Simple Exclusion Process Research Articles

Two-dimensional ASEP model to study density profiles in CVD growth

The growth of two-dimensional (2D) transition metal dichalcogenides using chemical vapor deposition has been an area of intense study, primarily due to the scalability requirements for potential device applications. One of the major challenges of such growths is the large-scale thickness variation of the grown film. To investigate the role of different growth parameters computationally, we use a 2D asymmetric simple-exclusion process (ASEP) model with open boundaries as an approximation to the dynamics of deposition on the coarse-grained lattice. The variations in concentration of particles (growth profiles) at the lattice sites in the grown film are studied as functions of parameters like injection and ejection rate of particles from the lattice, time of observation, and the right bias difference between the hopping probabilities towards right and towards left) imposed by the carrier gas. In addition, the deposition rates at a given coarse-grained site is assumed to depend on the occupancy of that site. The effect of the maximum deposition rate, i.e., the deposition rate at a completely unoccupied site on the substrate, has been explored. The growth profiles stretch horizontally when either the evolution time or the right bias is increased. An increased deposition rate leads to a step-like profile, with the higher density region close to the left edge. In 3D, the growth profiles become more uniform with the increase in the height of the precursor with respect to the substrate surface. These results qualitatively agree with the experimental observations.

Lifted TASEP: A Solvable Paradigm for Speeding up Many-Particle Markov Chains

Virtually all Markov-chain Monte Carlo algorithms used for sampling a given distribution are reversible, and they satisfy the detailed-balance condition. For local chains, this leads to a slow, diffusive exploration of sample space. Significant speedups can be achieved through nonreversible algorithms with the given distribution as a targeted steady state. However, nonreversible algorithms for sampling are difficult to set up and to analyze, and exact speedup results for interacting many-particle systems are very rare. Here, we introduce the “lifted” totally asymmetric simple exclusion process (TASEP) as an exactly solvable paradigm for nonreversible many-particle Markov chains. It samples the same hard-sphere distribution as the Metropolis algorithm for symmetrically diffusing hard-core particles on a one-dimensional lattice. We solve the lifted TASEP by an unusual kind of coordinate Bethe ansatz and show that it exhibits polynomial (in particle number) speedups in the relaxation time for the asymptotic approach of the steady state, as well as the nonasymptotic mixing time, compared to both Metropolis and Kardar-Parisi-Zhang-based dynamics. The lifted TASEP is the reduction onto the one-dimensional lattice of the successful hard-sphere event-chain Monte Carlo algorithm, and we discuss that it can likewise be generalized to soft interaction potentials. Published by the American Physical Society 2024

An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics

We prove a multidimensional ergodic theorem with weighted averages for the action of the group Zd on a probability space. At level n weights are of the form n−dψ(j/n), j∈Zd, for real functions ψ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from Faggionato (2023) [Theorem 4.4]. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in Faggionato (2022) [Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).

On perturbation around closed exclusion processes

We derive the formula for the stationary states of particle-number conserving exclusion processes infinitesimally perturbed by inhomogeneous adsorption and desorption. The formula not only proves but also generalises the conjecture proposed in [Phys. Rev. E 97, 032135] to account for inhomogeneous adsorption and desorption. As an application of the formula, we draw part of the phase diagrams of the open asymmetric simple exclusion process with and without Langmuir kinetics, correctly reproducing known results.

Semi-infinite Simple Exclusion Process: From Current Fluctuations to Target Survival.

The symmetric simple exclusion process (SEP), where diffusive particles cannot overtake each other, is a paradigmatic model of transport in the single-file geometry. In this model, the study of currents has attracted a lot of attention, but so far most results are restricted to two geometries: (i)a finite system between two reservoirs, which does not conserve the number of particles but reaches a nonequilibrium steady state, and (ii)an infinite system which conserves the number of particles but never reaches a steady state. Here, we obtain an expression for the full cumulant generating function of the integrated current in the important intermediate situation of a semi-infinite system connected to a reservoir, which does not conserve the number of particles and never reaches a steady state. This results from the determination of the full spatial structure of the correlations, which we infer to obey the very same closed equation recently obtained in the infinite geometry and argue to be exact. Besides their intrinsic interest, these results allow us to solve two open problems: the survival probability of a fixed target in the SEP, and the statistics of the number of particles injected by a localized source.

A generalized dynamic asymmetric exclusion process: orthogonal dualities and degenerations

Abstract In this paper, a generalized version of dynamic asymmetric simple exclusion process (ASEP) is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate q-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra U q ( sl 2 ) . In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from q-Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with q-Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP.

Novel ASEP-inspired solutions of the Yang-Baxter equation

Abstract We explore the algebraic structure of a particular ansatz of the Yang-Baxter equation (YBE), which is inspired by the Bethe Ansatz treatment of the asymmetric simple exclusion process spin-model. Various classes of Hamiltonian density arriving from the two types of R-matrices are found, which also appear as solutions of the constant YBE. We identify the idempotent and nilpotent categories of such constant R-matrices and perform a rank-1 numerical search for the lowest dimension. A summary of the final results reveals general non-Hermitian spin-1/2 chain models.

Non-equilibrium fluctuations for SEP([formula omitted]) with open boundary

We analyze the non-equilibrium fluctuations of the partial symmetric simple exclusion process, SEP(α), which allows at most α∈N particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter θ∈R. Setting α=1, we find the results of Landim et al. (2008), Franco et al. (2019) and Gonçalves et al. (2020) and extend the known results to cover all range of θ.

Diameters of Symmetric and Lifted Simple Exclusion Models

We determine diameters of Markov chains describing one-dimensional N-particle models with an exclusion interaction, namely the symmetric simple exclusion process (Ssep) and one of its non-reversible liftings, the lifted totally asymmetric simple exclusion process (Tasep). The diameters provide lower bounds for the mixing times, and we discuss the implications of our findings for the analysis of these models.

Mirror symmetry breakdown in the Kardar-Parisi-Zhang universality class.

The current/height fluctuation statistics of Kardar-Parisi-Zhang (KPZ) universality in 1+1 dimensions are sensitive to the initial state. We find that the averages over the initial states exhibit universal and scale-invariant patterns when conditioning on fluctuations. To establish universality of our findings, we demonstrate scale invariance at different times and heights using large-scale Monte Carlo simulations of the totally asymmetric simple exclusion process, which belongs to the KPZ universality class. Here we focus on current (or height) fluctuations in the steady-state regime described by the Baik-Rains distribution. The conditioned probability distribution of an initial-state order parameter shows a transition from uni- to bimodal. Bimodality occurs for negative current/height fluctuations that are dominated by superdiffusive shock dynamics. It is caused by two possible point-symmetric shock profiles and the KPZ mirror symmetry breakdown. Similar surprising relations between initial states and fluctuations might exist in other universality classes as well.

Importance sampling for counting statistics in one-dimensional systems.

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

Large deviations in the symmetric simple exclusion process with slow boundaries: A hydrodynamic perspective

We revisit the one-dimensional model of the symmetric simple exclusion process slowly coupled with two unequal reservoirs at the boundaries. In its non-equilibrium stationary state, the large deviations functions of density and current have been recently derived using exact microscopic analysis by Derrida, Hirschberg and Sadhu in [J. Stat. Phys. 182, 15 (2021)]. We present an independent derivation using the hydrodynamic approach of the macroscopic fluctuation theory (MFT). The slow coupling introduces additional boundary terms in the MFT-Action, which modifies the spatial boundary conditions for the associated variational problem. For the density large deviations, we explicitly solve the corresponding Euler-Lagrange equations using a simple local transformation of the optimal fields. For the current large deviations, our solution is obtained using the additivity principle. In addition to recovering the expression of the large deviations functions, our solution describes the most probable path for these rare fluctuations.

Totally asymmetric simple exclusion process on a dynamic lattice with local inhomogeneity

In this manuscript, we introduce a totally asymmetric simple exclusion process on a dynamic lattice with inhomogeneity to model motor-based long-range transport along microtubules (MTs) which exhibit cycles of growth, shortening and regrowth at their growing tips. By mean-field approximation and numerical simulations, we explore phase diagrams for the particle density near the dynamic end of the lattice as well as over the entire lattice. In particular, we find seven different phases for the density over the entire lattice and explore the corresponding phase diagram by analytical analysis. Interestingly, we find that the density may have a linear profile in part of the lattice. When the lattice has a defect, we separate the lattice into two subsystems: a right subsystem with fixed length and a defect in the middle and a left subsystem with dynamic length. When the right subsystem is in maximum-current phase, we calculate an analytical approximation of the density away from the boundaries and the defect site and the density is in association with the reduced hopping rate at the defect site; the density at the left side of the defect site serves as the effective entry rate of the left subsystem. We find that with defects, the linear profiles no longer appear at the left end of the left subsystem and only five possible phases could occur. This is a step-forward to a more comprehensive mathematical description of intracellular long-range transport of organelles along MTs with instability.

Approximating the Stationary Distribution of the ASEP with Open Boundaries

We investigate the stationary distribution of asymmetric and weakly asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment. Depending on the boundary parameters of the exclusion process, we provide conditions such that the stationary distribution projected onto a subinterval is close in total variation distance to a product measure.

Structured random matrices and cyclic cumulants: A free probability approach

In this paper, we introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise nonlinear operations. We then formulate an efficient method, based on an extremization problem, for computing the spectrum of subblocks of such large structured random matrices. We present different proofs — combinatorial or algebraic — of the validity of this method, which all have some connection with free probability. We illustrate this method with well-known examples of unstructured matrices, including Haar randomly rotated matrices, as well as with the example of structured random matrices arising in the quantum symmetric simple exclusion process.

Exact steady states in the asymmetric simple exclusion process beyond one dimension

The asymmetric simple exclusion process (ASEP) is a paradigmatic nonequilibrium many-body system that describes the asymmetric random walk of particles with exclusion interactions in a lattice. Although the ASEP is recognized as an exactly solvable model, most of the exact results obtained so far are limited to one-dimensional systems. Here, we construct the exact steady states of the ASEP with closed and periodic boundary conditions in arbitrary dimensions. This is achieved through the concept of transition decomposition, which enables the treatment of the multidimensional ASEP as a composite of the one-dimensional ASEPs. Published by the American Physical Society 2024

KPZ fluctuations in finite volume

These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects describing the relaxation to its stationary state of a finite system starting from a given initial condition. The second part is more specifically devoted to an approach allowing to express in a simple way the statistics of the current in the totally asymmetric simple exclusion process in terms of a contour integral on a compact Riemann surface, whose infinite genus limit leads to KPZ fluctuations in finite volume.

Askey–Wilson Signed Measures and Open ASEP in the Shock Region

Abstract We introduce a family of multi-dimensional Askey–Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey–Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.

Combining Reinforcement Learning and Tensor Networks, with an Application to Dynamical Large Deviations.

We present a framework to integrate tensor network (TN) methods with reinforcement learning (RL) for solving dynamical optimization tasks. We consider the RL actor-critic method, a model-free approach for solving RL problems, and introduce TNs as the approximators for its policy and value functions. Our "actor-critic with tensor networks" (ACTeN) method is especially well suited to problems with large and factorizable state and action spaces. As an illustration of the applicability of ACTeN we solve the exponentially hard task of sampling rare trajectories in two paradigmatic stochastic models, the East model of glasses and the asymmetric simple exclusion process, the latter being particularly challenging to other methods due to the absence of detailed balance. With substantial potential for further integration with the vast array of existing RL methods, the approach introduced here is promising both for applications in physics and to multi-agent RL problems more generally.

Time-dependent properties of run-and-tumble particles. II. Current fluctuations.

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of L sites, for arbitrary tumbling rate γ=τ_{p}^{-1} and density ρ; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of L large, the fluctuation of cumulative current Q_{i}(T,L) across the ith bond in a time interval T≫1/D grows first subdiffusively and then diffusively (linearly) with T: 〈Q_{i}^{2}〉∼T^{α} with α=1/2 for 1/D≪T≪L^{2}/D and α=1 for T≫L^{2}/D, where D(ρ,γ) is the collective- or bulk-diffusion coefficient; at small times T≪1/D, exponent α depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations D〈Q_{i}^{2}(T,L)〉/2χL≡W(y) as a function of scaled variable y=DT/L^{2} collapse onto a universal scaling curve W(y), where χ(ρ,γ) is the collective particle mobility. In the limit of small density and tumbling rate, ρ,γ→0, with ψ=ρ/γ fixed, there exists a scaling law: The scaled mobility γ^{a}χ(ρ,γ)/χ^{(0)}≡H(ψ) as a function of ψ collapses onto a scaling curve H(ψ), where a=1 and 2 in models I and II, respectively, and χ^{(0)} is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H(ψ) is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W(y) and H(ψ). We also calculate spatial correlation functions for the current and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length ξ∼τ_{p}^{1/2} diverging with persistence time τ_{p}≫1. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)1539-375510.1103/PhysRevE.109.024124].