Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus

TL;DR

This paper studies the zero-temperature behavior of Kawasaki dynamics in a 2D Ising lattice gas, showing that particles form a square shape and the center of mass performs Brownian motion as temperature approaches zero.

Contribution

It provides a rigorous analysis of the shape evolution and center of mass dynamics of particles under Kawasaki dynamics at low temperatures in a large 2D lattice.

Findings

Particles form an almost perfect square shape.

Center of mass exhibits Brownian motion at zero temperature.

Results hold in a proper time scale as temperature vanishes.

Abstract

We consider the Kawasaki dynamics at inverse temperature $\beta$ for the Ising lattice gas on a two-dimensional square of length $2L+1$ with periodic boundary conditions. We assume that initially the particles form a square of length $n$, which may increase, as well as $L$, with $\beta$. We show that in a proper time scale the particles form almost always a square and that the center of mass of the square evolves as a Brownian motion when the temperature vanishes.