Phase Separation and Coarsening in One-Dimensional Driven Diffusive Systems: Local Dynaimcs Leading to Long-Range Hamiltonians

TL;DR

This paper investigates a one-dimensional driven diffusive system with three particle species, revealing phase separation, long-range interactions, and slow coarsening dynamics, with implications for ergodicity breaking and metastability.

Contribution

It demonstrates how local dynamics can lead to long-range Hamiltonians and phase separation in a simple one-dimensional model, providing insights into ergodicity breaking.

Findings

Model obeys detailed balance when densities are equal.

Steady-state distribution governed by a long-range Hamiltonian.

Domain size grows logarithmically over time.

Abstract

A driven system of three species of particle diffusing on a ring is studied in detail. The dynamics is local and conserves the three densities. A simple argument suggesting that the model should phase separate and break the translational symmetry is given. We show that for the special case where the three densities are equal the model obeys detailed balance and the steady-state distribution is governed by a Hamiltonian with asymmetric long-range interactions. This provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension. The steady state of finite-size systems is studied using a generalized matrix product ansatz. The coarsening process leading to phase separation is studied numerically and in a mean-field model. The system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size. The typical domain size is shown to grow logarithmically in time. Generalizations to a larger number of species are discussed.