q) are. Thismeans that to compare the two snapshots, one shouldmonitor appropriate observables reflecting the reductionof available states in glass formation, instead of simplestructural changes.

V. CONFIGURATIONAL ENTROPY FROM REALSPACE CORRELATIONS

A. A real space view of metastability

In finite dimensions, the long-lived metastable statesenvisioned by mean-field theory do not exist since thesystem will eventually undergo structural relaxation in afinite time. Therefore, metastable states can at best existover a finite timescale.18,212 In the construction of Franz-Parisi,91,92 metastable states are therefore explored byintroducing a global constraint on the system via a fieldconjugate to the macroscopic overlap. This strategy al-lows one to estimate the number of free energy minimafor a given temperature.

The constraint envisioned in the Franz-Parisi is globaland acts on the bulk system. In this section, we in-troduce another type of constraint that again allows asharp distinction between the vicinity of a given config-uration (the glass basin), and the rest of the free energylandscape. The key difference with the Franz-Parisi con-straint is that we impose a spatially resolved constraintto the system using a cavity construction.65 We shall ar-gue that this provides a real space interpretation of therarefaction of metastable states in terms of a growingspatial correlation length, the so-called point-to-set cor-relation length. This correlation length cannot emergefrom the observation of the density profile in a singleconfiguration, but stems once again from the compari-son between the distinct density profiles available undersome constraint.

The main idea is illustrated in Fig. 10. We prepare anequilibrium configuration of the system, which we takeas a reference configuration, rNref . We then consider aconfiguration rN which is constrained to be equal to thereference configuration outside a cavity of radius R, butcan freely fluctuate inside the cavity. Therefore, the con-straint from the reference configuration is now only felt atthe frozen amorphous boundary of the cavity. By vary-ing the cavity size R, one can then infer how far the con-straint propagates inside the cavity. As quantified below,

17

FIG. 10. Sketch of the cavity construction to determine thepoint-to-set lengthscale. The positions of the particles out-side a cavity of radius R are given by a reference equilibriumconfiguration and are frozen, while particles inside the cav-ity evolve freely at thermal equilibrium in the presence of thefrozen amorphous boundaries. The overlap profile Q(r) is de-fined by comparing the density profile inside the cavity to thereference configuration.

one expects a crossover between small cavities where theconstraint is so strong that particles inside the cavity canonly remain close to the reference configuration, whereasfor very large cavities particles inside the cavity will ex-plore a large number of distinct states. The crossoverbetween these two regimes is used to define the point-to-set correlation length.65,213,214

Why is this crossover length directly connected to theconfigurational entropy? This can be understood follow-ing a simple thermodynamic argument. Suppose the par-ticles inside the cavity explore states that are very dif-ferent from the reference configuration. This will allowthem to sample states that have a low overlap Q withthe reference configuration. The free energy gained bythis exploration is directly given by the Franz-Parisi freeenergy, ∆F− = V (Qglass)vdR

d, where vd is the volume ofthe unit sphere in spatial dimension d. There is howevera free energy cost to explore those states, as the radialoverlap profile inside the cavity Q(r) will present an in-terface between Q(r = 0) ≈ 0 and Q(r = R) ≈ Qglass.This interface in the profile of the order parameter hasa free energy cost, and a simple estimate is given by∆F+ = Y sdR

d−1, where sd is the surface area of theunit sphere in dimension d and Y a surface tension be-tween two distinct glassy states. In many disordered sys-tems, the interfacial terms take a more general expres-sion, ∆F+ = ΥRθ, where Υ is a generalized surface ten-sion and the non-trivial exponent θ ≤ d− 1 accounts foradditional fluctuations in directions transverse to the in-terface.18,215 Physically, these fluctuations arise becausethe system can decrease the interfacial cost by allowingthe position of the interface to fluctuate and take advan-

tage of the weakest spots.The competition between exploring many states, that

decreases the free energy by ∆F−, and the interfacialcost of an inhomogeneous overlap profile, that increasesthe free energy by ∆F+, leads to a well-defined crossoverradius for the cavity where the two terms balance eachother,

∆F = ΥRθ − V (Qglass)vdRd = 0. (47)

This crossover radius defines the point-to-set correlationlengthscale ξpts, given by

ξpts =

(Υ

V (Qglass)vd

)1/(d−θ)

. (48)

This equation directly connects the decrease of the Franz-Parisi free energy to the growth of a spatial correlationlengthscale. It is important to notice that since V (Qglass)is unambiguously defined and can be measured in com-puter simulations, the same is true for the point-to-setcorrelation lengthscale whose existence and physical in-terpretation does not require any type of approximation.In particular, there is no need to assume the existence oflong-lived free-energy metastable states.

A connection between the point-to-set correlationlengthscale defined in Eq. (48) and the configurationalentropy can be established by using Eq. (45) expressingthe Franz-Parisi free energy V (Qglass) as an estimate ofthe configurational entropy. We realize that the growthof the point-to-set correlation lengthscale as temperaturedecreases is equivalent to a decrease of V (Qglass), andthus to a decrease of the Sconf , assuming a modest tem-perature dependence of Υ.18,216 Therefore, the growthof the point-to-set correlation lengthscale is a direct realspace consequence of the decrease of the configurationalentropy.18,65 If a Kauzmann transition where Sconf → 0intervenes, then it must be accompanied by a divergenceof the point-to-set correlation lengthscale, ξpts →∞.

The relation between the point-to-set lengthscale andthe configurational entropy can be used both ways.135

First, it provides a useful interpretation of the entropycrisis in terms of a diverging correlation lengthscale, asput forward in the early development of the random firstorder transition theory.18 We find it equally convenient touse this connection in the opposite direction and deducefrom the growth of the point-to-set correlation length aquantitative determination of the variation of the config-urational entropy.16,135 Using the above scaling relations,the measurement of ξpts provides another estimate of theconfigurational entropy

Sconf = N

(ξ0ξpts

)d−θ, (49)

where ξ0 is an unknown factor that results from conver-sion between entropy and lengthscale. At this stage, thevalue of the exponent θ is undetermined. It could bemeasured by comparing measurements of Sconf following

18

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0 T = 0.400 T = 0.140 T = 0.101 T = 0.070 T = 0.050 T = 0.039 T = 0.033 T = 0.028

Q center

R(a )

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0(b )

R pts

R < pts

P(Q center)

Qcenter

R > pts

FIG. 11. Measurement of the point-to-set lengthscale in a 2dsystems of polydisperse soft repulsive spheres.16 (a) Evolutionof the overlap at the center of the cavity, Qcenter, with thecavity radius R for different temperatures. (b) Evolution ofthe probability distribution of the overlap, P (Qcenter), withcavity radius R for a given low temperature T = 0.035 ≈0.3Tg.

Eq. (49) to an independent estimate, for example fromEq. (45). The two supported values for the exponent arethe simple value θ = d − 1,216–218 and the renormalizedvalue θ = d/218,219 stemming from the random inter-face analogy. They respectively lead to Sconf ∼ 1/ξpts

and Sconf ∼ 1/ξd/2pts , which are equivalent in d = 2 and

not very different in d = 3 given the relatively modestvariation of the configurational entropy reported in ex-periments.

B. Computational measurement

To determine the point-to-set correlation length nu-merically,134,220–223 we essentially follow the theoreticalconstruction described above and illustrated in Fig. 10.First, an equilibrium reference configuration rNref is pre-pared. We define a cavity of radius R, centered on arandomly-chosen position in the reference configuration.

We then define a configuration rN : the particles lyingoutside the cavity are frozen at the same positions as inthe reference configuration, whereas particles inside thecavity can thermalize freely.

The key observable is the overlap profile Q(r) betweenconfigurations rN and rNref inside the cavity. It is numer-ically convenient to focus on the value of the overlap atthe center of the cavity Qcenter ≡ Q(r = 0), which de-pends both on the cavity size R, and the temperatureT . Figure 11(a) shows the evolution of Qcenter with thecavity size R for polydisperse soft disks in d = 2.16 Atsmall R, Qcenter ≈ Qglass meaning that the system isconstrained to remain in the same state as the referenceconfiguration. The overlap is not strictly one becausethermal fluctuations allow small deviations around thereference configuration rNref . At larger R, however, Qcenter

decays to a small value, which implies that the systemcan freely explore states that have different density pro-files. The cavity size at which the transition from highto low overlap occurs determines the point-to-set length-scale ξpts. In practice, one can define ξpts when Qcenter

reaches a specific value, or from an empirical fitting ofthe whole function, such as Qcenter ≈ exp[−(R/ξpts)

b],where b is a fitting parameter. The temperature evolu-tion of Qcenter(R) is very interesting as it directly revealsthat the amorphous boundary condition constrains morestrongly the interior of the cavity as the temperature de-creases. Physically, it indicates that as temperature de-creases, the point-to-set correlation lengthscale grows, orequivalently that the configurational entropy decreases,in virtue of Eq. (49).

An interesting alternative view of the free-energy com-petition captured by Eq. (47) emerges by considering theevolution of the free-energy gain ∆F− of the configura-tion rN inside the cavity, as the cavity size is decreasedat constant T . For a very large cavity, the particles inrN are pinned at the boundaries, but those at the centerof the cavity evolve as freely as in the bulk equilibriumsystem. Since the free-energy gain ∆F− scales as Rd, itdecreases as the cavity size decreases, making it increas-ingly difficult for the configuration rN to explore otherstates inside cavity. As the cavity size approaches thepoint-to-set lengthscale, the entropic driving force to ex-plore many states inside the cavity becomes comparableto the free energy cost ∆F+ of the amorphous boundary.For even smaller cavities, the system is frozen in a sin-gle state. The scenario that we have just described forthe cavity is nothing but the entropy crisis predicted bythe random first transition theory for the bulk systemas T → TK . In other words, decreasing the cavity sizefor a given T > TK has an effect similar to approachingthe Kauzmann transition in a bulk system. The qualita-tive difference is that the Kauzmann transition is a sharpthermodynamic transition happening for N →∞ in thebulk, whereas the entropy crisis in the cavity takes placefor a finite system comprising N ∼ ξdpts particles. Thecrossover from small to large overlap observed aroundR ∼ ξpts(T ) in the profiles of Fig. 11(a) is conceptually

19

analogous to a Kauzmann transition rounded by the fi-nite size of the system.20

This analogy is even more striking when the fluctu-ations of the overlap are recorded,223 and not only itsaverage value. Figure 11(b) shows the probability dis-tribution of Qcenter, denoted P (Qcenter), for a fixed tem-perature as R is varied. For large R, the distributionpeaks at low values of the overlap, whereas for small R itpeaks near Qglass. Interestingly, at the crossover betweenthese two regimes, P (Qcenter) is clearly bimodal, whichis reminiscent of the distribution of the order parame-ter near a first-order phase transition in a finite system.These observations suggest that it is interesting to mon-itor the variance of these distributions, which is a mea-sure of the susceptibility χ associated with this roundedKauzmann transition. For a given T , it is found that χhas a maximum when R = ξpts, which provides a fitting-free numerical definition of the point-to-set correlationlengthscale.223

Despite the conceptual simplicity of the measurementsdescribed above, it is not straightforward to obtain statis-tically meaningful numerical measurements of the over-lap and of its fluctuations inside finite cavities. Thereare several reasons for this. First, to obtain a value forξpts at a given temperature, one needs to analyze a rangeof cavity sizes that encompasses the crossover shown inFig. 11. For each cavity size R, a large number of inde-pendent cavities need to be studied, and the overlap ineach individual cavity needs to be carefully monitored toensure that its equilibrium fluctuations have been prop-erly recorded. All in all, the number of required simula-tions is quite substantial.

The second major computational obstacle naturallystems from the physics at play as R is reduced. Be-cause the confined system undergoes a finite-size analogof the Kauzmann transition, a major slowing down arisesin the thermalization process. This amounts to studyingan ‘ideal’ glass transition in equilibrium conditions, anobviously daunting task. This is however possible in thepresent case because only a finite number of particles arecontained in the cavity. This allows the use of paralleltempering (or replica exchange) methods, first developedin the context of spin glasses to overcome thermalizationissues in systems with complex landscapes.208 With thesetechniques, the study of a given set of parameters (T,R)requires simulating a large number of copies of the sys-tem interpolating between the original system and a statepoint at which thermalization is fast. During the courseof the simulations, exchanges between neighboring statesare performed, so that each copy performs a random walkin parameter space. This method, developed in Ref. 223has proven sufficiently efficient and versatile to analyzepoint-to-set correlations in a broad range of model sys-tems down to very low temperetures.135

VI. PERSPECTIVE

We presented a short review of the configurational en-tropy in supercooled liquids approaching their glass tran-sition. We first described why and how configurationalentropy became a central thermodynamic quantity to de-scribe glassy materials, both from experimental and the-oretical viewpoints. We then offered our views on severalparadoxes surrounding the configurational entropy. Inparticular, we explained that there is no reason to tryand avoid an entropy crisis, that available data neitherdiscard nor disprove its existence, and that there existsno fundamental reason, published proof, or general argu-ments showing that it must be avoided. In other words,the Kauzmann transition remains an useful and valid hy-pothesis to interpret glass formation. We also insistedthat this is still a hypothesis, but in no way a proven ornecessary fact.

The biggest paradox of all, is perhaps that the con-figurational entropy, which represents the key signatureof the entropy crisis occurring in the modern mean-fieldtheory of the glass transition, cannot be rigorously de-fined in finite dimensions as a complexity that enumer-ates free energy minima. We have presented several com-putational schemes which are meant to provide at thesame time an estimate of the configurational entropy innumerical models of glass-forming liquids, and a physicalinterpretation that is valid in finite dimensions.

We started with the historical method based on inher-ent structures, which enumerates the number of poten-tial energy minima as well-defined, but incorrect proxies,for free energy minima. It is unclear that the inherentstructure configurational entropy can in fact vanish at aKauzmann transition, and Stillinger provided argumentsthat it cannot. This method is a computationally cheapmethod to remove a vibrational contribution to the totalentropy, but it cannot be used for simple models such ashard spheres or continuously polydisperse glass-formers.

We then showed that a generalized method elaborat-ing on earlier ideas introduced by Frenkel and Ladd forcrystals provides a better estimate of the configurationalentropy, as it naturally includes both the glass mixing en-tropy and finite temperature anharmonicities. Addition-ally, the method can be applied to all types of models,including hard spheres, at a relatively cheap computa-tional cost.

More recent methods were developed as direct applica-tions of the mean-field theory to computer works, whichboth bypass the need to mathematically define free en-ergy minima. Free energy measurements, based on theFranz-Parisi free energy, provide an estimate for the con-figurational entropy that is the closest to the originalmean-field definition. This method relies on the defini-tion of a global order parameter for the glass transition,the overlap, which quantifies the similarity between pairsof configurations. Conventional methods employed in thecontext of equilibrium phase transitions are combined tothese measurements.

20

Finally, we showed that the decrease of the entropycan be given a real space interpretation in terms of agrowing correlation lengthscale that is directly related tothe configurational entropy.

This brief summary shows that there now exist con-ceptually solid estimates of the configurational entropythat could truly provide a direct access to the thermody-namic behavior of supercooled liquids. Given the recentprogress of computer simulations to efficiently equilibratemodel systems down to temperatures that are matching,and in several cases, outperforming experimental work,we feel that this is an exciting moment for glass physics,since a direct demonstration of the relevance of an en-tropy crisis and increasingly precise localizations of theputative Kauzmann transition appear possible.

ACKNOWLEDGMENTS

This perspective is based on two sets of lectures. Thefirst one was given during the 2017 Boulder SummerSchool on “Disordered and Frustrated systems”. Thesecond one was given during the 2018 Bangalore SummerSchool “Entropy, information and order in soft matter”.We thank G. Biroli, P. Charbonneau, D. Coslovich, M.D. Ediger, A. Ikeda, W. Kob, K. Miyazaki, A. Ninarello,G. Parisi, G. Tarjus, and S. Yaida for collaborations anddiscussions on the topics discussed in this paper. Wethank S. Tatsumi and O. Yamamuro for providing uswith experimental data and discussions on experimen-tal measurement of the configurational entropy. The re-search leading to these results has received funding fromthe Simons Foundation (#454933, Ludovic Berthier).

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