Improvement of analysis for relaxation of fluctuations by the use of Gaussian process regression and extrapolation method

TL;DR

This paper introduces a Gaussian process regression and bootstrap-based extrapolation method to enhance the accuracy and reliability of nonequilibrium relaxation analysis for critical phenomena, validated on Ising models.

Contribution

The paper presents a novel approach combining Gaussian process regression with bootstrap and extrapolation to improve critical exponent estimation in NER analysis.

Findings

Accurate estimation of critical exponents for 2D Ising model.

Consistent critical exponents for 3D Ising model.

Enhanced reliability and reproducibility of NER analysis.

Abstract

The nonequilibrium relaxation (NER) method, which has been used to investigate equilibrium systems via their nonequilibrium behavior, has been widely applied to various models to estimate critical temperatures and critical exponents. Although the estimation of critical temperatures has become more reliable and reproducible, that of critical exponents raises concerns about the method's reliability. Therefore, we propose a more reliable and reproducible approach using Gaussian process regression. In addition, the present approach introduces statistical errors through the bootstrap method by combining them using the extrapolation method. Our estimation for the two-dimensional Ising model yielded $\beta = 0.12504(6)$, $\gamma = 1.7505(10)$, and $\nu = 1.0003(6)$, consistent with the exact values. The value $z = 2.1669(9)$ is reliable because of the high accuracy of these exponents. We also obtained the critical exponents for the three-dimensional Ising model and found that they are close to those reported in a previous study. Thus, for systems undergoing second-order transitions, our approach improves the accuracy, reliability, and reproducibility of the NER analysis. Because the proposed approach requires the relaxation of some observables from Monte Carlo simulations, its simplicity imparts it with significant potential.