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A novel stochastic approach of thermalization and symmetry breaking

Boubaker Smii

TL;DR

The paper addresses thermalization and symmetry breaking for a damped stochastic Klein–Gordon field on a spatial lattice with nonlinear coupling and Gaussian noise. It introduces a systematic perturbative framework based on retarded Green functions and Duhamel's principle, yielding a formal power series in the coupling constant $\lambda$ and expressing perturbative terms through space-time convolutions. A novel diagrammatic calculus using rooted trees and Feynman-type rules is developed to organize and interpret the perturbative expansion, with explicit examples and symmetry factors that clarify the underlying combinatorics for second-order damped dynamics. Numerical simulations in one dimension corroborate the theory, showing relaxation to stationary states and symmetry-broken patterns under the combined influence of damping and noise, and demonstrating how these factors shape domain formation and thermalization.

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.

A novel stochastic approach of thermalization and symmetry breaking

TL;DR

The paper addresses thermalization and symmetry breaking for a damped stochastic Klein–Gordon field on a spatial lattice with nonlinear coupling and Gaussian noise. It introduces a systematic perturbative framework based on retarded Green functions and Duhamel's principle, yielding a formal power series in the coupling constant and expressing perturbative terms through space-time convolutions. A novel diagrammatic calculus using rooted trees and Feynman-type rules is developed to organize and interpret the perturbative expansion, with explicit examples and symmetry factors that clarify the underlying combinatorics for second-order damped dynamics. Numerical simulations in one dimension corroborate the theory, showing relaxation to stationary states and symmetry-broken patterns under the combined influence of damping and noise, and demonstrating how these factors shape domain formation and thermalization.

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.
Paper Structure (12 sections, 8 theorems, 57 equations)

This paper contains 12 sections, 8 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Perturbative Solution on the Lattice for the Damped Stochastic Klein--Gordon Equation
  3. Functional setting and notations
  4. Duhamel formula and Perturbative expansion
  5. Feynman rules, worked trees and graphical representations
  6. Feynman diagrams and analytic value of the Tree
  7. Example A1 (all-noise leaves: $\xi\xi\xi$).
  8. Example A2 (one $f$ and two $\xi$ leaves: $f\,\xi\,\xi$).
  9. Example A5 (all-noise, nested).
  10. Numerical Simulations
  11. First-order formulation and simulations parameters
  12. Observables

Key Result

Lemma 1

The operator $\Delta_\delta$ is diagonal in Fourier space: Consequently, $-\Delta_\delta$ has positive symbol

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Definition 3
  • Proposition 2
  • ...and 13 more