Juan P. Garrahan

Dynamics on the way to forming glass: bubbles in space-time.

We review a theoretical perspective of the dynamics of glass-forming liquids and the glass transition, a perspective developed during this past decade based on the structure of trajectory space. This structure emerges from spatial correlations of dynamics that appear in disordered systems as they approach nonergodic or jammed states. It is characterized in terms of dynamical heterogeneity, facilitation, and excitation lines. These features are associated with a newly discovered class of nonequilibrium phase transitions. Equilibrium properties have little, if anything, to do with it. The broken symmetries of these transitions are obscure or absent in spatial structures, but they are vivid in space-time (i.e., trajectory space). In our view, the glass transition is an example of this class of transitions. The basic ideas and principles we review were originally developed through the analysis of idealized and abstract models. Nevertheless, the central ideas are easily illustrated with reference to molecular dynamics of more realistic atomistic models, and we use that illustrative approach here.

Geometrical Explanation and Scaling of Dynamical Heterogeneities in Glass Forming Systems

We show how dynamical heterogeneities in glass forming systems emerge as a consequence of the existence of dynamical constraints, and we offer an interpretation of the glass transition as an entropy crisis in trajectory space (space-time) rather than in configuration space. To illustrate our general ideas, we analyze the one-dimensional (d = 1) Fredrickson-Andersen and East models. Dynamics of such dynamically constrained systems are shown to be isomorphic to the statics of ( d + 1)-dimensional dense mixtures of polydisperse noninterpenetrating domains. The domains coincide with arrested regions in trajectory space.

Dynamic Order-Disorder in Atomistic Models of Structural Glass Formers

The glass transition is the freezing of a liquid into a solid state without evident structural order. Although glassy materials are well characterized experimentally, the existence of a phase transition into the glass state remains controversial. Here, we present numerical evidence for the existence of a novel first-order dynamical phase transition in atomistic models of structural glass formers. In contrast to equilibrium phase transitions, which occur in configuration space, this transition occurs in trajectory space, and it is controlled by variables that drive the system out of equilibrium. Coexistence is established between an ergodic phase with finite relaxation time and a nonergodic phase of immobile molecular configurations. Thus, we connect the glass transition to a true phase transition, offering the possibility of a unified picture of glassy phenomena.

Coarse-grained microscopic model of glass formers

We introduce a coarse-grained model for atomic glass formers. Its elements are physically motivated local microscopic dynamical rules parameterized by observables. Results of the model are established and used to interpret the measured behaviors of supercooled fluids approaching glass transitions. The model predicts the presence of a crossover from hierarchical super-Arrhenius dynamics at short length scales to diffusive Arrhenius dynamics at large length scales. This prediction distinguishes our model from other theories of glass formers and can be tested by experiment.

Random Tiling and Topological Defects in a Two-Dimensional Molecular Network

A molecular network that exhibits critical correlations in the spatial order that is characteristic of a random, entropically stabilized, rhombus tiling is described. Specifically, we report a random tiling formed in a two-dimensional molecular network of p-terphenyl-3,5,3′,5′-tetracarboxylic acid adsorbed on graphite. The network is stabilized by hexagonal junctions of three, four, five, or six molecules and may be mapped onto a rhombus tiling in which an ordered array of vertices is embedded within a nonperiodic framework with spatial fluctuations in a local order characteristic of an entropically stabilized phase. We identified a topological defect that can propagate through the network, giving rise to a local reordering of molecular tiles and thus to transitions between quasi-degenerate local minima of a complex energy landscape. We draw parallels between the molecular tiling and dynamically arrested systems, such as glasses.

Corresponding States of Structural Glass Formers

The earlier paper of this same title demonstrated a collapse of relaxation data of fragile supercooled glass forming liquids [J. Phys. Chem. B 2009, 113, 5563-5567]. For temperature T below that of the onset to supercooled behavior, T(o), the logarithm of structural relaxation time, log τ, is given by the parabolic form log(τ/τ(o)) = J(2)(1/T - 1/T(o))(2), where J and τ(o) are temperature-independent. This paper presents further applications of this formula. In particular, it is shown that the effects of attractive forces in numerical simulation of glass-forming liquids can be logically organized in terms of J and T(o). Further, analysis of experimental data for several systems suggests that J and T(o) are material properties. In contrast, values of similar parameters for other fitting formulas are shown to depend not only upon the material but also upon the range of data used in fitting these formulas. Expressions demonstrated to fail in this way include the Vogel-Fulcher-Tammann formula, a double-exponential formula, and a fractional exponential formula.

Excitation lines and the breakdown of Stokes-Einstein relations in supercooled liquids

By applying the concept of dynamical facilitation and analyzing the excitation lines that result from this facilitation, we investigate the origin of decoupling of transport coefficients in supercooled liquids. We illustrate our approach with two classes of models. One depicts diffusion in a strong glass former, and the other in a fragile glass former. At low temperatures, both models exhibit violation of the Stokes-Einstein relation, D approximately tau(-1), where D is the self-diffusion constant and tau is the structural relaxation time. In the strong case, the violation is sensitive to dimensionality d, going as D approximately tau(-2/3) for d=1 and as D approximately tau(-0.95) for d=3. In the fragile case, however, we argue that dimensionality dependence is weak, and show that for d=1, D approximately tau(-0.73). This scaling for the fragile case compares favorably with the results of a recent experimental study for a three-dimensional fragile glass former.

Dynamical first-order phase transition in kinetically constrained models of glasses.

We show that the dynamics of kinetically constrained models of glass formers takes place at a first-order coexistence line between active and inactive dynamical phases. We prove this by computing the large-deviation functions of suitable space-time observables, such as the number of configuration changes in a trajectory. We present analytic results for dynamic facilitated models in a mean-field approximation, and numerical results for the Fredrickson-Andersen model, the East model, and constrained lattice gases, in various dimensions. This dynamical first-order transition is generic in kinetically constrained models, and we expect it to be present in systems with fully jammed states.

Thermal Model for Adaptive Competition in a Market

New continuous and stochastic extensions of the minority game, devised as a fundamental model for a market of competitive agents, are introduced and studied in the context of statistical physics. The new formulation reproduces the key features of the original model, without the need for some of its special assumptions and, most importantly, it demonstrates the crucial role of stochastic decision making. Furthermore, this formulation provides the exact but novel nonlinear equations for the dynamics of the system.

Length scale for the onset of Fickian diffusion in supercooled liquids

The interplay between self-diffusion and excitation lines in space-time was recently studied in kinetically constrained models to explain the breakdown of the Stokes-Einstein law in supercooled liquids. Here, we further examine this interplay and its manifestation in incoherent scattering functions. In particular, we establish a dynamic length scale below which Fickian diffusion breaks down, as is observed in experiments and simulations. We describe the temperature dependence of this length scale in liquids of various fragilities, and provide analytical estimates for the van Hove and self-intermediate scattering functions.