# Calorimetric glass transition in a mean-field theory approach

^{a}Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland;^{b}Dipartimento di Fisica, Sapienza Università di Roma, I-00185 Rome, Italy;^{c}Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, Istituto per I Processi Chimico-Fisici, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy; and^{d}Laboratoire de Physique Théorique, École Normale Supérieure, UMR 8549 CNRS, 75005 Paris, France

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Contributed by Giorgio Parisi, January 7, 2015 (sent for review July 12, 2014)

## Significance

Understanding the properties of glasses is one of the major open challenges of theoretical physics. Making analytical predictions is usually very difficult for the known glassy models. Moreover, in experiments and numerical simulations thermalization of glasses cannot be achieved without sophisticated procedures, such as the vapor deposition technique. In this work we study a glassy model that is simple enough to be analytically solved and that can be thermalized in the glassy phase with a simple numerical method, opening the door to an intensive comparison between replica theory predictions and numerical outcomes.

## Abstract

The study of the properties of glass-forming liquids is difficult for many reasons. Analytic solutions of mean-field models are usually available only for systems embedded in a space with an unphysically high number of spatial dimensions; on the experimental and numerical side, the study of the properties of metastable glassy states requires thermalizing the system in the supercooled liquid phase, where the thermalization time may be extremely large. We consider here a hard-sphere mean-field model that is solvable in any number of spatial dimensions; moreover, we easily obtain thermalized configurations even in the glass phase. We study the 3D version of this model and we perform Monte Carlo simulations that mimic heating and cooling experiments performed on ultrastable glasses. The numerical findings are in good agreement with the analytical results and qualitatively capture the features of ultrastable glasses observed in experiments.

The theoretical interpretation of the properties of glasses is highly debated. There are two extreme viewpoints:

• One approach, the random first-order transition (RFOT) theory (1), which uses mostly the replica method (2) as its central tool, assumes that the dynamical properties of glasses do reflect the properties of the appropriate static quantities [such as the Franz–Parisi potential (3); for a review see refs. 2 and 4].

• The other approach, kinetically constrained models (KCMs), assumes that the glass transition is a purely dynamical phenomenon without any counterpart in static quantities (5⇓–7).

The mean-field version of the RFOT approach predicts the presence of a dynamical transition [identified with the mode-coupling transition (8)] at a nonzero temperature *T _{d}*, the relaxation time increases rapidly and becomes comparable with human timescales, leading to the phenomenological glass transition. In the KCM approach this slowdown is a phenomenon originated only by constraints on the dynamics, whereas the RFOT picture views the off-equilibrium states as metastable, thermodynamic states, that can be identified with the minima of a suitable equilibrium free-energy functional and then studied using a modified equilibrium formalism, generally built on the replica method.

According to replica formalism, the system explores the whole collection of possible states, with lower and lower free energy, as the temperature is lowered from

To test this scenario, it would be necessary to perform experiments and simulations at various temperatures in this range, but then one must face the problem of equilibrating the glass former at temperatures

Some progress in this direction has been made recently both in experiments (9) and numerical simulations (10), with the introduction of the so-called vapor deposition technique, which allows one to obtain extraordinarily stable glasses [usually referred to as ultrastable glasses (10⇓⇓–13)] in a relatively short time, even for temperatures much lower than

The recent introduction (17) of a semirealistic soluble model for glasses, the Mari–Kurchan (MK) model, gives us the possibility of addressing both the equilibration and the theoretical problem. It allows us to obtain equilibrated configurations also beyond the dynamical transition and deep into the glass phase, using the so-called planting method (18). Moreover, it is in principle solvable in the replica method, allowing us to study the metastable glassy states with a static formalism, without having to solve the dynamics.

Our aim is to use this model to simulate slow annealing experiments usually performed on glasses and ultrastable glasses, to compare the numerical outcomes with experimental results and theoretical predictions in the replica method.

## The Model

We consider the potential energy of the family of models introduced by Mari and Kurchan (MK model) (17):*N d*-dimensional vectors, representing particles positions, and the particles move in a *d* dimensional cube of size *L*, with periodic boundary conditions. The main feature of the model are the variables *v* could be in principle any interesting short-ranged repulsive pairwise interaction.

The main effect of the random shifts is to destroy the direct correlation among the particles that interact with a given particle (17). This makes the computation of static quantities very simple, because in the Mayer expansion of the grand-canonical potential only the tree diagrams survive in the thermodynamic limit (17). The idea is quite old (19), and had important applications to turbulence, but has only recently been applied to glasses.

### Static Thermodynamic Properties in Liquid Phase.

Here we will summarize analytical and numerical results obtained by Mari and Kurchan for this model. In the following *D* will denote the diameter of spheres. In hard-sphere systems the potential *D* and the role of inverse temperature is played by the packing fraction *d*-dimensional sphere of diameter *D*; we will call it density absorbing the multiplicative factor in its definition.

The Hamiltonian contains random terms and the interesting quantities have then to be averaged over these parameters. We can define the annealed entropy

A more interesting quantity is the quenched entropy. In this model one finds that *ρ* that diverges in the infinite-density limit. This implies that the configurational entropy contains a term proportional to

Using standard termodynamic relations one can derive from Eq. **3** the liquid-phase equilibrium equation of state*P* is the pressure.

For what concerns the radial distribution function, one has to take the random shifts into account:*θ* is the usual Heaviside step function. This result is the same obtained with high-dimensional hard spheres (20), but the mean-field nature of the model has allowed us to get it in any number of spatial dimensions. The equilibrium pressure is related to density by the usual relation for hard spheres (21),**6**, the equilibrium equation of state (Eq. **4**) can be derived again.

### Glassy Properties.

The model is interesting because despite the extreme simplicity of the statics (a feature that it has in common with facilitated models) the dynamics is extremely complex. At high densities there is the glass phase that in the thermodynamic limit is separated from the liquid phase by a mode coupling transition. This transition exists only if we embed the model in a space with an infinite number of dimensions *d*; when

Accurate simulations (22) give a higher value for the mode-coupling dynamical density (i.e.,

## Numerical Simulations

When a glass is gradually heated during DSC experiments thermodynamic quantities, like the internal energy, continue to follow the glassy behavior also in the liquid phase, until the so-called onset temperature

We aim to study numerically this deviation from equilibrium in the liquid region and the subsequent relaxation process in the MK model. In the MK model we are able to obtain equilibrium configurations beyond the dynamic transition via a special procedure, allowed only by the presence of random shifts, the so-called planting (18) method. Basically, planting consists of two steps: the generation of a random configuration of sphere positions, independently drawn out from the uniform distribution over the volume, and the generation of the random shifts configuration **1**) is satisfied for every pair of spheres (see *Supporting Information* for details). The mean-field nature of the interaction guarantees that planted configurations are equilibrated (18). The planted glass in the MK model, like vapor-deposited ultrastable glasses in real world, is the best possible starting point for the study of the deviation from equilibrium in the liquid phase. We start from a planted configuration and mimic DSC heating experiments by running adiabatic stepwise decompression scans, where the system performs jumps between different density values and a large number of Monte Carlo steps for each density value, to reach thermalization.

We refer to *Supporting Information* for all other simulation details. We present now the results of numerical simulations, based on the Monte Carlo method, of a system composed of

### Decompression Jump and Spheres Contact Region Emptying.

The outcome of the planting technique is a thermalized initial configuration at a certain density

We consider now the following decompression protocol: We start from a planted configuration at *φ*. We do not see structural relaxation for

### Mean Square Displacement and Structural Relaxation Time.

To study the behavior of the relaxation time as a function of density in the liquid phase and evaluate the dynamic glass transition density *φ* we are studying. We stress that in Eq. **8** *i* immediately after the density jump. We let the system evolve for *φ*. For *Supporting Information* for details). We fitted the resulting curve of *φ*, displayed in Fig. 3, with the power law behavior *γ* is not too different from the one obtained in ref. 17 performing a similar analysis on the relaxation time, whereas the value of

### Decompression and Compression Scans: Qualitative Comparison with Experiments.

In Fig. 4 we represent the behavior of the reduced pressure *φ*, we see a deviation from equilibrium independent from the decompression rate. For values well above

The decompression protocol adopted for our system, composed by hard spheres, is equivalent to the typical DSC’s heating scans, with two crucial differences. (*i*) In DSC experiments we move toward the glassy phase by decreasing the temperature: In the case of hard spheres the inverse of the density plays the same role of the temperature. (*ii*) The starting configurations of the dynamics are fully equilibrated and this corresponds only to the case of DSC with infinitely slow cooling speed and relatively fast heating speed. The observed deviation of pressure from equilibrium for

In Fig. 5 curves for different values of

## Replica Computation of Metastable States Curves

So far we have shown how the MK model allows one to prepare the system in a glass state, even at densities much higher that the dynamical one, without incurring the problem of extremely large equilibration times. In addition, this model has another remarkable advantage: It is in principle solvable, thanks to its mean-field nature. The interaction network is tree-like (or, alternatively, without loops) in the thermodynamic limit, like in Van der Waals liquids (17, 21), and thus it also allows for a ready comparison between numerics and analytic computations. In particular, it allows us to perform computations in the replica method. Although the MK model is soluble, its actual analytic solution is exceedingly complex (23), so we have to resort to some kind of approximation: Here we assume that the cages have a Gaussian shape (22).

In the replica approach to the glass transition (1, 4), it is assumed that for densities

The most important feature of these metastable states is that they are degenerate, that is, they can have the same free entropy. In fact, if one fixes a density *s* for the free entropy, it is possible to see that the number of states that share it (in the functional picture, the number of minima that all have the same height *s*) scales exponentially with the size of the system,

### The Replica Method.

The replica method provides us with a standard procedure to compute the complexity and also the in-state entropy (24). Its concrete application to hard-sphere systems is described in full detail in section III of ref. 4; here we recall it briefly. It consists of introducing *m* independent replicas of the system and forcing them to occupy the same metastable state. The entropy of the replicated system becomes then*s* is the free entropy of the state. In the thermodynamic limit, the partition function will be dominated with probability 1 only by the states with the entropy

### Isocomplexity Approximation.

The replica formalism has been applied to the study of infinite dimensional hard spheres in a series of papers (20, 25, 26) with remarkable success. However, those results concern only the properties of the glass former after equilibration, whereas our numerical results concern the glass former when it is still trapped inside a metastable state, before equilibration takes place. Indeed, one could argue that, for experimental and practical purposes, getting predictions for this regime is even more important than the study of the equilibrium solution for infinite waiting times. This program, however, poses a challenge because in principle it requires solving the dynamics for different preparation protocols. To this day, the only first-principles dynamical theory for glass formers is the mode coupling theory (8), which performs well near the dynamical transition but notoriously fails at higher densities, forcing one to use phenomenological models for the description of the high-density (or low-temperature) regime, as done by Keys et al. (14). We present here a computation that has the advantage of being both fairly simple and static in nature.

Because the system is trapped in a single metastable state during the simulation, it is clear that its physical properties are determined only by the in-state entropy **9** allows us to compute only quantities related to the states that dominate the partition function. Indeed, we can see that for every density *φ* we can choose the value of *m*, but in principle we still have no way of knowing what is actually the state the system is trapped into, that is, we lack a criterion to choose a function

To overcome this difficulty, we assume that every state can be followed in density without any crossings between states, or bifurcations, or spinodal points (28); this means that the number (and thus the complexity) of states that share the same value *s* of the in-state entropy is a conserved quantity during the experiment and can then be used as a label for the states. This method is usually referred to as isocomplexity (27, 28).

In summary, to choose

We refer to *Supporting Information* for the details of the computation of the isocomplexity lines displayed in Fig. 6. Once the potential

## Conclusions

We studied a mean-field model of glass transition, the MK model. We were able both to obtain a stable glass, thanks to the planting technique, and to study numerically and analytically (within replica method and isocomplexity assumption) the variations of pressure caused by relatively fast changes of density. We showed, both numerically and analytically, that qualitatively this model displays the same behavior of experimental ultrastable glasses reported in refs. 9 and 10. Our model seems to show a first-order phase transition when evading from metastable equilibrium (see ref. 29 and *Supporting Information*). This is in qualitative agreement with experiments, which show that the melting of ultrastable glasses (12, 13) has some features in common with first-order transitions.

We have also shown that the RFOT approach, together with the replica method, is able to qualitatively describe the process of glass formation through a slow annealing, with very little computational cost and without resorting to a posteriori phenomenological considerations. Our results can be compared with the DSC experiments where cooling is much slower than heating and as a result the cooled configurations (before heating) may be approximated with equilibrium configurations. We can study this situation in the MK model just because we can plant a thermalized equilibrium configuration at the density we prefer. The very interesting problem of understanding the behavior of DSC experiments when the cooling speed is the same (or faster) than the heating speed is not studied in this paper: In this situation analytic computations could be done only if we had the dynamics under analytic control, a goal that has not yet been reached.

## Acknowledgments

We thank Francesco Zamponi for useful discussions. This work was supported by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant 247328.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: giorgio.parisi{at}roma1.infn.it or Corrado.Rainone{at}roma1.infn.it.

Author contributions: G.P. designed research; M.S.M., G.P., and C.R. performed research; and M.S.M., G.P., and C.R. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1500125112/-/DCSupplemental.

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