Table of contents

This special issue of Journal of Physics: Condensed Matter contains the proceedings of the workshop `Unifying Concepts in Glass Physics', 15-18 September 1999 at the Abdus Salam International Center for Theoretical Physics, Trieste, Italy. As the title suggests, the goal was to bring together active scientists working on elucidating theoretical concepts common to systems exhibiting glassy behaviour. The past decade has seen a renaissance of sorts in the field of glass physics, due to the advancement and development of new theoretical ideas and experimental findings made possible by new or refined techniques and especially to the use of computer simulations in testing theories and providing insight into the nature of glass-forming systems. This workshop provided theoreticians with an opportunity to debate many of the currently invoked theoretical concepts, and discuss their similarities and differences. Although the major emphasis was on phenomena related to structural glass formers and the phenomenology of glassy dynamics and the glass transition in supercooled liquids, many contributions concerned related problems in spin systems, granular materials, etc. Among the themes represented were the role of energy landscapes of glass formers, thermodynamic aspects of glassy behaviour, aging, mode coupling theory and spatially heterogeneous dynamics.

Participants were asked to submit preliminary contributions prior to the workshop, and these manuscripts, together with comments by co-participants, are available on the web athttp://www.ictp.trieste.it/~franz/workshop.html. Ample time was provided in each session for open discussion. The many participants who attended the workshop represented most of the dominant themes of research in the physics of glassy systems.

While no workshop proceedings can be truly comprehensive, the peer-reviewed contributions in this volume together provide a representative survey of the state of the art in current theoretical concepts regarding glassy behaviour. We have insisted as much as possible on clarity of exposition, in addition to rigorous standards of scientific merit. We hope it will be a useful compendium both for people who seek an introduction to this exciting area of research and those who wish to keep abreast of the latest developments.

Finally, we would like to thank the Abdus Salam International Center for Theoretical Physics, which financed and hosted the workshop; in particular, the director of the center Professor Miguel Virasoro, and Ms Elizabeth Brancaccio, who attended to the numerous details of organizing the workshop. We would like to thank the Center for Theoretical and Computational Materials Science at the National Institute of Standards and Technology, USA, for financial support for these proceedings. We wish to thank Jason Wilde of IOP Publishing for reviewing and accepting our proposal to publish the proceedings inJournal of Physics: Condensed Matter, and Sharon D'Souza and Jacky Mucklow at IOP Publishing for assisting us on editorial and production matters. Most of all, we thank the participants of the workshop for a lively and stimulating discourse on theoretical concepts in glass physics.

Silvio Franz (ICTP, Trieste, Italy)

Sharon C Glotzer (NIST, Gaithersburg, USA)

Srikanth Sastry (JNCASR, Bangalore, India)

In the context of the instantaneous normal mode approach, the spectrum of the Hessian of Hamiltonian is a key quantity to describe a liquids behaviour. The determination of the spectrum represents a major task for theoretical studies, and has recently been addressed in various works. In this work a new approach for the analytic computation of the Hessian spectrum is presented. The one-dimensional case for a system of particles interacting via a purely repulsive potential at low density is analysed in detail and, also in the localized sector, the spectrum is computed exactly. Finally the possible extensions of the method are discussed, together with a comparison with different approaches to the problem.

It is now well established that molecular dynamics simulations realistically capture the onset of glassy behaviour in simple liquids. Despite the extensive information with which these models can, provide us concerning the cooperative fluctuations responsible for the special properties of glass-forming liquids, the literature remains dominated by the same speculative models that characterized the field 20 years ago. The difficulty in exploiting simulations lies in articulating questions concerning cooperative behaviour in forms explicit enough to be addressed by the calculations. This difficulty will be discussed in the context of dynamic heterogeneities, subdiffusive behaviour and anomalous heat capacities in simulations of a glass-forming mixture.

We briefly review the molecular mode-coupling theory which takes into account both the translational and the orientational degrees of freedom. Depending on the symmetry of the physical system, several scenarios of ideal glass transition from ergodic to nonergodic behaviour may occur. The validity of the scaling laws for the α- and β-relaxation, originally derived from mode-coupling theory for simple liquids, will be discussed. These predictions will be compared with results from molecular dynamics simulations. We will demonstrate which of the predictions are consistent with these numerical results and which are not. Particular attention will be given to the rotational diffusion tensor.

We first review our recent simulation work on dynamic heterogeneity and supercooled liquid rheology. We then treat a supercooled polymer melt to study the stress relaxation function, transient stress evolution, shear thinning, and elongation of chains.

In this paper, we discuss theoretically the behaviour of the four-point non-linear susceptibility and its associated correlation length for supercooled liquids close to the mode-coupling instability temperature T_c. We work in the theoretical framework of the glass transition as described by mean-field theory of disordered systems, and the hypernetted-chain approximation. Our results give an interpretation framework for recent numerical findings on heterogeneities in supercooled liquid dynamics.

The traditional strategy for dealing with slow dynamics is summarized. Direct application of this strategy to supercooled liquids and glasses gives rise to problems. Two types of reduced description, namely, microcanonical and canonical versions of dynamical density functional theories, are explained. The non-equilibrium projector technique is used to develop this canonical reduced description. Some speculative proposals as regards how to deal with the barrier crossing regime are outlined.

A formally exact set of equations is derived for the description of non-equilibrium phenomena exhibited by classical liquids and glasses. With the help of a non-equilibrium projection operator formalism, the correlation functions and fluctuation propagators are expressed in terms of memory functions and time-dependent collective frequencies. This formally exact set of equations is approximated by applying mode-coupling approximations to the memory functions. The resulting set of equations for wave-vector-dependent correlation functions, fluctuation propagators and one-time structure factors S_q(t) generalizes the well known mode-coupling theory of the glass transition to situations far away from equilibrium.

This paper reports results of molecular dynamics simulations for a glassy polymer melt consisting of short, non-entangled chains. The temperature region studied covers the supercooled state of the melt above the mode-coupling critical temperature. The analysis focuses on the interplay of simple-liquid and polymer-specific effects. One can clearly distinguish two regimes: a regime of small and one of large monomer displacements. The first regime corresponds to motion of a monomer in its local environment. It is dominated by the cage effect and well described by the idealized mode-coupling theory. The second regime is governed by the late-β/early-α process. In this regime the connectivity of the monomers begins to interfere with the cage dynamics and finally becomes dominant. The monomer displacement is compared with simulation results for a binary Lennard-Jones mixture to highlight the differences which are introduced by the connectivity of the particles.

In this paper we give the results of our recent work on the dynamical TAP (Thouless-Anderson-Palmer) approach to mean-field glassy systems. Our aim is to clarify the connection between the free-energy landscape and the out-of-equilibrium dynamics in solvable models.

Frequently, qualitative explanations of glassy behaviour are based on the `free-energy landscape paradigm'. While its relevance for equilibrium properties may be clear, the relationship between the free-energy landscape and out-of-equilibrium dynamics is not well understood yet. In this paper we clarify this relationship for the class of spin-glass models which reproduce phenomenologically some features of structural glasses. The method that we use is a generalization of the dynamics of the Thouless, Anderson and Palmer approach to the thermodynamics of mean-field spin glasses. Within this framework, we show to what extent the dynamics can be represented as an evolution in the free-energy landscape. In particular, the relationship between the long-time dynamics and the local properties of the free-energy landscape is explicitly apparent using this approach.

Using molecular dynamics computer simulations we investigate the out-of-equilibrium dynamics of a Lennard-Jones system after a quench from a high temperature to one below the glass transition temperature. By studying the radial distribution function we obtain evidence that during the aging the system is very close to the critical surface of mode-coupling theory. Furthermore, we show that two-time correlation functions show a strong dependence on the waiting time elapsed since the quench and that their shape is very different from the one in equilibrium. By investigating the temperature and time dependence of the frequency distribution of the normal modes we show that the energy of the inherent structures can be used to define an effective (time-dependent) temperature of the aging system.

We report on a refined version of our spin-glass-type approach to the low-temperature physics of structural glasses. Its key idea is based on a Born-von Karman expansion of the interaction potential about a set of reference positions in which glassy aspects are modelled by taking the harmonic contribution within this expansion to be random. Within the present refined version the expansion at the harmonic level is reorganized so as to respect the principle of global translational invariance. By implementing this principle, we have for the first time a mechanism that fixes the distribution of the parameters characterizing the local potential energy configurations responsible for glassy low-temperature anomalies solely in terms of assumptions about interactions at a microscopic level.

Long-time (~1 ns region) molecular dynamics (MD) simulations of lithium metasilicate (Li₂SiO₃) and a mixed-alkali silicate (LiKSiO₃) glass have been performed to confirm the mechanism of the `mixed-alkali effect'. The motion of lithium ions in lithium metasilicate (Li₂SiO₃) glass is divided into slow (A) and fast (B) categories in the glassy state. The waiting time distribution of the jump motion of each component shows a power-law behaviour with different exponents. The slow dynamics is caused by localized jump motions and by the long waiting time. On the other hand, the fast dynamics of the lithium ions in Li₂SiO₃ is characterized as Lévy flight caused by cooperative jumps. Short intervals between jump events also occur in fast dynamics, in the short-time region. The main diffusion and conduction processes of silicate glasses are not the single jumps but the cooperative jumps. A component with accelerated dynamics is almost absent in the mixed-alkali system. The contributions of the temporal and spatial aspects of the particle dynamics are separated. A large change in the spatial parameters has been observed on mixing, and interception of the jump path by other kinds of ion path suppresses the cooperative jump process. On the other hand, in longer-time regions, motion of the framework has been found to accompany the small number of events where alkali metal ions jump to unlike-ion sites. Thus both loosening of the glass structure and a reduction of ionic diffusion coefficients occur in the mixed-alkali system.

We analyse the properties of the energy landscape of finite-size fully connected p-spin-like models. In the thermodynamic limit the high-temperature phase is described by the schematic mode-coupling theory of supercooled liquids. In this limit, the barriers between different basins are infinite below the critical dynamical temperature at which the ergodicity is broken in infinite time. We show that finite-size fully connected p-spin-like models, where activated processes are possible, exhibit properties typical of real supercooled liquid when both are near the critical glass transition. Our results support the conclusion that fully connected p-spin-like models are the natural statistical mechanical models for studying the glass transition in supercooled liquids.

Liquid can sustain shear waves up to a minimum wavenumber, indicating solid-like behaviour over short length scales. With the increase of density this critical wavenumber decreases, indicating a growth of a dynamic length scale. The speed of the propagating shear waves goes to zero approaching a critical wavenumber. The maximum wavelength shows an initial enhancement approaching the mode-coupling transition and finally grows at a slower rate as the sharp transition is cut off. The growth of this dynamic length scale in the supercooled regime is studied using the transverse correlation function and by computing the feedback effects of dynamic correlation in an extended mode-coupling model where the structure factor of the liquid is used as an input.

We propose a model for reorientational motions of molecules associated with secondary beta-relaxation in supercooled liquids. The secondary relaxation is attributed to relaxation within a given local minimum, while the primary relaxation is attributed to transitions between distinct free-energy minima. We find that (i) at the temperature where the peak frequency of the extrapolated beta-relaxation intersects the alpha-relaxation, the actual and the extrapolated spectra differ in their time constants by approximately one decade; (ii) there is no clear division between the imaginary part of the dielectric susceptibility for the alpha- and the beta-relaxation for temperatures larger than 1.1 T_g. Thus, one must proceed with caution to extrapolate low temperature data of beta-relaxation to higher temperatures in order to estimate the temperature at which the time scales for the two processes cross. The relaxation times for the alpha- and the beta-processes cannot cross except at high temperature, where only the primary relaxation remains.

The coupling model, based on fundamental properties of nonintegrable Hamiltonian systems (chaos), captures one important aspect of the many-body relaxation dynamics, namely the existence of a temperature insensitive crossover time, t_c, when in its neighbourhood the fast independent relaxation of an individual unit at earlier times is slowed down to become the cooperative and dynamically heterogeneous relaxation at longer times. For molecular systems, t_c is of the order of a picosecond and thus the coupling model bridges relaxation dynamics at microscopic times to that at macroscopic times. One result, which relates the slowed-down cooperative relaxation time to the independent relaxation time of an individual unit, has spawned many applications. A nonlinearly coupled arrays of oscillators model, which has all the features of the coupling model, is further exploited to show that the strength of intermolecular interaction determines the nonexponentiality and fragility of the long-time dynamics of glass-formers. A related model is invoked to consider the relaxation of the (precursor) vibration attempting the structural relaxation. The fast relaxation that originates from such vibrational relaxation is shown to be a plausible cause of the susceptibility minimum seen in dielectric relaxation, light and neutron scattering experiments. The model predicts that the degree of nonexponentiality of the structural relaxation is correlated with the strength of the fast precursor vibrational relaxation, in agreement with experimental observation.

We propose damage spreading (DS) as a tool for investigating the topological features related to the ruggedness of the free-energy landscape. We argue that DS measures the positiveness of the largest Lyapunov exponent associated with the basins of attraction visited by the system during its dynamical evolution. We discuss recent results obtained in the framework of mode-coupling theory and comment on possible extensions to the study of realistic glasses. Preliminary results are presented for purely repulsive soft-sphere glasses.

We formulate ten questions, covering outstanding aspects of the phenomenology of glassforming liquids, which we believe must be properly answered by any successful theory of structural glassformers. The questions range across thermodynamic, mass transport and vibrational dynamics phenomena. While these questions will only be addressed properly by a collective variables approach (many aspects of which are reported in these proceedings) a number of them can be dealt with by use of simple physical models of appropriate form. Here we discuss one such model in which the existence of elementary configurational excitations of the amorphous quasilattice is proposed. These states, which may range from broken bonds to packing defects, can be excited independently in the majority of cases, or cooperatively in others. We summarize essential results of this model. These suggest that the source of the different fragilities in liquids (and the reason that structural glasses, alone among `glassy' systems, have marked heat capacity jumps at T_g) may lie largely in the way these configurational excitations couple to the vibrational modes of the system. The generation of low frequency modes in the density of vibrational states, as a direct consequence of the excitation of configurational states, explains why the quasi-elastic scattering from fragile liquids is so much stronger near and above T_g than in the case of strong liquids, and why the normal glass transition can be detected in picosecond time scale experiments.

Interactions among the `excitations', or `defects', are taken into account using the one component system equivalent of the binary system `regular solution' model (which keeps only the first order term of the free energy of mixing expansion). We show that a liquid-liquid first order transition must occur at sufficiently strong defect-defect interactions. The highly overconstrained amorphous silicon quasilattice is a strong candidate for such a transition. We identify the `first order melting' of amorphous silicon, and the sudden, reproducible, termination of supercooling in experimental liquid silicon and germanium, with the phase transition predicted by the model. Many more cases of this phase transition may be anticipated, and a corresponding range of glasses with low residual entropies - approaching the `perfect' glass state - are predicted.

Kinetic lattice-gas models display fragile-glass behaviour, in spite of their trivial Gibbs-Boltzmann measure. This suggests that the nature of glass transition might be, at least in some cases, understood in purely kinetic or dynamical terms.

In an effort to understand the glass transition, the kinetics of a spin model with frustration but no quenched randomness has been analysed. The phenomenology of the spin model is remarkably similar to that of structural glasses. Analysis of the model suggests that defects play a major role in dictating the dynamics as the glass transition is approached.

We propose that the salient feature to be explained about the glass transition of supercooled liquids is the temperature-controlled super-Arrhenius activated nature of the viscous slowing down, more strikingly seen in weakly bonded, fragile systems. In the light of this observation, the relevance of simple models of spherically interacting particles and that of models based on free-volume congested dynamics are questioned. Finally, we discuss how the main aspects of the phenomenology of supercooled liquids, including the crossover from Arrhenius to super-Arrhenius activated behaviour and the heterogeneous character of the α-relaxation, can be described by an approach based on frustration-limited domains.

A general framework is given for deriving the jump rate distribution function and the long-time behaviour of the waiting time distribution for the jump is shown to be a power-law function whose exponent is related to the contribution of fast modes relative to that of slow modes in the density fluctuation. A unified explanation for the vitrification process and glass transition singularities is presented.

Computer simulations have been employed in recent years to evaluate the configurational entropy changes in model glass-forming liquids. We consider two methods, both of which involve the calculation of the `intra-basin' entropy as a means for obtaining the configurational entropy. The first method involves the evaluation of the intra-basin entropy from the vibrational frequencies of inherent structures, by making a harmonic approximation to the local potential energy topography. The second method employs simulations that confine the liquid within a localized region of configuration space by the imposition of constraints; apart from the choice of the constraints, no further assumptions are made. We compare the configurational entropies estimated for a model liquid (binary mixture of particles interacting via the Lennard-Jones potential) for a range of temperatures, at fixed density.

In this article we review the thermodynamics of liquids in the framework of the inherent-structure formalism. We then present calculations of the distribution of the basins in the potential energy of a binary Lennard-Jones mixture as a function of temperature. The comparison between the numerical data and the theoretical formalism allows us to evaluate the degeneracy of the inherent structures in a bulk system and to estimate the energy of the lowest-energy disordered state (which we define as the Kauzmann energy). We find that, around the mode-coupling temperature, the partition function of the liquid is approximated well by the product of two loosely coupled partition functions, one depending on the inherent-structure quantities (depths of the basins and their degeneracy) and one describing the free energy of the liquid constrained in one typical basin.

The thermodynamics of glass-forming systems can be expressed in terms of the density of inherent structures, which correspond to the minima of the potential energy landscape. In previous work this approach has been applied to Lennard-Jones-type systems, yielding a density of inherent structures which to a very good approximation turned out to be Gaussian. In this work we clarify whether the Gaussian distribution is just a consequence of the central-limit theorem or whether it also contains information about the local structure of the glass-forming system.

A picture for the thermodynamics of the glassy state is introduced. It assumes that one extra parameter, the effective temperature, is needed to describe the glassy state. This explains the classical paradoxes concerning the Ehrenfest relations and the Prigogine-Defay ratio. As a second feature, the approach connects the response of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized non-equilibrium way.

Properties of the free-energy landscape in phase space of a dense hard-sphere system characterized by a discretized free-energy functional of the Ramakrishnan-Yussouff form are investigated numerically. A considerable number of glassy local minima of the free energy are located and the distribution of an appropriately defined `overlap' between minima is calculated. The process of transition from the basin of attraction of a minimum to that of another one is studied using a new `microcanonical' Monte Carlo procedure, leading to a determination of the effective height of free-energy barriers that separate different glassy minima. The general appearance of the free-energy landscape resembles that of a putting green: deep minima separated by a fairly flat structure. The growth of the effective free-energy barriers with increasing density is consistent with the Vogel-Fulcher law, and this growth is primarily driven by an entropic mechanism.

For a binary liquid mixture of hard discs well above the freezing density of the one-component liquid the size dependence of the self-diffusion coefficients of small and large particles is calculated by molecular dynamics simulation. The observed size effect can be satisfactorily explained by the size-dependent cut-off of long-wavelength fluctuations of transverse collective flow and gives no indication of the importance of more `cooperative' processes at high densities.

Within the framework of the mode coupling theory (MCT) of structural relaxation, mechanisms and properties of non-ergodicity transitions in rather dilute suspensions of colloidal particles characterized by strong short-ranged attractions are studied. Results building on the virial expansion for particles with hard cores and interacting via an attractive square-well potential are presented, and their relevance to colloidal gelation is discussed.

In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this random solid state, particles are permanently but randomly localized in space, and a rigidity to shear deformations emerges. Owing to the permanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the conventional glass transition) in the sense that it is amenable to treatment by established techniques of equilibrium statistical mechanics. In this paper I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities of and differences between such systems and those exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correlations and susceptibilities that signal the approach of the random solidification transition, discussing the physical interpretation and values of these quantities both at the Gaussian level of approximation and, via a renormalization-group approach, beyond.

A statistical mechanical approach to granular material is proposed. Using lattice models from standard statistical mechanics and results from a mean-field replica approach we find a jamming transition in granular media closely related to the glass transition in supercooled liquids. These models reproduce the logarithmic relaxation in granular compaction and reversible-irreversible lines, in agreement with experimental data. The models also exhibit aging effects and breakdown of the usual fluctuation-dissipation relation. It is shown that the glass transition may be responsible for the logarithmic relaxation and may be related to the cooperative effects underlying many phenomena exhibited by granular materials such as the Reynolds transition.

A scenario for systems with slow dynamics is characterized by stating that there are several temperatures coexisting for the sample, with a single temperature shared by all observables at each (widely separate) timescale.

In preparation for the study of granular rheology, we show within this framework that glassy systems with driving and friction that are generic and do not correspond to a thermal bath - and whose microscopic `fast' motion is hence not thermal - have a well-defined macroscopic temperature associated with the slow degrees of freedom.

This temperature is what a thermometer coupled to the system will measure if tuned to respond to low frequencies, and, since it can be related to the number of stationary configurations, it is the formalization of Edwards' `compactivity' ideas.

Two models are presented for studying the influence of slow dynamics on granular compaction. It is found in both cases that high values of packing fraction are achieved only by the slow relaxation of cooperative structures. Ongoing work investigating the full implications of these issues is discussed.

Certain simple models of structural glasses (Cugliandolo L F, Kurchan J, Parisi G and Ritort F 1995 Phys. Rev. Lett. 74 1012, Parisi G 1997 Statistical properties of random matrices and the replica method Preprint cond-mat/9701032) map onto random-matrix models. These random-matrix models have gaps in their eigenvalue distributions. It turns out that matrix models with gaps in their eigenvalue distributions have the unusual property of multiple solutions or minima of the free energy at the same point in phase space. I present evidence for the presence of multiple solutions in these models both analytically and numerically. The multiple solutions have different free energies and observable correlation functions, the differences arising at higher order in {1/N}. The system can get trapped into different minima depending upon the path traversed in phase space to reach a particular point. The thermodynamic limit also depends upon the sequence by which N is taken to infinity (e.g. odd or even N), which is reminiscent of the structure discussed for another model for glasses (Marinari E, Parisi G and Ritort F 1994 J. Phys. A: Math. Gen. 27 7615). Hence it would be of interest to study the landscape of these multiple solutions and determine whether it corresponds to a supercooled liquid or glass.

We study the distribution of attraction basins as a function of energy in simple glasses. We find that it is always broad. Furthermore, we identify two types of glass, both with an exponentially large number of metastable states. In one type the largest attraction basin is exponentially small, whereas in the other it is polynomially small in the system size N. If there exists a tuning parameter that connects one regime with another, then these two phases are separated by a critical point. We discuss implications for optimization problems.

This paper gives an introduction to and brief overview of some of our recent work on the equilibrium thermodynamics of glasses. We have focused on first-principles computations for simple fragile glasses, starting from the two-body interatomic potential. A replica formulation translates this problem into that of a gas of interacting molecules, each molecule being built of m atoms, and having a gyration radius (related to the cage size) which vanishes at zero temperature. We use a small-cage expansion, valid at low temperatures, which allows us to compute the cage size, the specific heat (which follows the Dulong and Petit law), and the configurational entropy. The no-replica interpretation of the computations is also briefly described. The results, particularly those concerning the Kauzmann temperature and the configurational entropy, are compared to recent numerical simulations.

We calculate `inherent structures' (configurations corresponding to local potential energy minima) in the ±J Ising spin glass over a wide range of temperature T in two and three dimensions. We find that the T-dependence of the average value of the inherent-structure energy E is strikingly similar to that shown recently for a glass-forming liquids. E decreases with T only weakly at high T, but begins to decrease much more rapidly as the spin-glass transition temperature T<sub>sg</sub> is approached. As in the liquid, we find that the rapid decrease of E with decreasing T occurs in the regime of T in which the relaxation of the spin autocorrelation function becomes increasingly non-exponential. In addition, we show that the inherent structures of the spin glass can be used to identify clusters of strongly correlated `frozen' spins, and that an incipient percolating cluster of these spins appears (within numerical error) at T = T_sg.