Open interacting particle systems and Ising measures

TL;DR

The article advances the theory of open stochastic interacting particle systems by introducing a generalized open KLS model that preserves an Ising-type invariant measure, enabling exact computation of the stationary current. It demonstrates current reversal as a density-dependent phenomenon and uses the extremal-current principle to motivate a phase diagram featuring two extremal-current phases, extending beyond classic ASEP and KLS results. The work develops a constructive framework for open boundary dynamics, derives precise boundary-rate conditions ensuring invariance of the Ising measure, and provides detailed proofs via ground-state transformations that illuminate when the dynamics are reversible or invariant. Together, these results offer a rigorous, exact handle on boundary-driven nonequilibrium behavior and point to broad avenues for extending to multi-species, long-range, and quantum settings with explicit invariant measures.

Abstract

We first survey some open questions concerning stochastic interacting particle systems with open boundaries. Then an asymmetric exclusion process with open boundaries that generalizes the lattice gas model of Katz, Lebowitz, and Spohn (KLS) is introduced and invariance of the one-dimensional Ising measure is proved. The stationary current is computed in explicit form and is shown to exhibit current reversal at some density. Based on the extremal-current principle for one-dimensional driven diffusive systems with one conservation law, the phase diagram for boundary-induced phase transitions is conjectured for this case. There are two extremal-current phases, unlike in the conventional open asymmetric simple exclusion process, which exhibits only one extremal-current phase or the previously considered conventional open KLS model with one or three extremal-current phases.

Figures (4)

• Figure 1: Stationary current $j$ as a function of the particle density $\rho$ for repulsive static interaction with $\epsilon = 2/3$ and with kinetic interaction parameters $\ell =0.9, \lambda = 1/9$ and different values of $r$ and $\kappa$. Curves from the bottom to top: $(r,\kappa) = (1.606, 0.650), (1.720, 0.389), (1.900, 1/9), (2.140, -0.123)$.
• Figure 2: Stationary current $j$ as a function of the particle density $\rho$ for attractive static interaction with $\epsilon = - 2/3$ and with kinetic interaction parameters $\ell =4.5, \lambda = -7/9$ and different values of $r$ and $\kappa$. Curves from the bottom to top: $(r,\kappa) = (4.3, - 0.697), (4.6, -0.722), (5.5, -0.778), (6.7, -0.825)$.
• Figure 3: Stationary current $j$ as a function of the particle density $\rho$ for strong attractive static interaction with $\epsilon = - 0.980$ with kinetic interaction parameters $\ell =150.75$, $\lambda = -0.987$ and different values of $r$ and $\kappa$. Curves from the bottom to top: $(r,\kappa) = (56.55, - 0.982 ), (66.65, -0.985), (76.75, -0.987), (81.8, -0.988), (96.95, -0.989)$.
• Figure 4: Schematic phase diagram of the generalized KLS model for current reversal, with left boundary density $\rho_{-}$ on the $x$-axis and right boundary density $\rho_{+}$ on the $y$-axis. The minimal (maximal) current phase is marked by Min (Max). Both are bounded by straight second-order phase transition lines. In the central region marked by $\rho_{+}$ the bulk density is equal to the right boundary density. Curved first-order phase transition lines separate it from two regions with bulk density equal to the left boundary density (lower left and upper right regions marked by $\rho_{-}$).

Theorems & Definitions (14)

• Theorem 4.1
• Theorem 4.2
• Definition 4.1
• Theorem 4.3
• Remark 4.1
• Remark 4.2
• Theorem 4.4
• Remark 4.3
• Remark 4.4
• Remark 4.5