Facilitated spin models, mode coupling theory, and ergodic–nonergodic transitions

Ergodicity and mixing properties of the northeast model

The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at pc = 1 - βc, where βc is the critical parameter for oriented percolation. For p < pc, the trivial measure, δ0, that puts mass one on the configuration with all spins set at 0 is the unique ergodic, translation-invariant, stationary measure. For p ≥ pc, the product Bernoulli-p measure on configuration space is the unique nontrivial, ergodic, translation-invariant, stationary measure for the system, and it is mixing. For p > ⅔, it is shown that there is exponential decay of correlations.

Crossover from β to α Relaxation in Cooperative Facilitation Dynamics

β and α relaxation processes are dynamical scaling regimes of glassy systems occurring on two separate time scales which both diverge as the glass state is approached. We study here the crossover scaling from β to α relaxation in the cooperative facilitation scenario (CFS) and show that it is quantitatively described, with no adjustable parameter, by the leading order asymptotic formulas for scaling predicted by the mode-coupling theory (MCT). These results establish (i) the mutual universality of the MCT and CFS, and (ii) the existence of a purely dynamic realization of MCT, which is distinct from the well-established random first order transition scenario for disordered systems. Some implications of the emerging kinetic-static duality are discussed.

Dynamic facilitation picture of a higher-order glass singularity

We show that facilitated spin mixtures with a tunable facilitation reproduce, on a Bethe lattice, the simplest higher-order singularity scenario predicted by the mode-coupling theory (MCT) of liquid-glass transition. Depending on the facilitation strength, they yield either a discontinuous glass transition or a continuous one, with no underlying thermodynamic singularity. Similar results are obtained for facilitated spin models on a diluted Bethe lattice. The mechanism of dynamical arrest in these systems can be interpreted in terms of bootstrap and standard percolation and corresponds to a crossover from a compact to a fractal structure of the incipient spanning cluster of frozen spins. Theoretical and numerical simulation results are fully consistent with MCT predictions.

Irreducible memory function and slow dynamics in disordered systems

We show how the irreducible memory function can be obtained in a rather straightforward way, and that it can be expressed in terms of two contributions representing two parallel decay channels. This representation should be useful for treating systems with a slow time dependence and where eventually some internal degrees of freedom enters in the relaxation process, and cuts off an underlying ideal ergodic to nonergodic transition. We also show how the irreducible memory function under certain mild conditions defines a regenerative stochastic process, or a two level stochastic system. This leads to a picture with dynamical heterogeneities, where the statistical properties asymptotically are ruled by limit processes. This can explain the universal behavior observed in many glass-forming systems.

Ergodicity and mixing properties of the northeast model

The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at p c = 1 - βc, where βc is the critical parameter for oriented percolation. For p &lt; p c, the trivial measure, δ0, that puts mass one on the configuration with all spins set at 0 is the unique ergodic, translation-invariant, stationary measure. For p ≥ p c, the product Bernoulli-p measure on configuration space is the unique nontrivial, ergodic, translation-invariant, stationary measure for the system, and it is mixing. For p &gt; ⅔, it is shown that there is exponential decay of correlations.

Mode-Coupling Theory for Multiple-Time Correlation Functions of Tagged Particle Densities and Dynamical Filters Designed for Glassy Systems

The theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system developed by Van Zon and Schofield [Phys. Rev. E 2002, 65, 011106] is extended here to include tagged particle densities. Such densities have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques which are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite basis composed of the products of slow variables of the system. The resulting formally exact mode-coupling expressions for multiple-point and multiple-time correlation functions are made tractable by applying the so-called N-ordering method. This theory is used to derive for moderate densities the leading mode coupling expressions for indicators of relaxation type and domain relaxation, which use dynamical filters that lead to multiple-time correlations of a tagged particle density. The mode coupling expressions for higher order correlation functions are also successfully tested against simulations of a hard sphere fluid at relatively low density.

Two-Gaussian excitations model for the glass transition

We develop a modified "two-state" model with Gaussian widths for the site energies of both ground and excited states, consistent with expectations for a disordered system. The thermodynamic properties of the system are analyzed in configuration space and found to bridge the gap between simple two-state models ("logarithmic" model in configuration space) and the random energy model ("Gaussian" model in configuration space). The Kauzmann singularity given by the random energy model remains for very fragile liquids but is suppressed or eliminated for stronger liquids. The sharp form of constant-volume heat capacity found by recent simulations for binary mixed Lennard-Jones and soft-sphere systems is reproduced by the model, as is the excess entropy and heat capacity of a variety of laboratory systems, strong and fragile. The ideal glass in all cases has a narrow Gaussian, almost invariant among molecular and atomic glassformers, while the excited-state Gaussian depends on the system and its width plays a role in the thermodynamic fragility. The model predicts the possibility of first-order phase transitions for fragile liquids. The analysis of laboratory data for toluene and o-terphenyl indicates that fragile liquids resolve the Kauzmann paradox by a first-order transition from supercooled liquid to ideal-glass state at a temperature between T(g) and Kauzmann temperature extrapolated from experimental data. We stress the importance of the temperature dependence of the energy landscape, predicted by the fluctuation-dissipation theorem, in analyzing the liquid thermodynamics.

Glassy dynamics and domains: explicit results for the East model

A general matrix-based scheme for analyzing the long-time dynamics in kinetically constrained models such as the East model is presented. The treatment developed here is motivated by the expectation that slowly relaxing spin domains of arbitrary size govern the highly cooperative events that lead to spin relaxation at long times. To account for the role of large spin domains in the dynamics, a complete basis expressed in terms of domains of all sizes is introduced. It is first demonstrated that accounting for single domains of all possible sizes leads to a simple analytical result for the two-time single-spin correlation function in the East model that is in excellent quantitative agreement with simulation data for equilibrium spin-up density values c > or = 0.6. It is then shown that including also two neighboring domains leads to a closed expression that describes the slow relaxation of the system down to c approximately 0.3. Ingredients of generalizing the method to lower values of c are also provided, as well as to other models. The main advantage of this approach is that it gives explicit analytical results and that it requires neither an arbitrary closure for the memory kernel nor the construction of an irreducible memory kernel. It also allows one to calculate quantities that measure heterogeneity in the same framework, as is illustrated on the neighbor-pair correlation function, the average relaxation time, and the width of the distribution of relaxation times.

Is a "homogeneous" description of dynamic heterogeneities possible?

We study the simplest model of dynamic heterogeneities in glass forming liquids: one-spin facilitated kinetic Ising model introduced by Fredrickson and Andersen [G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. 53, 1244 (1984); J. Chem. Phys. 83, 5822 (1985)]. We show that the low-temperature, long-time behavior of the density autocorrelation function predicted by a scaling approach can be obtained from a self-consistent mode-couplinglike approximation.

Diagrammatic kinetic theory for a lattice model of a liquid. II. Comparison of theory and simulation results

We compare the predictions of the mean field, the two site multiple scattering, and the simple mode coupling approximation developed in the previous paper for the dynamics of a tagged particle in an excluded volume lattice gas with the results of computer simulations. The tagged particle has a transition rate of gamma while the background particles have transition rates of alphagamma. We consider the tracer diffusion coefficient and the incoherent intermediate scattering function (IISF) for low, intermediate, and high concentrations of particles and for simple square and cubic lattices. In general, the approximate kinetic theories are more accurate in predicting simulations results at low concentrations, high dimensions, and large alpha. For the tracer diffusion coefficient, the mean field approximation is the least accurate, the two site multiple scattering approximation is more accurate, and the simple mode coupling approximation is the most accurate; all three approximate theories overestimate the simulation results. For the IISF, the mean field approximation is quantitatively accurate in the limit of small concentration and large alpha but in general decays too quickly. The two site multiple scattering approximation is quantitatively accurate at low and intermediate concentrations for large wave vectors; it is always more accurate than the mean field approximation and always decays more quickly than the simulation results. The simple mode coupling approximation is the most accurate of the three approximations in most cases and especially so for small wave vectors, high concentration, and small alpha; unfortunately, its predictions are not quantitatively accurate in these highly nonmean field regimes. We discuss the implications of these results for developing diagrammatic kinetic theories.

Glassy dynamics in the asymmetrically constrained kinetic Ising chain

We study the dynamics of the East model, comprising a chain of uncoupled spins in a downward-pointing field. Glassy effects arise at low temperatures T from the kinetic constraint that spins can only flip if their left neighbor is up. We give details of our previous solution of the nonequilibrium coarsening dynamics after a quench to low T [Phys. Rev. Lett. 83, 3238 (1999)], including the anomalous coarsening of down-spin domains with typical size d approximately t(T ln 2), and the pronounced "fragile glass" divergence of equilibration times as t(*)=exp(1/T(2) ln 2). We also link the model to the paste-all coarsening model, defining a family of interpolating models that all have the same scaling distribution of domain sizes. We then proceed to the problem of equilibrium dynamics at low T. Based on a scaling hypothesis for the relation between time scales and length scales, we propose a model for the dynamics of "superdomains" which are bounded by up-spins that are frozen on long time scales. From this we deduce that the equilibrium spin correlation and persistence functions should exhibit identical scaling behavior for low T, decaying as g(t). The scaling variable is t=(t/t(*))(T ln 2), giving strongly stretched behavior for low T. The scaling function g(.) decays faster than exponential, however, and in the limit T-->0 at fixed t reaches zero at a finite value of t.