标题： 强非线性系统的平衡分布函数 显示英文标题

标题： The equilibrium distribution function for strongly nonlinear systems

摘要： 平衡分布函数决定了统计物理中的宏观可观测量。 虽然传统方法在弱非线性或接近可积系统中可以修正平衡分布，但在强非线性区域则失效。 我们开发了一个框架，以在强非线性多体系统中获得平衡分布和色散关系，结合了超出随机相位近似的修正，并捕捉了内在的非线性效应。 该理论在非线性薛定谔方程、Majda-McLaughlin-Tabak模型和FPUT-beta模型上得到了验证，展示了其在不同类型的非线性系统中的准确性。 数值结果表明，即使在强非线性区域，该方法也显著优于现有方法。 这项工作为强非线性系统中的平衡统计特性建立了理论基础。 显示英文摘要

摘要： The equilibrium distribution function determines macroscopic observables in statistical physics. While conventional methods correct equilibrium distributions in weakly nonlinear or near-integrable systems, they fail in strongly nonlinear regimes. We develop a framework to get the equilibrium distributions and dispersion relations in strongly nonlinear many-body systems, incorporating corrections beyond the random phase approximation and capturing intrinsic nonlinear effects. The theory is verified on the nonlinear Schrodinger equation, the Majda-McLaughlin-Tabak model, and the FPUT-beta model, demonstrating its accuracy across distinct types of nonlinear systems. Numerical results show substantial improvements over existing approaches, even in strong nonlinear regimes. This work establishes a theoretical foundation for equilibrium statistical properties in strongly nonlinear systems.