Kinetic Flat-Histogram Simulations of Non-Equilibrium Stochastic Processes with Continuous and Discontinuous Phase Transitions

TL;DR

This paper introduces a kinetic flat-histogram algorithm based on a generalized Wang-Landau method to efficiently sample stationary distributions of non-equilibrium stochastic processes, including those with phase transitions.

Contribution

It presents a novel flat-histogram algorithm for non-equilibrium processes, extending Wang-Landau sampling to continuous and discontinuous phase transitions.

Findings

Algorithm accurately reproduces known stationary distributions.

Effective in sampling bistable phase transitions.

Applicable to models on lattices and networks.

Abstract

As far as we know, there is no flat-histogram algorithm to sample the stationary distribution of non-equilibrium stochastic processes. The present work addresses this gap by introducing a generalization of the Wang-Landau algorithm, applied to non-equilibrium stochastic processes with local transitions. The main idea is to sample macroscopic states using a kinetic Monte Carlo algorithm to generate trial moves, which are accepted or rejected with a probability that depends inversely on the stationary distribution. The stationary distribution is refined through the simulation by a modification factor, leading to convergence toward the true stationary distribution. A visitation histogram is also accumulated, and the modification factor is updated when the histogram satisfies a flatness condition. The stationary distribution is obtained in the limit where the modification factor reaches a threshold value close to unity. To test the algorithm, we compare simulation results for several stochastic processes with theoretically known behavior. In addition, results from the kinetic flat-histogram algorithm are compared with standard exact stochastic simulations. We show that the kinetic flat-histogram algorithm can be applied to phase transitions in stochastic processes with bistability, which describe a wide range of phenomena such as epidemic spreading, population growth, chemical reactions, and consensus formation. With some adaptations, the kinetic flat-histogram algorithm can also be applied to stochastic models on lattices and complex networks.