Deterministic replica-exchange method without pseudo random numbers for simulations of complex systems

TL;DR

This paper introduces a deterministic replica-exchange method (DETREM) for complex system simulations that eliminates the need for pseudo random numbers by using a differential equation approach, maintaining accurate thermal equilibrium sampling.

Contribution

The paper presents DETREM, a novel replica-exchange method that replaces pseudo random numbers with a differential equation, ensuring detailed balance and accurate sampling without randomness.

Findings

DETREM reproduces Boltzmann distribution results.

DETREM avoids seed selection issues in pseudo random number generation.

Equivalent results to traditional REM were confirmed with the 2D Ising model.

Abstract

We propose a replica-exchange method (REM) which does not use pseudo random numbers. For this purpose, we first give a conditional probability for Gibbs sampling replica-exchange method (GSREM) based on the heat bath method. In GSREM, replica exchange is performed by conditional probability based on the weight of states using pseudo random numbers. From the conditional probability, we propose a new method called deterministic replica-exchange method (DETREM) that produces thermal equilibrium distribution based on a differential equation instead of using pseudo random numbers. This method satisfies the detailed balance condition using a conditional probability of Gibbs heat bath method and thus results can reproduce the Boltzmann distribution within the condition of the probability. We confirmed that the equivalent results were obtained by REM and DETREM with two-dimensional Ising model. DETREM can avoid problems of choice of seeds in pseudo random numbers for REM using a differential equation.