Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant measures of exclusion processes with a look-ahead rule

Lam Thi Nhung, Ngo Phuoc Nguyen Ngoc, Huynh Anh Thi

Abstract

We study a one-dimensional exclusion process with a fixed jump length $I \ge 1$ in which a particle may advance or retreat $I$ sites provided all intermediate sites are vacant, with hopping rates of Arrhenius type depending on the local headway. We identify the class of rates admitting an explicit Ising-Gibbs invariant measure, with stationarity governed by pairwise balance rather than detailed balance. In the thermodynamic limit, we derive a closed-form stationary current that recovers the mean-field prediction for look-ahead traffic flow models exactly when particles are uncorrelated, and quantifies the correlation-induced correction for non-trivial interactions, illustrated with two explicit families of interaction potentials.

Invariant measures of exclusion processes with a look-ahead rule

Abstract

We study a one-dimensional exclusion process with a fixed jump length in which a particle may advance or retreat sites provided all intermediate sites are vacant, with hopping rates of Arrhenius type depending on the local headway. We identify the class of rates admitting an explicit Ising-Gibbs invariant measure, with stationarity governed by pairwise balance rather than detailed balance. In the thermodynamic limit, we derive a closed-form stationary current that recovers the mean-field prediction for look-ahead traffic flow models exactly when particles are uncorrelated, and quantifies the correlation-induced correction for non-trivial interactions, illustrated with two explicit families of interaction potentials.

Paper Structure

This paper contains 20 sections, 3 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and invariant measure
  3. Lattice, configurations, and dynamics
  4. Ising-Gibbs invariant measure
  5. Stationary current
  6. Exact stationary current
  7. Special cases and comparison with mean-field theory
  8. Non-interacting limit and recovery of mean-field.
  9. Effect of interactions.
  10. Equilibrium case.
  11. Non-concavity and inflection point.
  12. Examples
  13. Model 1: finite-range interaction.
  14. Model 2: Gaussian preferred-spacing interaction.
  15. Discussion
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let the $I$-SEP be governed by the generator generator with jump rates right_jump--left_jump. Then the probability measure is invariant for the process. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Stationary current $J(\rho)$ for model \ref{['eq:finite_range_J']}. Solid lines: exact formula \ref{['eq:current_exact']}; dashed lines: mean-field \ref{['eq:mf_sun_tan']}. (a) Fixed $I=2$, $J\in\{-1,-0.5,0,0.5,1\}$; at $J=0$ the two curves coincide. (b) Fixed $J=0.5$, $I\in\{1,2,3,4\}$; dashed lines of the same color show the corresponding mean-field curves.
  • Figure 2: Gaussian preferred-spacing model \ref{['eq:gaussian_J']} with $I=2$, $A=2$, $\mu=0.5$, and $g_0\in\{3,5,8\}$. (a) Interaction potential $J_g$. (b) Exact stationary current $J(\rho)$ (solid) vs. mean-field (dashed black); the peak density $\rho^\star$ is annotated. The mean-field prediction is independent of $g_0$, while the exact formula captures the shift of $\rho^\star$ with preferred spacing.

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 2.1: Ising-Gibbs invariant measure
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1: Equivalence of ensembles
  • Remark 3.1
  • Proposition 3.1: Exact stationary current