A novel stochastic approach of thermalization and symmetry breaking

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.