非线性随机 Klein-Gordon 方程在空间晶格上的热化与对称破缺新方法
A novel stochastic approach of thermalization and symmetry breaking
AI总结：
本文提出了一种非线性随机 Klein-Gordon 方程在空间晶格上的热化与对称破缺新方法，通过根树和费曼图展开，实现了对复杂因果关系的建模。
摘要：
我们研究了非线性随机 Klein-Gordon 方程在空间晶格上的热化与对称破缺现象，考虑了阻尼、非线性相互作用以及随机激励项，基于延迟格林函数和杜哈梅原理的微扰解来建立一个耦合常数的级数展开。所得表达式以根树和费曼型图的形式进行可视化，其结构模式将解释展开中的组合因子。这些表示方法为第二阶阻尼 Klein-Gordon 设置提供了新的应用和解释，能够更清晰地建模复杂的因果关系，相较于一阶随机方法具有技术创新性。该模型考虑了来自确定性方法的初始数据以及随机源，例如高斯噪声。仿真在对称破缺区域进行了展示，以显示向稳态状态的弛豫以及对称破缺模式。
英文摘要：
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.
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