Event-Driven Monte Carlo: exact dynamics at all time-scales for discrete-variable models

TL;DR

This paper introduces an exact event-driven Monte Carlo algorithm for simulating real-time dynamics of discrete-variable classical systems, capable of handling ultra-fast regimes without assuming specific statistical distributions.

Contribution

The proposed algorithm uniquely does not rely on predefined distributions, enabling precise simulation of fast and ultra-fast dynamics in classical many-body systems.

Findings

Successfully tested against known solutions for Ising models

Accurately captures non-equilibrium dynamics and thermodynamics

Outperforms standard kinetic Monte Carlo in ultra-fast regimes

Abstract

We present an algorithm for the simulation of the exact real-time dynamics of classical many-body systems with discrete energy levels. In the same spirit of kinetic Monte Carlo methods, a stochastic solution of the master equation is found, with no need to define any other phase-space construction. However, unlike existing methods, the present algorithm does not assume any particular statistical distribution to perform moves or to advance the time, and thus is a unique tool for the numerical exploration of fast and ultra-fast dynamical regimes. By decomposing the problem in a set of two-level subsystems, we find a natural variable step size, that is well defined from the normalization condition of the transition probabilities between the levels. We successfully test the algorithm with known exact solutions for non-equilibrium dynamics and equilibrium thermodynamical properties of Ising-spin models in one and two dimensions, and compare to standard implementations of kinetic Monte Carlo methods. The present algorithm is directly applicable to the study of the real time dynamics of a large class of classical markovian chains, and particularly to short-time situations where the exact evolution is relevant.