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Modern computational studies of the glass transition

Ludovic Berthier, David R. Reichman

arXiv:2208.02206v3cond-mat.stat-mechcond-mat.dis-nncond-mat.soft

TL;DR

Understanding glass formation is difficult because simulations face severe timescale barriers, while important questions span liquid-to-glass statistical mechanics and the properties of amorphous solids. This review surveys enhanced-sampling and equilibration algorithms and the advances they have supported, while identifying unresolved research challenges.

Abstract

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The physics of the glass transition and amorphous materials continues to attract the attention of a wide research community after decades of effort. Supercooled liquids and glasses have been studied numerically since the advent of molecular dynamics and Monte Carlo simulations in the last century. Computer studies have greatly enhanced both experimental discoveries and theoretical developments and constitute an active and continually expanding research field. Our goal in this review is to provide a modern perspective on this area. We describe the need to go beyond canonical methods to attack a problem that is notoriously difficult in terms of time scales, length scales, and physical observables. We first summarise recent algorithmic developments to achieve enhanced sampling and faster equilibration using replica exchange methods, cluster and swap Monte Carlo algorithms, and other techniques. We then review some major recent advances afforded by these novel tools regarding the statistical mechanical description of the liquid-to-glass transition as well as the mechanical, vibrational and thermal properties of the glassy solid. We finally describe some important challenges for future research.

I. INTRODUCTION

Glass formation is a major condensed-matter problem marked by extreme dynamical slowing, collective behavior, and disordered solids whose microscopic excitations remain poorly understood. Simulations face severe timescale and sampling barriers, while newer algorithms enable access to more deeply cooled glassy states and phenomena.

I. INTRODUCTION

  • Cooling to a glass spans about fifteen orders of magnitude in relaxation time, alongside dynamical heterogeneity, non-Arrhenius viscosity, and Stokes-Einstein violations.These features accompany a glass state that lacks the structural hallmarks of an ordered crystal.Source: §I · p. 1
  • Glass formation connects to problems in spin glasses, protein folding, and satisfiability, and remains an outstanding problem in condensed-matter science.A glass is mechanically stable yet structurally disordered, and its microscopic elementary excitations remain an open problem.Source: §I · p. 1
  • Simulations face sluggish dynamics, finite-size effects, ergodicity breaking, sample-to-sample fluctuations, and self-induced heterogeneity.Historically, standard simulations reached only mild supercooling and roughly the first five decades of dynamical slowdown, leaving conditions near Tg and realistic glass studies inaccessible.Source: §I · p. 1
  • Novel algorithms have enabled glassy states effectively cooled as slowly as laboratory-prepared states, opening computational access to new phenomena and properties.The review highlights growing static and dynamical length scales near the transition, brittle-to-ductile yielding, and microscopic information relevant to two-level-system theories.Source: §I · p. 1

A. Basic tools and glass-forming models

The review distinguishes local-dynamics simulation from Monte Carlo methods that can use non-local moves to accelerate equilibrium sampling. It focuses chiefly on spherical, short-range glass-former models as a practical compromise, while excluding properties tied specifically to non-spherical degrees of freedom.

A. Basic tools and glass-forming models

  • Molecular dynamics and realistic-dynamics Monte Carlo use local particle moves, whereas equilibration Monte Carlo can use non-local or cluster moves to explore configuration space more efficiently.Molecular dynamics calculates forces from the interaction potential; Monte Carlo generally requires only the potential energy.Source: A. Basic tools and glass-forming models · p. 2
  • Off-lattice molecular models represent detailed molecular interactions, but non-spherical units, rotations, and long-ranged interactions make them more computationally intensive.Examples include the BKS silica force field and a coarse-grained three-site representation of ortho-terphenyl.Source: A. Basic tools and glass-forming models · p. 2
  • Spherical short-range models, including hard-sphere and Lennard-Jones systems, can reproduce many glass-former behaviors while being more efficient to simulate.Polydispersity and tuned interactions are generally used to prevent facile crystallization.Source: A. Basic tools and glass-forming models · p. 2
  • Lattice glass models enable direct use of mathematical tools such as the replica method, but can crystallize easily unless crystallization is frustrated.They provide a bridge between glass-transition theories and more complex off-lattice models.Source: A. Basic tools and glass-forming models · p. 2
  • The review focuses largely on spherical short-range models as a compromise between molecular complexity and lattice-model realism, but they cannot describe rotational Stokes-Einstein behavior or realistic dielectric relaxation.Their shared behaviors with laboratory glass-formers include Stokes-Einstein violations, growing timescales, and non-exponential, non-Arrhenius relaxation.Source: A. Basic tools and glass-forming models · pp. 2–3

B. Equilibration tools in complex systems

Local moves and standard computing resources cannot equilibrate samples near the glass transition, motivating tailored non-local and enhanced-sampling approaches. Their performance is mixed: replica exchange has limited bulk-liquid benefits, while specialized approaches offer distinct possibilities but retain important constraints.

B. Equilibration tools in complex systems

  • Local moves and processor-speed gains are insufficient to equilibrate samples near the glass transition, so tailored non-local Monte Carlo and related techniques are central to the review.An algorithmic speedup of one or two orders of magnitude can be useful but does not radically change the physics probed when relaxation times span many orders of magnitude.Source: B. Equilibration tools in complex… · p. 3
  • Parallel tempering and replica-exchange methods exchange temperatures or configurations between replicas to sample rough energy landscapes, though their performance for bulk glasses is modest.Fig. 1 sketches parallel tempering alongside cluster and swap moves; in the general approach, exchanges preserve equilibrium.Source: B. Equilibration tools in complex… · p. 3, Figure 1 · p. 4
  • Replica-exchange molecular dynamics was up to 100 times more efficient at phase-space exploration in the studied regimes, but equilibration below Tc was not attempted.The reported efficiency used effective particle diffusion as its measure; a later, stricter potential-energy-minimum criterion found RXMD no more efficient than standard molecular dynamics.Source: B. Equilibration tools in complex… · p. 4
  • Replica exchange appears most efficient for supercooled liquids in confined geometries or with some particles frozen, settings used to extract growing static length scales.Some studies combine replica exchange with other sampling techniques, making its individual contribution difficult to isolate.Source: B. Equilibration tools in complex… · pp. 4–5
  • Physical-vapor-deposition-inspired simulations can help interpret experiments, but their surface-to-bulk speedup is only 1–2 orders of magnitude and equilibration is not guaranteed.In experiments, surface mobility can permit relaxation at temperatures where bulk material is arrested, producing films with properties nearly equivalent to very slowly prepared bulk glasses.Source: B. Equilibration tools in complex… · p. 5
  • Biased sampling of rare dynamic trajectories can produce configurations with properties unlike bulk equilibrium systems that appear deeper in the energy landscape.The configurations are described as very stable, but their stability has not been quantified; the methods scale poorly and are currently limited to relatively small systems.Source: B. Equilibration tools in complex… · p. 5

C. Advanced Monte Carlo techniques: Cluster Monte Carlo and swap Monte Carlo

Cluster and swap Monte Carlo methods offer alternatives to local moves for equilibrating glassy systems, but their efficiency depends strongly on the model. Carefully tuned swap models can reach laboratory-level annealing, while cluster methods have enabled equilibration beyond the mode-coupling regime in two dimensions.

Cluster Monte Carlo

  • Dress and Krauth’s cluster method identifies overlapping particles in a rotated copy and exchanges clusters through detailed-balance-preserving Metropolis moves.The approach can accelerate configuration-space exploration, but identifying and moving clusters efficiently is challenging for off-lattice systems.Source: C. Advanced Monte Carlo techniques:… · p. 5
  • Cluster Monte Carlo equilibrated polydisperse hard disks beyond the mode-coupling density, without evidence of a thermodynamic glass-transition signature.The results establish that the mode-coupling transition is not a thermodynamic singularity in this system; they do not locate a thermodynamic transition.Source: C. Advanced Monte Carlo techniques:… · pp. 5–6

Comparison and scope

  • SMC is more efficient than cluster Monte Carlo for simple hard disks, but its application is limited for monodisperse and complex molecular glass-formers.Combining cluster moves with rejection-free event-chain Monte Carlo has enhanced cluster sampling, while high-throughput molecular dynamics and Anton remain powerful for complex liquids.Source: C. Advanced Monte Carlo techniques:… · p. 6

Swap Monte Carlo

  • SMC adds potentially long-ranged pair swaps to local Metropolis moves, and its usefulness depends sensitively on swap acceptance relative to local moves.Move frequency and the range of particle sizes or types selected for swaps require optimization, although the Metropolis construction itself is simple.Source: C. Advanced Monte Carlo techniques:… · p. 6
  • A factor larger than 1010 speedup at experimental Tg lets tuned polydisperse models reach laboratory-level annealing without crystallization.The comparison is for three-dimensional soft spheres; equilibration can also be achieved below Tg to produce ultrastable computer glasses.Source: Figure 2 · p. 7, C. Advanced Monte Carlo techniques:… · p. 7
  • Approximately two orders of magnitude was the earlier SMC speedup; later tuning enabled efficient equilibration without crystallization.Earlier studies also found SMC more efficient than pivot-based cluster Monte Carlo for polydisperse two-dimensional hard disks, but observed crystallization in other investigated systems.Source: C. Advanced Monte Carlo techniques:… · p. 7

A. Statistical mechanics analysis of glass transition

Enhanced equilibration has brought configurational-entropy measurements into the temperature range around and below the experimental glass transition. The review reports dimensional differences in extrapolated Kauzmann temperatures and discusses phase-transition and length-scale tests of glass thermodynamics.

Configurational entropy

  • Swap Monte Carlo enables many independent equilibrium configurations across temperatures encompassing the experimental glass transition, supporting ensemble-averaged measurements and thermodynamic calculations.New computational estimates of Sconf are conceptually closer to its rigorous theoretical definition and can be measured where theory predicts validity and experimental estimates exist.Source: A. Statistical mechanics analysis of… · pp. 7–8
  • In mean-field theory, Sconf quantifies the complexity of the rugged free-energy landscape, and it vanishes at the Kauzmann temperature in an entropy crisis.Earlier computational estimates instead used potential-energy minima as an approximation; those measurements covered mostly temperatures above the mode-coupling crossover, where RFOT theory says Sconf cannot be defined.Source: A. Statistical mechanics analysis of… · p. 8
  • Configurational entropy falls steeply in all studied models; extrapolations suggest finite TK in three dimensions but TK = 0 in two dimensions.The dimensional contrast is in line with RFOT theory’s prediction that finite-dimensional fluctuations destroy the Kauzmann transition for d ≲ 2.Source: A. Statistical mechanics analysis of… · p. 8, Figure 3 · p. 8

Thermodynamic phase diagram

  • Three-dimensional glass-formers show a first-order transition line in the (T, ϵ) phase diagram and a critical endpoint consistent with random-field Ising universality.In two dimensions, no critical endpoint is found, and scaling agrees with a zero-temperature Kauzmann transition.Source: A. Statistical mechanics analysis of… · p. 9

Length scales

  • The point-to-set length scale is measured by freezing particles outside a cavity and tracking the interior as cavity size changes.Parallel tempering is needed to measure its thermodynamic properties because frozen boundaries strongly constrain the cavity interior; its temperature evolution agrees with Sconf.Source: A. Statistical mechanics analysis of… · p. 9
  • Randomly pinning a fraction of particles reduces available configuration space while preserving global isotropy and spatial homogeneity.A sharp change with increasing pinned fraction appears consistent with an incipient phase transition, but lower-temperature studies and rigorous finite-size scaling remain needed to establish an equilibrium Kauzmann transition.Source: A. Statistical mechanics analysis of… · p. 9

B. Rheology of amorphous solids

Improved glass preparation enabled simulations to capture brittle yielding in stable glasses, rather than only the ductile behavior seen in earlier work. The review also describes how deformation protocols address timescale limits while leaving questions about the brittle-to-ductile transition and other deformation conditions open.

B. Rheology of amorphous solids

  • Atomic and molecular glasses are studied through linear elasticity, initial plasticity, and possible shear-band failure because they fracture under large deformations.Dense colloidal suspensions, by contrast, are suited to steady-state flow-curve studies at large deformation.Source: B. Rheology of amorphous solids · p. 10
  • Athermal quasi-static deformation addresses the timescale mismatch by following incremental strain steps with global energy minimization, but neglects thermal fluctuations.This makes the effective deformation rate zero.Source: B. Rheology of amorphous solids · p. 10
  • Stable glasses shift from ductile yielding to abrupt brittle failure, with finite-size scaling indicating failure in a single deformation step.The stress–strain relation is discontinuous in the thermodynamic limit, and the transition is compatible with a discontinuous nonequilibrium phase transition.Source: Figure 4 · p. 10, B. Rheology of amorphous solids · p. 10
  • Earlier work suggested the brittle-to-ductile change could be a second-order critical point, but finite-size concerns leave that picture unsettled.Further work with off-lattice and more coarse-grained models is proposed to clarify the transition [136].Source: B. Rheology of amorphous solids · pp. 10–11
  • Recent work examines periodic deformation and finite deformation rates, while thermal fluctuations and the microscopic origins of plastic events remain targets for further study.Source: B. Rheology of amorphous solids · p. 11

C. Vibrational and thermal properties of glasses

The review examines glass vibrations, low-temperature tunneling defects, and the predicted Gardner transition. Improved equilibration has revealed how vibrational and tunneling properties vary with glass stability, while several questions about low-energy excitations and the transition’s generality remain open.

C. Vibrational and thermal properties of glasses

  • Quasi-localized modes contribute D_ql(ω) ∼ ω^4, and better annealing makes their cores more localized while rapidly reducing the law’s prefactor.The extended-mode spectrum approaches Debye behavior at low frequency; the ultimate fate of these features as quench rates decrease remains open.Source: C. Vibrational and thermal properties… · p. 11, Figure 5 · p. 11
  • Near 1 K, glass specific heat is much larger than in the parent crystal and nearly linear rather than cubic in temperature; two-level-system theory attributes the excess excitations to tunneling defects.The excess contribution from quasi-localized modes does not account for the specific-heat data.Source: C. Vibrational and thermal properties… · pp. 11–12
  • SMC-enabled studies directly extracted many tunneling systems, found distributions consistent with two-level-system theory, and reported sharply lower densities with increased glass stability [162].Rare, highly collective tunneling events occur, especially in more poorly annealed systems.Source: Figure 5 · p. 11, C. Vibrational and thermal properties… · p. 12
  • Open problems include the SMC limits for studying two-level-system universality, a rigorous multidimensional treatment of tunneling, and quantitative sampling protocols aligned with experimental quenching.Earlier simulations also lacked an SMC-like approach and studied unrealistically prepared glasses, alongside small-system limitations.Source: C. Vibrational and thermal properties… · p. 12
  • The Gardner transition predicts two glassy states with distinct properties; signs are reported in three-dimensional hard spheres, a strong crossover in two-dimensional hard disks, and no apparent transition in Lennard-Jones systems.Its behavior in generic glass-formers remains unresolved.Source: C. Vibrational and thermal properties… · p. 12

IV. PERSPECTIVES FOR FUTURE RESEARCH

Future research must extend simulation methods to more complex glass formers and address unresolved thermodynamic and dynamical questions near the glass transition. The review also points to machine learning as a route to investigating how liquid structure relates to heterogeneous dynamics.

Machine learning

  • Machine-learning methods seek structural motifs and predictors of heterogeneous slow dynamics, with the goal of improving predictions and identifying useful structural descriptors.The review highlights potential for mechanistic understanding of how structures that resemble the high-temperature liquid encode heterogeneous dynamics; deformation-related plasticity is also heterogeneous in space and time.Source: §IV · p. 13

More complex glass formers

  • Monte Carlo approaches currently cannot simulate molecular glass-forming liquids, leaving results from simple point-particle models unconfirmed in more experimentally typical systems.Extended SMC could first be attempted for coarse-grained molecules or short polymeric chains.Source: §IV · p. 13

Thermodynamic tests

  • RFOT theory appears to describe thermodynamics in two- and three-dimensional glass formers, but the proposed Kauzmann transition remains an extrapolation.Equilibrating close enough to T_K to observe the entropy crisis may require smarter Monte Carlo moves, potentially combined with parallel tempering.Source: §IV · p. 13

Long-time dynamics

  • Long-time dynamics of deeply supercooled liquids in the experimental temperature regime remain largely unexplored, despite SMC enlarging the analyzable time window.Four-point S4(q, t) measurements require extensive ensemble averaging and large systems, while direct long-time dynamics remain difficult for highly annealed samples.Source: §IV · pp. 13–14
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