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A Class of Exclusion Processes Capable of Exhibiting Current Reversal

Ngo Phuoc Nguyen Ngoc, Lam Thi Nhung

TL;DR

This work develops a generalized ASEP framework with an Ising-like invariant measure $\hat{\pi}(\boldsymbol{\eta}) \propto x^{\sum_i \eta_i \eta_{i+1}}$ and analyzes stationary transport, revealing current reversal and nonmonotonic current-density profiles. By imposing recursive relations on jump rates, the authors prove the invariant measure remains short-range while the dynamics can be long-range, and they derive explicit current expressions using a headway-based grand-canonical formalism. The results show current reversal can occur even at $x=1$ and that the current profile lacks the symmetry of ASEP, with Models 2 and 3 exhibiting multiple extrema; Model 1 remains reversal-free. Overall, the framework unifies ASEP, the KLS model, and several related exclusion processes, advancing understanding of non-equilibrium transport in driven systems and clarifying how long-range dynamics interact with stationary weights.

Abstract

A century after Ising introduced the Ising measure to study equilibrium systems, its relevance has expanded well beyond equilibrium contexts, notably appearing in non-equilibrium frameworks such as the Katz--Lebowitz--Spohn (KLS) model. In this work, we investigate a class of generalized asymmetric simple exclusion processes (ASEP) for which the Ising measure serves as the stationary state. We show that the average stationary current in these models can display current reversal and other unconventional behaviors, offering new insights into transport phenomena in non-equilibrium systems. Moreover, although long-range interaction rates often give rise to long-range interactions in the potential function, our model provides a counterexample: even with long-range interactions in the dynamics, the resulting potential remains short-ranged. Finally, our framework encompasses several well-known models as special cases, including ASEP, the KLS model, the facilitated exclusion process, the cooperative exclusion process, and the assisted exchange model.

A Class of Exclusion Processes Capable of Exhibiting Current Reversal

TL;DR

This work develops a generalized ASEP framework with an Ising-like invariant measure and analyzes stationary transport, revealing current reversal and nonmonotonic current-density profiles. By imposing recursive relations on jump rates, the authors prove the invariant measure remains short-range while the dynamics can be long-range, and they derive explicit current expressions using a headway-based grand-canonical formalism. The results show current reversal can occur even at and that the current profile lacks the symmetry of ASEP, with Models 2 and 3 exhibiting multiple extrema; Model 1 remains reversal-free. Overall, the framework unifies ASEP, the KLS model, and several related exclusion processes, advancing understanding of non-equilibrium transport in driven systems and clarifying how long-range dynamics interact with stationary weights.

Abstract

A century after Ising introduced the Ising measure to study equilibrium systems, its relevance has expanded well beyond equilibrium contexts, notably appearing in non-equilibrium frameworks such as the Katz--Lebowitz--Spohn (KLS) model. In this work, we investigate a class of generalized asymmetric simple exclusion processes (ASEP) for which the Ising measure serves as the stationary state. We show that the average stationary current in these models can display current reversal and other unconventional behaviors, offering new insights into transport phenomena in non-equilibrium systems. Moreover, although long-range interaction rates often give rise to long-range interactions in the potential function, our model provides a counterexample: even with long-range interactions in the dynamics, the resulting potential remains short-ranged. Finally, our framework encompasses several well-known models as special cases, including ASEP, the KLS model, the facilitated exclusion process, the cooperative exclusion process, and the assisted exchange model.

Paper Structure

This paper contains 12 sections, 2 theorems, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Dynamics and invariant measure
  4. Stationary average current and discussion
  5. Average current
  6. Discussion
  7. Conclusions
  8. Headway distribution
  9. Average current of Models 1,2,3
  10. Proofs
  11. Proof of Theorem \ref{['Th1']}
  12. Proof of Theorem \ref{['Th2']}

Key Result

Theorem 2.1

Consider a type exclusion process in which the jump rates of particles satisfy the following recursive relations: Here, the quantities $r_k := r(k,0), \ell_k := \ell(k,0)$ are free parameters. Under these conditions, the invariant measure of the process takes an Ising-like form: where $\pi(\boldsymbol{\eta}) = x^{\sum_{i=1}^{L}\eta_i\eta_{i+1}}$ is the Boltzmann weight with nearest-neighbor inte

Figures (5)

  • Figure 1: Stationary current $j$ of Model 1 versus particle density $\rho$ in the repulsive ($x = 0.05$ and $0.0005$), non-interacting ($x = 1.0001$), and attractive ($x = 10$) regimes. The jump rates are $r_1 = 2$ and $\ell_1 = 1$.
  • Figure 2: Stationary current $j$ of Model 2 versus particle density $\rho$ in the repulsive ($x = 0.005$), non-interacting ($x = 1.0001$), and attractive ($x = 10$) regimes. The jump rates are $r_1 = 4$, $r_2 = 2$, and $\ell_1 = \ell_2 = 1$.
  • Figure 3: Stationary current $j$ of Model 3 versus particle density $\rho$ in the repulsive ($x = 0.005$), non-interacting ($x = 1.0001$), and attractive ($x = 10$) regimes. The jump rates are $r_1 = 1$, $r_2 = 2$, $r_3 = 4$, and $\ell_1 = 2$, $\ell_2 = 0.005$, $\ell_3 = 4$.
  • Figure 4: Stationary current $j$ of Model 3 versus particle density $\rho$ in the repulsive ($x = 0.005$), non-interacting ($x = 1.0001$), and attractive ($x = 10$) regimes. The jump rates are $r_1 = 0.0001$, $r_2 = 0.1$, $r_3 = 5$, and $\ell_1 = 4$, $\ell_2 = 2$, $\ell_3 = 1$.
  • Figure 5: Stationary current $j$ of Model 3 versus particle density $\rho$ in the repulsive ($x = 0.005$), non-interacting ($x = 1.0001$), and attractive ($x = 10$) regimes. The jump rates are $r_1 = r_2 = r_3 = 1$, and $\ell_1 = 4$, $\ell_2 = 2$, $\ell_3 = 1$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2.2