A Cavity Master Equation for the continuous time dynamics of discrete spins models

TL;DR

This paper introduces a novel method using Random Point Processes to derive a master equation for the continuous time dynamics of discrete spin models, validated through analytical and numerical comparisons.

Contribution

The paper presents a new approach to close the Master Equation for Ising spin dynamics using Random Point Processes, enabling analytical and numerical analysis of complex spin systems.

Findings

Analytical solutions for mean field ferromagnet and 1D Ising dynamics.

Numerical validation against Monte Carlo simulations.

Effective modeling of spin systems on random graphs.

Abstract

We present a new method to close the Master Equation representing the continuous time dynamics of Ising interacting spins. The method makes use of the the theory of Random Point Processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean field ferromagnet and the dynamics of the one dimensional Ising system. We then present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the Random Field Ising model, and the Viana-Bray spin-glass model.