Totally Asymmetric Simple Exclusion Process Research Articles

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

Open 2–TASEP with integrable boundaries

In this paper, we explore a two-species extension of the totally asymmetric simple exclusion process (TASEP) known as ‘2–TASEP’ with open boundaries. In this model, carriers on a one-dimensional lattice exhibit distinct behaviors: loaded carriers move right, empty carriers move left, and they exchange positions at a unit rate. At the boundaries carriers are loaded at the left and unloaded at the right. We focus on the stationary state of this model. Using the integrability of this model, we introduce spectral parameters to deform it. This process uncovers an underlying algebraic structure linked to the Weyl group CL . Exploiting this structure, we analytically compute the stationary state of the inhomogeneous model, which permits, after spectral parameter specialization, to derive the exact average particle current of the original 2–TASEP.

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

Interaction vs inhomogeneity in a periodic TASEP

We study the non-equilibrium steady states in a totally asymmetric simple exclusion process with periodic boundary conditions, also incorporating (i) an extra (nearest-neighbour) repulsive interaction and (ii) hopping rates characterized by a smooth spatial inhomogeneity. We make use of a generalized mean-field approach (at the level of nearest-neighbour pair clusters), in combination with kinetic Monte Carlo simulations. It turns out that the so-called shock phase can exhibit a lot of qualitatively different subphases, including multiple-shock phases, and a minimal-current shock phase. We argue that the resulting, considerably rich phase diagram should be relatively insensitive to minor details of either interaction or spatial inhomogeneity. As a consequence, we also expect that our results help elucidate the nature of shock subphases detected in previous studies.

The second class particle process at shocks

We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities λ to the left of the origin and ρ to the right of it and λ<ρ. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.

Codebase release 1.1 for TASEPy

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic lattice model for one-dimensional particle transport subject to excluded-volume interactions. Solving the inhomogeneous TASEP in which particles’ hopping rate vary across the lattice is a long-standing problem. In recent years, a power series approximation (PSA) has been developed to tackle this problem, however no computer algorithm currently exists that implements this approximation. This paper addresses this issue by providing a Python-based package TASEPy that finds the steady state solution of the inhomogeneous TASEP for any set of hopping rates using the PSA truncated at a user-defined order.

TASEPy: A Python-based package to iteratively solve the inhomogeneous exclusion process

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic lattice model for one-dimensional particle transport subject to excluded-volume interactions. Solving the inhomogeneous TASEP in which particles’ hopping rate vary across the lattice is a long-standing problem. In recent years, a power series approximation (PSA) has been developed to tackle this problem, however no computer algorithm currently exists that implements this approximation. This paper addresses this issue by providing a Python-based package TASEPy that finds the steady state solution of the inhomogeneous TASEP for any set of hopping rates using the PSA truncated at a user-defined order.

Infinite order phase transition in the slow bond TASEP

AbstractIn the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz posed the well‐known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of is 0. This was ultimately resolved rigorously in Basu‐Sidoravicius‐Sly which established that .Here we study the effect of the current as tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal.

Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder.

We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.

The suppressed high density phase in three-lane asymmetric exclusion processes with narrow entrances

This paper studies totally asymmetric simple exclusion processes (TASEPs) on three lanes with narrow entrances. Three types of particles move on three parallel lanes, and changing lanes is forbidden. One particle cannot enter one lane if exit site of the neighboring lane is occupied. Simple mean field approach in which correlation of sites is ignored is firstly carried out in the analysis of the system. It is shown that high density is suppressed in the middle lane, and the other two lanes are always symmetric. In addition, the analysis indicates that there are only four phases in the system, and spontaneous symmetry breaking phenomenon does not exist. Extensive Monte Carlo simulations are carried out to confirm the analysis. However, the simple mean field analytical results of phases boundaries deviate that of simulations in the phase diagram. Motivated by this, cluster mean field approach is carried out, and correlations of vertical cluster sites are considered. It is found that the cluster mean field analysis performs much better than the simple mean field analysis.

Shock fluctuations in TASEP under a variety of time scalings

We consider the totally asymmetric simple exclusion process (TASEP) with two different initial conditions with shock discontinuities formed by blocks of fully packed particles. Initially a second class particle is at the left of a shock discontinuity. Using multicolored TASEP we derive exact formulas for the distribution of the second class particle and colored height functions. These are given in terms of the height function at different positions of a single TASEP configuration. We study the limiting distributions of second class particles (and colored height functions). The result depends on how the width blocks of particles scale with the observation time; we study a variety of such scalings.

Analytical investigation on totally asymmetric simple exclusion process with Langmuir kinetics and a parallel update with two sub-steps

Motivated by the fact that parallel update rules need to be employed to study the traffic dynamic of vehicles and pedestrians with bulk on-off ramps. This paper studies the totally asymmetric simple exclusion process (TASEP) with Langmuir kinetics and a parallel update with two sub-steps. Moreover, this new model can be analyzed analytically. Based on the mean-field method, the phase diagrams are obtained and the influences of model parameters are investigated. Compared to the system under the random update, there do not exist the phases related to the maximum current phase and Meissner phase in the system under the parallel update.

Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

We study a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder’s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.

Absorbing time asymptotics in the oriented swap process

The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multispecies version of the totally asymmetric simple exclusion process (TASEP) on a finite interval. An open problem from a 2009 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process as the number of particles goes to infinity. We resolve this question by proving that this random variable satisfies GOE Tracy–Widom asymptotics. As a central ingredient of our proof, we reexamine a distributional identity relating the behavior of the oriented swap process to last passage percolation, conjectured in a recent paper of Bisi, Cunden, Gibbons, and Romik. We use a shift-invariance principle for multispecies TASEPs, obtained by exploiting recent results of Borodin, Gorin, and Wheeler for the stochastic colored six-vertex model, to prove a weakened form of the Bisi et al. conjectural identity, that is nonetheless sufficient for proving the asymptotic result for the absorbing time.

Two-species totally asymmetric simple exclusion process model: From a simple description to intermittency and traveling traffic jams.

We extend the paradigmatic and versatile totally asymmetric simple exclusion process (TASEP) for stochastic 1D transport to allow for two different particle species, each having specific entry and exit rates. We offer a complete mean-field analysis, including a phase diagram, by mapping this model onto an effective one-species TASEP. Stochastic simulations confirm the results, but indicate deviations when the particle species have very different exit rates. We illustrate that this is due to a phenomenon of intermittency, and formulate a refined "intermittent" mean-field theory for this regime. We discuss how nonstationary effects may further enrich the phenomenology.

From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

Existence and uniqueness of solution of the differential equation describing the TASEP-LK coupled transport process

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C∞[0,1] steady-state solution by the method of upper and lower solution, and the uniqueness in both W1,2(0,1) and L∞(0,1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C([0,1]×[0,+∞))∩C2,1([0,1]×(0,+∞)), which, for any continuous initial value, converges to the steady-state solution in C[0,1] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species.

Upper-tail large deviation principle for the ASEP

We consider the asymmetric simple exclusion process (ASEP) on Z started from step initial data and obtain the exact Lyapunov exponents for H0(t), the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for −H0(t). Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [40].

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

Limiting Directions for Random Walks in Classical Affine Weyl Groups

Abstract Let $W$ be a finite Weyl group and $\widetilde W$ the corresponding affine Weyl group. A random element of $\widetilde W$ can be obtained as a reduced random walk on the alcoves of $\widetilde W$. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of $|W|$ many directions. We compute these directions when $W$ is $B_n$, $C_n$, and $D_n$ and the random walk is weighted by Kac and dual Kac labels. This settles Lam’s questions for types $B$ and $C$ in the affirmative and for type $D$ in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the $D^*$-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.