Monte Carlo simulation of classical spin models with chaotic billiards

TL;DR

This paper investigates the use of chaotic billiard dynamics as a deterministic alternative to traditional Monte Carlo methods for sampling from classical spin models, demonstrating its effectiveness in various models and critical phenomena.

Contribution

It introduces and numerically validates chaotic billiard dynamics as a novel deterministic sampling method for classical spin models, including Ising and Potts models.

Findings

Chaotic billiard dynamics can produce samples converging to the true distribution.

The method accurately identifies critical points and exponents.

It extends successfully to multi-state spin models like the Potts model.

Abstract

It has recently been shown that the computing abilities of Boltzmann machines, or Ising spin-glass models, can be implemented by chaotic billiard dynamics without any use of random numbers. In this paper, we further numerically investigate the capabilities of the chaotic billiard dynamics as a deterministic alternative to random Monte Carlo methods by applying it to classical spin models in statistical physics. First, we verify that the billiard dynamics can yield samples that converge to the true distribution of the Ising model on a small lattice, and we show that it appears to have the same convergence rate as random Monte Carlo sampling. Second, we apply the billiard dynamics to finite-size scaling analysis of the critical behavior of the Ising model and show that the phase transition point and the critical exponents are correctly obtained. Third, we extend the billiard dynamics to spins that take more than two states and show that it can be applied successfully to the Potts model. We also discuss the possibility of extensions to continuous-valued models such as the XY model.