We study a one-dimensional classical stochastic kinetically constrained model (KCM) inspired by Rydberg atoms in their "facilitated" regime, where sites can flip only if a single of their nearest neighbours is excited. We call this model "XOR-FA" to distinguish it from the standard Fredrickson-Andersen (FA) model. We describe the dynamics of the XOR-FA model, including its relation to simple exclusion processes in its domain wall representation. The interesting relaxation dynamics of the XOR-FA is related to the prominence of large dynamical fluctuations that lead to phase transitions between active and inactive dynamical phases as in other KCMs. By means of numerical tensor network methods we study in detail such transitions in the dynamical large deviation regime.
We calculate the corrections to the thermodynamic limit of the critical density for jamming in the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models, and find them to be finite-density corrections, and not finite-size corrections. We do this by introducing a new numerical algorithm, which requires negligible computer memory since contrary to alternative approaches, it generates at each point only the necessary data. The algorithm starts from a single unfrozen site and at each step randomly generates the neighbors of the unfrozen region and checks whether they are frozen or n…
We analyse a first-order dynamical phase transition that takes place in the Fredrickson--Andersen (FA) model. We construct a two-dimensional spin system whose thermodynamic properties reproduce the dynamical large deviations of the FA model and we analyse this system numerically, comparing our results with finite-size scaling theory. This allows us to rationalise recent results for the FA model, including the exponential divergence of its susceptibility at phase coexistence. We also discuss a simple interfacial model that reproduces quantitatively the behaviour of the FA model at coexistence.
The Fredrickson-Andersen (FA) model---a kinetically constrained lattice model---displays an ergodic to non-ergodic transition with a slow two-step relaxation of dynamical correlation functions close to the transition point. We derive an asymptotically exact solution for the dynamical occupation correlation function of the FA model on the Bethe lattice by identifying an exact expression for its memory kernel. The exact solution fulfills a scaling relation between critical exponents and allows to calculate the exponents explicitly. In addition, we propose an approximate dynamics that describes n…
We investigate the dynamics of the randomly pinned Fredrickson-Andersen model on the Bethe lattice. We find a line of random pinning dynamical transitions whose dynamical critical properties are in the same universality class of the $A_2$ and $A_3$ transitions of Mode Coupling Theory. The $A_3$ behavior appears at the terminal point, where the relaxation becomes logarithmic and the relaxation time diverges exponentially. We explain the critical behavior in terms of self-induced disorder and avalanches, strengthening the relationship discussed in recent works between glassy dynamics and Random …
We study the effects of random pinning on the Fredrickson-Andersen model on the Bethe lattice. We find that the nonergodic transition temperature rises as the fraction of the pinned spins increases and the transition line terminates at a critical point. The freezing behavior of the spins is analogous to that of a randomly pinned p-spin mean-field spin glass model which has been recently reported. The diverging behavior of correlation lengths in the vicinity of the terminal critical point is found to be identical to the prediction of the inhomogeneous mode-coupling theory at the A3 singularity …
We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability $p\in[0,1]$ or $q=1-p$ respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $ν$ different from the stationary product $p$-Bernoulli measure $μ$. We assume that, under $ν$, the mean distance between two nearest empty sites is uniformly bounded. We then prove conve…
We study noise-averaged observables for a system of exchange-coupled quantum spins (qubits), each subject to a stochastic drive, by establishing mappings onto stochastic models in the strong-noise limit. Averaging over noise yields Lindbladian equations of motion; when these are subjected to a strong-noise perturbative treatment, classical master equations are found to emerge. The dynamics of noise averages of operators displays diffusive behaviour or exponential relaxation, depending on whether the drive conserves one of the spin components or not. In the latter case, the second moment of ope…
We investigate the relation between the cooperative length and the relaxation time, represented respectively by the culling time and the persistence time, in the Fredrickson-Andersen, Kob-Andersen and spiral kinetically-constrained models. By mapping the dynamics to diffusion of defects, we find a relation between the persistence time, $τ_{p}$, which is the time until a particle moves for the first time, and the culling time, $τ_{c}$, which is the minimal number of particles that need to move before a specific particle can move, $τ_{p}=τ^γ_{c}$, where $γ$ is model- and dimension dependent. We …
We derive expressions for the critical density for jamming in a hyper-rhomboid system of arbitrary shape in any dimension for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find that changing the system's shape without altering its total volume or particle density may induce jamming. We also find a transition between shapes in which the correlation length between jammed particles is infinite, and shapes that have a finite correlation length.
Is a "homogeneous" description of dynamic heterogeneities possible?
本文研究了玻璃形成液体动力学非均匀性的最简单模型：Fredrickson和Andersen提出的单自旋易化动力学伊辛模型[G. H. Fredrickson和H.C.安德森，物理评论快报。53，1244（1984）; J.Chem.Phys.83，5822（1985）]。我们表明，低温下，长时间的行为的密度自相关函数预测的缩放方法可以从一个自洽的模式耦合近似。
We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: for long and wide channels it decreases algebraically $v_{c}\sim W^{-1/2}$ with the system's …
On the example of a mean-field Fredrickson-Andersen kinetically constrained model, we focus on the known property that equilibrium dynamics take place at a first-order dynamical phase transition point in the space of time-realizations. We investigate the finite-size properties of this first order transition. By discussing and exploiting a mapping of the classical dynamical transition -an argued glassiness signature- to a first-order quantum transition, we show that the quantum analogy can be exploited to extract finite-size properties, which in many respects are similar to those in genuine mea…
Supercooled liquids are kinetically trapped materials in which the transition to a thermodynamically more stable state with long-range order is strongly suppressed. To assess the glass-forming abilities of a liquid empirical rules exist, but a comprehensive microscopic picture of devitrification is still missing. Here we study the crystallization of a popular model glass former, the binary Kob-Andersen mixture, in small systems. We perform trajectory sampling employing the population of the locally favored structure as order parameter. While for large population a dynamical phase transition ha…
We determine the finite size corrections to the large deviation function of the activity in a kinetically constrained model (the Fredrickson-Andersen model in one dimension), in the regime of dynamical phase coexistence. Numerical results agree with an effective model where the boundary between active and inactive regions is described by a Brownian interface.
The dynamical behaviours of a kinetically constrained spin model (Fredrickson-Andersen model) on a Bethe lattice are investigated by a perturbation analysis that provides exact final states above the nonergodic transition point. It is observed that the time-dependent solutions of the derived dynamical systems obtained by the perturbation analysis become systematically closer to the results obtained by Monte Carlo simulations as the order of a perturbation series is increased. This systematic perturbation analysis also clarifies the existence of a dynamical scaling law, which provides a implica…
Ghent University（根特大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；National Research Council Canada（加拿大国家研究委员会）；Vector Institute for Artificial Intelligence（人工智能矢量研究所）
We use a neural network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We obtain the scaled cumulant-generating function for the dynamical activity of the Fredrickson-Andersen model, a prototypical kinetically constrained model, in one and two dimensions, and present the first size-scaling analysis of the dynamical activity in two dimensions. These results provide a new route to the study of dynamical large-deviation functions, and highlight the broad applicability of the neural-network state ansatz across d…
Here we demonstrate that tensor network techniques - originally devised for the analysis of quantum many-body problems - are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples we consider the Fredrickson-Andersen and East models, two paradigmatic KCMs relevant to the modelling of glasses. We show how variational matrix product states allow to numerically approximate - systematically and with high accuracy - the leading eigenstates of the tilted dynamical generators which encode the large deviation statistics of the dynamic…
We analyze large deviations of the time-averaged activity in the one dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multi-canonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase trans…
We introduce and investigate an open many-body quantum system in which kinetically constrained coherent and dissipative processes compete. The form of the incoherent dissipative dynamics is inspired by that of epidemic spreading or cellular-automaton-based computation related to the density-classification problem. It features two non-fluctuating absorbing states as well as a $\mathcal{Z}_2$-symmetric point in parameter space. The coherent evolution is governed by a kinetically constrained $\mathcal{Z}_2$-symmetric many-body Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen m…
The large deviation (LD) statistics of dynamical observables is encoded in the spectral properties of deformed Markov generators. Recent works have shown that tensor network methods are well suited to compute the relevant leading eigenvalues and eigenvectors accurately. However, the efficient generation of the corresponding rare trajectories is a harder task. Here we show how to exploit the MPS approximation of the dominant eigenvector to implement an efficient sampling scheme which closely resembles the optimal (so-called "Doob") dynamics that realises the rare events. We demonstrate our appr…
We add extreme driving to the Kob-Andersen kinetically-constrained lattice-gas model in order to mimic the effect of gravity on dense granular systems. For low particle densities, the current that develops in the system agrees at arbitrary field intensity with a mean field theory. At intermediate densities, spatial correlations give rise to non-monotonic dependence of the current on field intensity. At higher densities, the current ultimately vanishes at a finite, field-dependent jamming density. We supplement the study of this bulk behavior with an investigation of the current through a narro…
Recent work has shown the effectiveness of tensor network methods for computing large deviation functions in constrained stochastic models in the infinite time limit. Here we show that these methods can also be used to study the statistics of dynamical observables at arbitrary finite time. This is a harder problem because, in contrast to the infinite time case where only the extremal eigenstate of a tilted Markov generator is relevant, for finite time the whole spectrum plays a role. We show that finite time dynamical partition sums can be computed efficiently and accurately in one dimension u…
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb Z^d$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for a…
We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient $D$ as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behavior of $D$ when the density $1-q$ of the environment goes to $1$ in two classes of KCSM. For noncooperative models, the diffusion coefficient $D$ scales like a power of $q$, with an expone…
Kinetic facilitated models and the Mode Coupling Theory (MCT) model B are within those systems known to exhibit a discontinuous dynamical transition with a two step relaxation. We consider a general scaling approach, within mean field theory, for such systems by considering the behavior of the density correlator <q(t)> and the dynamical susceptibility <q^2(t)> -<q(t)>^2. Focusing on the Fredrickson and Andersen (FA) facilitated spin model on the Bethe lattice, we extend a cluster approach that was previously developed for continuous glass transitions by Arenzon et al (Phys. Rev. E 90, 020301(R…
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 …
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behaviour. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next ste…
We study the statistical properties of the long-time dynamics of the rule 54 reversible cellular automaton (CA), driven stochastically at its boundaries. This CA can be considered as a discrete-time and deterministic version of the Fredrickson-Andersen kinetically constrained model (KCM). By means of a matrix product ansatz, we compute the exact large deviation cumulant generating functions for a wide range of time-extensive observables of the dynamics, together with their associated rate functions and conditioned long-time distributions over configurations. We show that for all instances of b…
Thermodynamic phase transitions in lattice spin systems with severe kinetic constraints: Numerical simulation results
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Changchun Institute of Applied Chemistry, Chinese Academy of Sciences（中国科学院长春应用化学研究所）
