Asymmetric Simple Exclusion Process Research Articles

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.

Phase diagram of generalized totally asymmetric simple exclusion process on an open chain: Liggett-like boundary conditions.

The totally asymmetric simple exclusion process with generalized update is a version of the discrete time totally asymmetric exclusion process with an additional interparticle interaction that controls the degree of particle clustering. Though the model was shown to be integrable on the ring and on the infinite lattice, on the open chain it was studied mainly numerically, while no analytic results existed even for its phase diagram. In this paper, we introduce boundary conditions associated with the infinite translation invariant stationary states of the model, which allow us to obtain the exact phase diagram analytically. We discuss the phase diagram in detail and confirm the analytic predictions by extensive numerical simulations.

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.

Exit times of totally asymmetric simple exclusion processes.

We address the question of the time needed by N particles, initially located on the first sites of a finite one-dimensional lattice of size L, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations and numerical simulations, we show that when N≪L, the mean exit time of the particles is asymptotically given by T_{N}(L)∼L+β_{N}sqrt[L] for large lattices. Building upon exact results obtained for two particles, we devise an approximate continuous space and time description of the random motion of the particles that provides an analytical recursive relation for the coefficients β_{N}. The results are shown to be in very good agreement with numerical results. This approach sheds some light on the exit dynamics of N particles in the regime where N is finite while the lattice size L→∞. This complements previous asymptotic results obtained by Johansson [Commun. Math. Phys. 209, 437 (2000)0010-361610.1007/s002200050027] in the limit where both N and L tend to infinity while keeping the particle density N/L finite.

Current fluctuations in a partially asymmetric simple exclusion process with a defect particle.

We study an exclusion process on a ring comprising a free defect particle in a bath of normal particles. The model is one of the few integrable cases in which the bath particles are partially asymmetric. The presence of the free defect creates localized or shock phases according to parameter values. We use a functional approach to Bethe equationsresulting from a nested Bethe ansatz to calculate exactly the mean currents and diffusion constants. The results agree very well with Monte Carlo simulations and reveal the main modes of fluctuation in the different phases of the steady state.

The steady state of the boundary-driven multiparticle asymmetric diffusion model

We consider the multiparticle asymmetric diffusion model (MADM) introduced by Sasamoto and Wadati with integrability preserving reservoirs at the boundaries. In contrast to the open asymmetric simple exclusion process the number of particles allowed per site is unbounded in the MADM. Taking inspiration from the stationary measure in the symmetric case, i.e. the rational limit, we first obtain the length 1 solution and then show that the steady state can be expressed as an iterated product of Jackson q-integrals. In the proof of the stationarity condition, we observe a cancellation mechanism that closely resembles the one of the matrix product ansatz. To our knowledge, the occupation probabilities in the steady state of the boundary-driven MADM were not available before.

Totally asymmetric simple exclusion process with local resetting in a resource-constrained environment.

Inspired by the process of mRNA translation, in which the stochastic degradation of mRNA-ribosome machinery is modeled by the resetting dynamics, we study an open totally asymmetric simple exclusion process with local resetting at the entry site in a resource-constrained environment. The effect of constrained resources on the stationary properties of the system has been comprehended in the form of the filling factor. The mean-field approximations are utilized to obtain stationary state features, such as density profiles and phase diagrams. The phase diagram possesses pure phases as well as coexisting phases, including a low-density-high-density phase separation, which did not manifest under periodic boundary conditions despite the system being closed there as well. The role of the resetting rate has been investigated on the stationary properties of the system, depending on how the filling factor scales with the system size. In contrast to the resetting model for infinite resources, two distinct phase transitions are observed for the smaller values of the filling factor leading to a change in the topology of the phase diagram. The impact of the resetting rate along with the finite-size effect has also been examined on the shock dynamics. All the mean-field results are found in remarkable agreement with the Monte Carlo simulations.

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.

Boundary current fluctuations for the half‐space ASEP and six‐vertex model

AbstractWe study fluctuations of the current at the boundary for the half‐space asymmetric simple exclusion process (ASEP) and the height function of the half‐space six‐vertex model at the boundary at large times. We establish a phase transition depending on the effective density of particles at the boundary, with Gaussian symplectic ensemble (GSE) and Gaussian orthogonal ensemble (GOE) limits as well as the Baik–Rains crossover distribution near the critical point. This was previously known for half‐space last‐passage percolation, and recently established for the half‐space log‐gamma polymer and Kardar–Parisi–Zhang equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half‐space six‐vertex model and a half‐space Hall–Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half‐space ASEP and six‐vertex model, indicating a hidden free fermionic structure.

Mixing time for the asymmetric simple exclusion process in a random environment

Mixing time for the asymmetric simple exclusion process in a random environment

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.

General criterion applicable to exclusion processes with an Ising-like invariant measure

This paper explores extensions of the totally asymmetric simple exclusion process in one dimension, assuming periodic boundary conditions. Specifically, we investigate situations where a particle’s movement rate is influenced by the positions of its two nearest neighbors. We establish the necessary and sufficient conditions for these exclusion processes to maintain an Ising-like measure as their invariant measure. Our analysis covers several renowned models, such as the cooperative exclusion process (Antal and Schütz, 2000), its dual model (Gabel and Redner, 2011), the Katz–Lebowitz–Spohn model (Katz et al., 1984), and the facilitated exclusion process (Basu and Mohanty, 2009; Gabel et al., 2010). To compare the obtained models, we will examine the average particle velocity and flux. Upon doing so, we highlight that this research contributes to the comprehension of the collective pushing phenomenon observed, for example, in biological traffic models (Epshtein and Nudler, 2003; Epshtein et al., 2003).

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.

From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

A Bijection Between Evil-Avoiding and Rectangular Permutations

Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.

Re-appearance of phases in the phase diagram of asymmetrically coupled two-lane exclusion process

Inspired by the finite resources in transport systems, we propose a two-lane totally asymmetric simple exclusion process model with attachment, detachment, and a limited supply of particles. We use a generalized mean-field theory in collaboration with a finite difference scheme and boundary layer analysis to evaluate the significance of limited resources in our proposed system. The outcomes are verified using Monte Carlo simulations. The steady-state behavior of the system is determined using phase diagrams, density profiles, and phase transitions. The study reports a novel phenomenon in which a phase occupies two different regions far from each other in the phase diagrams for an intermediate value of the total number of particles in the system. We also observe unique single and multiple re-entrance transitions, double shock, and varied new mixed phases, leading to bulk-induced phase transitions as exciting findings in the present study.

Tradeoff of generalization error in unsupervised learning

Abstract Finding the optimal model complexity that minimizes the generalization error (GE) is a key issue of machine learning. For the conventional supervised learning, this task typically involves the bias-variance tradeoff: lowering the bias by making the model more complex entails an increase in the variance. Meanwhile, little has been studied about whether the same tradeoff exists for unsupervised learning. In this study, we propose that unsupervised learning generally exhibits a two-component tradeoff of the GE, namely the model error (ME) and the data error (DE)—using a more complex model reduces the ME at the cost of the DE, with the DE playing a more significant role for a smaller training dataset. This is corroborated by training the restricted Boltzmann machine to generate the configurations of the two-dimensional Ising model at a given temperature and the totally asymmetric simple exclusion process with given entry and exit rates. Our results also indicate that the optimal model tends to be more complex when the data to be learned are more complex.

A reverse duality for the ASEP with open boundaries

We prove a duality between the asymmetric simple exclusion process (ASEP) with non-conservative open boundary conditions and an asymmetric exclusion process with particle-dependent hopping rates and conservative reflecting boundaries. This is a reverse duality in the sense that the duality function relates the measures of the dual processes rather than expectations. Specifically, for a certain parameter manifold of the boundary parameters of the open ASEP this duality expresses the time evolution of a family of shock product measures with N microscopic shocks in terms of the time evolution of N particles in the dual process. The reverse duality also elucidates some so far poorly understood properties of the stationary matrix product measures of the open ASEP given by finite-dimensional matrices.