Fredrickson-Andersen Model Research Articles

Thermodynamics of the Fredrickson-Andersen model on the Bethe lattice

Thermodynamics of the Fredrickson-Andersen model on the Bethe lattice

Elucidating Dynamical Behaviors in Kinetically Constrained Models via Energy-Activity Double-Biased Matrix Product State Analysis.

We investigate dynamical phase transitions in two representative kinetically constrained models: the 1D Fredrickson-Andersen and East Models. A recently developed energy-activity double-bias approach utilizing both s and g fields, conjugated with dynamical activity and trajectory energy, is combined with matrix product state (MPS) methods. It is demonstrated that MPS methods facilitate the numerical approximation of large-deviation statistics of dynamics by determining the eigenvalues of tilted dynamical generators under the influence of double-biasing fields. Specifically, by focusing on the g-field, a nearly "half-filled" state at moderate negative g values is identified, indicating the potential existence of an anomalous phase. Additionally, dynamical quantities under various s, g, and T conditions are obtained via tensor networks, showing good qualitative consistency with mean-field results and our previous extensive numerical simulation results obtained by the path sampling method. Our study introduces novel methodologies for examining decoupled dynamical behaviors that, although energetically active, remain dynamically inactive within the system. This approach offers a fresh perspective on the theoretical framework and computational strategies for studying dynamical phase transitions in kinetically constrained systems.

Dynamical Phase Transition in Kinetically Constrained Models with Energy-Activity Double-Bias Trajectory Ensemble.

We investigate the dynamical phase transitions in two representative kinetically constrained models, the 1D Fredrickson-Andersen and East models, by utilizing a recently developed s,g double-bias ensemble approach. In this ensemble, the fields s and g are applied to bias the dynamical activity and trajectory energy, respectively, in the trajectory ensemble. We first confirm that the dynamical phase transitions are indeed first-order in both the models. The phase diagrams in (s, g, T) space obtained via extensive numerical simulations show good qualitative agreement with the mean-field results. We also demonstrate that the temperature-dependent dynamical phase transition is possible in the systems when both fields are applied simultaneously. The trajectory energy and dynamical activity exhibit strong correlations for both systems. From extensive finite-size scaling analyses using the system size and observation time, we obtain scaling functions for the susceptibility and field and find scaling exponents that are model-dependent.

Fredrickson–Andersen model in two dimensions

The present article overviews recent mathematical advances on the Fredrickson–Andersen kinetically constrained spin model in two dimensions. It was introduced in physics as a toy model for recovering the glassy phenomenology in supercooled liquids close to the glass transition via dynamic constraints as opposed to static interactions.

Exact solution of the “Rule 150” reversible cellular automaton

We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained "exclusive one-spin facilitated" (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.

Metastability associated with many-body explosion of eigenmode expansion coefficients

Metastable states in stochastic systems are often characterized by the presence of small eigenvalues in the generator of the stochastic dynamics. We here show that metastability in many-body systems is not necessarily associated with small eigenvalues. Instead, many-body explosion of eigenmode expansion coefficients characterizes slow relaxation, which is demonstrated for two models, interacting particles in a double-well potential and the Fredrickson-Andersen model, the latter of which is a prototypical example of kinetically constrained models studied in glass and jamming transitions. Our results provide new insights into slow relaxation and metastability in many-body stochastic systems.

Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz.

We use a neural-network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We use recurrent neural networks to describe the large deviations of the dynamical activity of model glasses, kinetically constrained models in two dimensions. We present the first finite size-scaling analysis of the large-deviation functions of the two-dimensional Fredrickson-Andersen model, and explore the spatial structure of the high-activity sector of the South-or-East model. 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 domains in physics.

Dynamics and large deviation transitions of the XOR-Fredrickson-Andersen kinetically constrained model.

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 neighbors 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.

Dynamical phase coexistence in the Fredrickson–Andersen model

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.

Kinetically constrained models with random constraints

We study two kinetically constrained models in a quenched random environment. The first model is a mixed threshold Fredrickson-Andersen model on $\mathbb{Z}^{2}$, where the update threshold is either $1$ or $2$. The second is a mixture of the Fredrickson-Andersen $1$-spin facilitated constraint and the North-East constraint in $\mathbb{Z}^{2}$. We compare three time scales related to these models - the bootstrap percolation time for emptying the origin, the relaxation time of the kinetically constrained model, and the time for emptying the origin of the kinetically constrained model - and understand the effect of the random environment on each of them.

Using Matrix Product States to Study the Dynamical Large Deviations of Kinetically Constrained Models.

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 modeling of glasses. We show how variational matrix product states allow us to numerically approximate-systematically and with high accuracy-the leading eigenstates of the tilted dynamical generators, which encode the large deviation statistics of the dynamics. Via this approach, we can study system sizes beyond what is possible with other methods, allowing us to characterize in detail the finite size scaling of the trajectory-space phase transition of these models, the behavior of spectral gaps, and the spatial structure and "entanglement" properties of dynamical phases. We discuss the broader implications of our results.

Noisy coupled qubits: Operator spreading and the Fredrickson-Andersen model

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 operators -- from which the average subsystem purity and out-of-time-order correlation functions can be extracted -- is described by the Fredrickson-Andersen model, originally introduced as a model of cooperative relaxation near the glass transition. It is known that fluctuations of a ballistically propagating front in the model are asymptotically Gaussian in one dimension. We extend this by conjecturing, with strong numerical evidence, that in two dimensions the long-time fluctuations are in the Kardar-Parisi-Zhang universality class, complementing a similar observation in random unitary circuits.

Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model.

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 multicanonical 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 transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.

The Fredrickson-Andersen model with random pinning on Bethe lattices and its MCT transitions

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 A2 and A3 transitions of the mode coupling theory. The A3 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 field Ising model.

Fredrickson-Andersen model on Bethe lattice with random pinning

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 point for the glass transition.

Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models.

We investigate the relation between the cooperative length and 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_{c}^{γ}, where γ is model- and dimension-dependent. We also show that the persistence function in the Kob-Andersen and Fredrickson-Andersen models decays subexponentially in time, P(t)=exp[-(t/τ)^{β}], but unlike previous works, we find that the exponent β appears to decay to 0 as the particle density approaches 1.

Finite-density effects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models.

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 not. Our results correspond to systems of size greater than 10(7) × 10(7), much larger than any simulated before, and are consistent with the rigorous bounds on the asymptotic corrections. We also find that the average number of sites that seed a critical droplet is greater than 1.

Jamming Transition in Kinetically Constrained Models with Reflection Symmetry

A class of kinetically constrained models with reflection symmetry is proposed as an extension of the Fredrickson–Andersen model. It is proved that the proposed model on the square lattice exhibits a freezing transition at a non-trivial density. It is conjectured by numerical experiments that the known mechanism of the singular behaviors near the freezing transition in a previously studied model (spiral model) is not responsible for that in the proposed model.

Jamming transition of kinetically constrained models in rectangular systems

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) ~ W(-1/2) with the system's width W, while in square systems it decreases logarithmically vc) ~ 1/lnL with length L. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as W ≈ 10.

Finite Size Scaling of the Dynamical Free-Energy in a Kinetically Constrained Model

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.