Renormalized waves and thermalization of the Klein-Gordon equation: What sound does a nonlinear string make?

TL;DR

This paper investigates the thermalization process of the nonlinear Klein-Gordon equation, revealing that in a renormalized wave basis, the system exhibits weakly nonlinear behavior with a Planck-like spectrum, even under strong nonlinearities.

Contribution

It introduces a local in time renormalized wave basis to analyze thermalization, showing weakly nonlinear oscillations and a shifted dispersion relation in the Klein-Gordon equation.

Findings

Renormalized waves oscillate around a single frequency.

The dispersion relation is nonlinearly shifted by the mean square field.

The spectrum exhibits a Planck-like distribution with energy equipartition at low frequencies.

Abstract

We study the thermalization of the classical Klein-Gordon equation under a u^4 interaction. We numerically show that even in the presence of strong nonlinearities, the local thermodynamic equilibrium state exhibits a weakly nonlinear behavior in a renormalized wave basis. The renormalized basis is defined locally in time by a linear transformation and the requirement of vanishing wave-wave correlations. We show that the renormalized waves oscillate around one frequency, and that the frequency dispersion relation undergoes a nonlinear shift proportional to the mean square field. In addition, the renormalized waves exhibit a Planck like spectrum. Namely, there is equipartition of energy in the low frequency modes described by a Boltzmann distribution, followed by a linear exponential decay in the high frequency modes.