Metastability for Kawasaki dynamics at low temperature with two types of particles

TL;DR

This paper analyzes the metastable transition in a two-particle-type lattice gas with Kawasaki dynamics at low temperature, identifying the critical droplet and transition time distribution in a finite box with open boundaries.

Contribution

It characterizes the metastable behavior, identifies the critical droplet, and computes the transition time distribution for a two-type particle lattice gas under Kawasaki dynamics.

Findings

Transition time is asymptotically exponential.

First entrance distribution on critical droplets is uniform.

Expected transition time is computed up to a multiplicative factor.

Abstract

This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a \emph{critical droplet} appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under \emph{three hypotheses} on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities -- the energy, the shape and the number of the critical droplets -- which are identified in the second and the third paper as well.