Invariant measures of exclusion processes with a look-ahead rule

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 otherwise. The resulting fundamental diagram is right-skewed and non-concave, as in empirical traffic data, and for a preferred-spacing interaction-where the stationary headway distribution becomes bimodal-the mean-field approximation fails qualitatively. The closed-form current is confirmed by kinetic Monte Carlo simulations and by exact finite-size computations.