Optimal linear Kawasaki model

TL;DR

This paper develops a linear approximation of the Kawasaki model's exchange rates to better understand the conservative dynamics of Ising systems, leading to accurate estimates of critical temperatures and exponents.

Contribution

It introduces a linear form for exchange rates in the Kawasaki model that minimizes the error in detailed balance, connecting to a Cahn-Hilliard equation for conservative dynamics.

Findings

Derived a linear approximation for exchange rates.

Estimated critical temperatures matching known values.

Calculated dynamic and critical exponents.

Abstract

The Kawasaki model is not exactly solvable as any choice of the exchange rate ($w_{jj'}$) which satisfies the detailed balance condition is highly nonlinear. In this work we address the issue of writing $w_{jj'}$ in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. In the continuum limit, our approach leads to a Cahn-Hilliard equation of conservative dynamics. The work presented in this paper will help us anticipate how the conservative dynamics of an arbitrary Ising system depends on the temperature and the coupling constants. In particular, for two and three dimensional systems, the critical temperatures estimated in our work are in good agreement with the actual values. We also calculate the dynamic and some of the critical exponents of the model.