Constrained Lattice Gas Research Articles

Self-diffusion coefficient in the Kob-Andersen model

The Kob-Andersen model is a fundamental example of a kinetically constrained lattice gas, that is, an interacting particle system with Kawasaki type dynamics and kinetic constraints. In this model, a particle is allowed to jump when sufficiently many neighboring sites are empty. We study the motion of a single tagged particle and in particular its convergence to a Brownian motion. Previous results showed that the path of this particle indeed converges in diffusive time-scale, and the purpose of this paper is to study the rate of decay of the self-diffusion coefficient for large densities. We find upper and lower bounds matching to leading behavior.

Constraint relaxation leads to jamming.

Adding transitions to an equilibrium system increases the activity. Naively, one would expect this to hold also in out-of-equilibrium systems. We demonstrate, using relatively simple models, how adding transitions to an out of equilibrium system may in fact reduce the activity and even cause it to vanish. This surprising effect is caused by adding heretofore forbidden transitions into less and less active states. We investigate six related kinetically constrained lattice gas models, some of which behave as naively expected while others exhibit this nonintuitive behavior. These models exhibit an absorbing state phase transition, which is also affected by the added transitions. We introduce a semi-mean-field approximation describing the models, which agrees qualitatively with our numerical simulation.

Quantum Slow Relaxation and Metastability due to Dynamical Constraints.

One of the general mechanisms that give rise to the slow cooperative relaxation characteristic of classical glasses is the presence of kinetic constraints in the dynamics. Here we show that dynamical constraints can similarly lead to slow thermalization and metastability in translationally invariant quantum many-body systems. We illustrate this general idea by considering two simple models: (i)a one-dimensional quantum analogue to classical constrained lattice gases where excitation hopping is constrained by the state of neighboring sites, mimicking excluded-volume interactions of dense fluids; and (ii)fully packed quantum dimers on the square lattice. Both models have a Rokhsar-Kivelson (RK) point at which kinetic and potential energy constants are equal. To one side of the RK point, where kinetic energy dominates, thermalization is fast. To the other, where potential energy dominates, thermalization is slow, memory of initial conditions persists for long times, and separation of timescales leads to pronounced metastability before eventual thermalization. Furthermore, in analogy with what occurs in the relaxation of classical glasses, the slow-thermalization regime displays dynamical heterogeneity as manifested by spatially segregated growth of entanglement.

Bulk diffusion in a kinetically constrained lattice gas

In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient encapsulating all microscopic details of the dynamics. This diffusion coefficient is, in principle, determined by a Green–Kubo formula. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be computed except when a lattice gas additionally satisfies the gradient condition. We develop a procedure to systematically obtain analytical approximations for the diffusion coefficient for non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn which is a version of the Green–Kubo formula particularly suitable for diffusive lattice gases. Restricting the variational formula to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We apply this approach to a kinetically constrained non-gradient lattice gas in two dimensions, viz. to the Kob–Andersen model on the square lattice.

Kinetically constrained lattice gases: Tagged particle diffusion

Les gaz reticulaires avec contrainte cinetique (KCLG) sont des systemes de particules en interaction sur le reseau $\mathbb{Z}^{d}$ avec au plus une particule par site et une dynamique de type Kawasaki. Leur particularite est que les sauts ne sont autorises que si la configuration satisfait une contrainte exigeant suffisamment de sites vides dans un certain voisinage local. Les KCLG ont ete introduits et massivement etudies dans la litterature physique comme modeles pour les systemes vitreux. Nous nous concentrons sur l’une des classes de KCLG les plus etudiees : les modeles de Kob–Andersen (KA). Nous analysons le comportement d’un traceur (c.-a-d. une particule marquee) a l’equilibre. Pour toute dimension $d\geq2$ et toute densite de particules, nous montrons qu’a l’echelle diffusive la trajectoire du traceur converge vers un mouvement brownien $d$-dimensionnel avec matrice de diffusion non degeneree. Par consequent nous demontrons la non-existence d’une transition (qui avait ete conjecturee dans la litterature physique) entre un regime diffusif et un regime non diffusif. Notre technique est assez flexible et peut etre etendue pour analyser le comportement du traceur sous d’autres choix de contraintes.

Kinetically Constrained Lattice Gases

Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularly in the so-called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L2) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the density-density time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints.

Relationship between vibrations and dynamical heterogeneity in a model glass former: Extended soft modes but local relaxation

We study the relation between short-time vibrational modes and long-time relaxational dynamics in a kinetically constrained lattice gas with harmonic interactions between neighbouring particles. We find a correlation between the location of the low- (high-) frequency vibrational modes and regions of high (low) propensity for motion. This is similar to what was observed in continuous force systems, but our interpretation is different: in our case relaxation is due to localised excitations which propagate through the system; these localised excitations act as background disorder for the elastic network, giving rise to anomalous vibrational modes. Our results provide an example whereby a correlation between spatially extended low-frequency modes and high-propensity regions does not imply that relaxational dynamics originates in extended soft modes but rather belies their common origin. We consider other measures of elastic heterogeneity, such as non-affine displacement fields and mode localisation lengths, and discuss implications of our results to interpretations of dynamic heterogeneity more generally.

First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories

We investigate the dynamics of kinetically constrained models of glass formers by analysing the statistics of trajectories of the dynamics, or histories, using large deviation function methods. We show that, in general, these models exhibit a first-order dynamical transition between active and inactive dynamical phases. We argue that the dynamical heterogeneities displayed by these systems are a manifestation of dynamical first-order phase coexistence. In particular, we calculate dynamical large deviation functions, both analytically and numerically, for the Fredrickson–Andersen model, the East model, and constrained lattice gas models. We also show how large deviation functions can be obtained from a Landau-like theory for dynamical fluctuations. We discuss possibilities for similar dynamical phase-coexistence behaviour in other systems with heterogeneous dynamics.

Dynamic facilitation explains ‘democratic’ particle motion of metabasin transitions

Transitions between metabasins in supercooled liquids seem to occur through rapid collective particle rearrangements. These events have been called ‘democratic’ as they appear homogeneous over a significant number of particles. This could suggest that ‘democratic’ rearrangements are fundamentally distinct to those leading to dynamic heterogeneity. Here we show, however, that this apparent homogeneous particle motion can be explained solely in terms of dynamic facilitation, and is therefore intriniscally heterogeneous. We do so by studying metabasin transitions in facilitated spin models and constrained lattice gases. We find that metabasin transitions occur through a sequence of locally facilitated events taking place over a relatively short time frame. When observed on small enough spatial windows these events appear sudden and homogeneous. Our results indicate that metabasin transitions, while apparently homogeneous and ‘democratic’, are yet another manifestation of dynamical heterogeneity in glass formers.

Hydrodynamic limit for a particle system with degenerate rates

We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists blocked configurations which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.

Dynamics of single polymers under extreme confinement

We study the dynamics of a single-chain polymer confined to a two-dimensional cell. Weintroduce a kinetically constrained lattice gas model that preserves the connectivity of thechain, and we use this kinetically constrained model to study the dynamics of the polymerat varying densities through Monte Carlo simulations. Even at densities close to the fullypacked configuration, we find that the monomers comprising the chain manage to diffusearound the box with a root mean square displacement of the order of the box dimensionsover timescales for which the overall geometry of the polymer is, nevertheless, largelypreserved. To capture this shape persistence, we define the local tangent fieldand study the two-time tangent–tangent correlation function, which exhibitsa glass-like behavior. In both closed and open chains, we observe reptationalmotion and reshaping through local fingering events which entail global monomerdisplacement.

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.

Dynamic propensity in a kinetically constrained lattice gas

We apply the concept of dynamic propensity to a simple kinetically constrained model ofglass formers, the two-vacancy assisted triangular lattice gas, or (2)-TLG. We find that thepropensity field, defined in our case as the local root-mean-square displacement averagedover the ensemble of trajectories with identical initial configurations, is a good measure ofdynamical heterogeneity. This suggests a configurational origin for spatial fluctuations ofthe dynamics, but just as in the case of atomistic systems, we find that propensity is notcorrelated to any simple structural property. We show instead that certain extendedclusters of particles connected to vacancies correlate well with propensity, indicatingthat these are the fundamental excitations of the (2)-TLG. We also discuss timecorrelations and the correlation between configurations within the propensityensemble.

Fluctuations in the coarsening dynamics of the O(N) model with N → ∞: are they similar to those in glassy systems?

We study spatio-temporal fluctuations in the non-equilibrium dynamics of thed-dimensionalO(N) model inthe large N limit. We analyse the invariance of the dynamic equations for the global correlation andresponse in the slow ageing regime under transformations of time. We find that theseequations are invariant under scale transformations. We extend this study to the action inthe dynamic generating functional, finding similar results. This model therefore fallsinto a different category to glassy problems in which full time reparametrizationinvariance, a larger symmetry that encompasses timescale invariance, is expected to berealized asymptotically. Consequently, the spatio-temporal fluctuations of the largeN O(N) model should follow a pattern different from that for glassy systems. We compute thefluctuations of local, as well as spatially separated, two-field composite operators andresponses, and we confront our results with the ones found numerically for the 3DEdwards–Anderson model and kinetically constrained lattice gases. We analyse thedependence of the fluctuations of the composite operators on the growing domain lengthand we compare to what has been found for supercooled liquids and glasses. Finally, weshow that the development of time reparametrization invariance in glassy systems isintimately related to a well-defined and finite effective temperature, specified from themodification of the fluctuation-dissipation theorem out of equilibrium. We then conjecturethat the global asymptotic time reparametrization invariance is broken down to timescaleinvariance in all coarsening systems.

Heterogeneity and growing length scales in the dynamics of kinetically constrained lattice gases in two dimensions

We study dynamical heterogeneity and growing dynamical length scales in two kinetically constrained models, namely, the one- and two-vacancy assisted triangular lattice gases. One of the models is a strong glassformer and the other is a fragile glassformer. Both exhibit heterogeneous dynamics with broadly distributed time scales as seen in the distribution of persistence times. We show that the Stokes-Einstein relation is violated, to a greater degree in the fragile glassformer, and show how this violation is related to dynamic heterogeneity. We extract dynamical length scales from structure factors of mobile particles and show, quantitatively, the growth of this length scale as density increases. We comment on how the scaling of lengths and times in these models relates to that in facilitated spin models of glasses.

Rotational correlation and dynamic heterogeneity in a kinetically constrained lattice gas

We study the dynamical heterogeneity and glassy dynamics in a kinetically constrained lattice-gas model which has both translational and rotational degrees of freedom. We find that the rotational relaxation time tracks the structural relaxation time as density is increased whereas the translational diffusion constant exhibits a strong decoupling. We investigate distributions of exchange and persistence times for both the rotational and translational degrees of freedom and compare our results on the distributions of rotational exchange times to recent single molecule studies.

Decoupling of Self‐Diffusion and Structural Relaxation during a Fragile‐to‐Strong Crossover in a Kinetically Constrained Lattice Gas

Decoupling crossover in glass formers: In normal liquids, the self-diffusion constant, Ds, and mean relaxation time, τα, are inversely proportional. However, in glass formers, Ds and τα are decoupled, and if fragile–strong crossovers exist, Dsτα versus temperature T can be non-monotonic. The authors illustrate such a decoupling theoretically using an asymmetry parameter b (see figure). A similar behavior is seen in experiment.

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.

Glassy dynamics of kinetically constrained models

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying ``equilibrium glass transition''. After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

A Monte Carlo simulation of the overlayer structure of PF 3 on Pt(111)

Infrared reflection-absorption spectroscopy of the PF symmetric stretch region of PF 3 chemisorbed on Pt(111) shows two absorption bands whose relative intensities change as a function of temperature and coverage. These changes in the infrared spectra are attributed to changes in the PF 3 overlayer structure. Here we have carried out Monte Carlo simulations of this system based on the Ising model. We find that when we consider the interaction energy between a molecule and up to its third-nearest neighbors, a constrained lattice gas model reproduces the experimental infrared spectroscopy results. One of the two bands arises from those molecules which condense into compact ordered clusters, and the other band from the rest of the molecules. Because the exact relationship between the PF stretch frequency and the adssorbate environment is not known, the contribution of each molecule to a particular band is not straightforward and a statistical method for assigning molecules to each infrared band is developed. Though the model is very simple, the simulation results are in quantitative agreement with the experimental data.