Exact solution of an integrable non-equilibrium particle system

TL;DR

This paper provides an exact solution for a non-equilibrium particle system with unbounded particles per site, revealing detailed steady-state properties and correlations, and connecting non-equilibrium states to equilibrium measures.

Contribution

It introduces an exact, closed-form solution for the steady state of an integrable boundary-driven particle system with unbounded occupation numbers, using probabilistic and integrable techniques.

Findings

Exact factorial moments of the steady state are derived.

Long-range correlations are computed explicitly.

System approaches local equilibrium in the thermodynamic limit.

Abstract

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of $N$ sites connected at its ends to two reservoirs can be solved exactly, i.e. the factorial moments of the non-equilibrium steady-state can be written in closed form for each $N$. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. Long-range correlations are computed in the finite-volume system. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.