Equilibrium Distributions for Strongly Nonlinear Many-Body Systems

Abstract

Obtaining equilibrium distributions of nonlinear systems is essential for accurately computing macroscopic observables. Conventional theoretical corrections are typically limited to weak nonlinearities, where interaction terms can be treated as effectively uncorrelated perturbations and the random phase approximation applies. In this Letter, we develop a framework to determine equilibrium distributions based on the generalized energy equipartition principle. Our approach recovers existing corrections in the weakly nonlinear regime and, crucially, remains valid for strong nonlinearities, where perturbative contributions become correlated and conventional approaches break down. Numerical simulations of the nonlinear Schr\"odinger equation, the Majda-McLaughlin-Tabak model, and the Fermi-Pasta-Ulam-Tsingou model demonstrate accurate corrections for nonlinearities more than an order of magnitude stronger than those accessible to conventional theories.