Analysis of the finite-size effect of the long-range Ising model under Glauber dynamics

TL;DR

This paper investigates finite-size effects in the long-range Ising model with Glauber dynamics, deriving differential equations for correlations and comparing analytical results with numerical simulations.

Contribution

It extends previous work on infinite-range models to finite-size systems, incorporating perturbation theory and Gaussian assumptions to analyze correlation dynamics.

Findings

Analytical results match simulations for slow decay interactions.

Different behaviors observed when interaction decay is rapid.

Derived differential equations describe two-body correlation evolution.

Abstract

We considered a long-range Ising model under Glauber dynamics and calculated the difference from the mean-field approximation in a finite-size system using perturbation theory. To deal with the BBGKY hierarchy, we assumed that certain types of extensive properties have a Gaussian distribution, which turned out to be equivalent to the Kirkwood superposition approximation within the range of first-order perturbation. After several calculations, ordinary differential equations that describe the time development of a two-body correlation were derived. This discussion is the generalization of our previous study which developed a similar consideration on the infinite-range Ising model. The results of the calculation fit those of the numerical simulations for the case in which the decay of the interaction was sufficiently slow; however, they exhibited different behaviors when the decay became rapid.