A novel stochastic approach of thermalization and symmetry breaking

TL;DR

This paper introduces a new diagrammatic stochastic method for analyzing thermalization and symmetry-breaking in a nonlinear damped Klein-Gordon system, combining perturbative solutions with visual representations to better model complex causal relationships.

Contribution

It develops a novel second-order diagrammatic expansion approach for stochastic Klein-Gordon equations, enhancing modeling of causal interactions and symmetry-breaking phenomena.

Findings

Simulations demonstrate relaxation to stationary states.

Identification of symmetry-broken patterns.

Enhanced understanding of causal structures in stochastic systems.

Abstract

We investigate thermalization and symmetry-breaking in a nonlinear stochastic Klein-Gordon equation on a spatial lattice, taking into account damping, nonlinear interaction, and stochastic forcing terms reduced by a perturbative solution based on retarded Green functions and the principle of Duhamel to establish a series expansion with the coupling constant. The obtained expressions have a visual representation in the form of rooted trees and Feynman-type diagrams, where their structural pattern will explain the combinatorial factors involved in the expansion. These representations offer a novel application and interpretation specifically tailored for the secondorder, damped Klein-Gordon setting, enabling more complex causal relationships to be explicitly modeled compared to first-order stochastic approaches, thus marking a technical innovation in diagrammatic expansions for such systems. The model takes into account both the initial data from a deterministic approach as well as stochastic sources, for instance, Gaussian noise. Simulations have been performed symmetry-breaking regime to show the relaxation towards stationary states and the symmetry-broken patterns.