Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

TL;DR

This paper analyzes the low-temperature tunneling behavior of a two-dimensional lattice gas under Kawasaki dynamics, demonstrating that the process effectively behaves like a Markov chain on the torus with uniform jump rates.

Contribution

It proves the tunneling behavior of the Kawasaki dynamics among minimal energy states and characterizes the effective Markov process at low temperatures.

Findings

Process is close to a Markov chain on the torus at time scale e^{2β}

Jumps occur between any two sites with positive rates

Jump rates relate to simple random walk transition rates

Abstract

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $\beta$ on a two dimensional torus $\Lambda_L=\{0,..., L-1\}^2$ . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are $n^2\ll L$ particles and that the initial state is the configuration in which all sites of the square $\mb x + \{0,..., n-1\}^2$ are occupied. We show that in the time scale $e^{2\beta}$ the process is close to a Markov process on $\Lambda_L$ which jumps from any site $\mb x$ to any other site $\mb y\not =\mb x$ at a strictly positive rate which can be expressed in terms of the jump rates of simple random walks.