# Universal law of thermalization for one-dimensional perturbed Toda lattices

## TL;DR

This paper demonstrates that in one-dimensional perturbed Toda lattices, the thermalization time universally scales as the inverse square of the perturbation strength, revealing a fundamental law governing thermalization in weakly nonlinear systems.

## Contribution

It establishes a universal law for thermalization time in perturbed Toda lattices, linking it to perturbation strength and extending to weak nonlinear lattices.

## Key findings

- Thermalization time scales as $oxed{T_{eq} \\sim \\epsilon^{-2}}$.
- Universal applicability to weak nonlinear lattices.
- Thermalization behavior derived in the thermodynamic limit.

## Abstract

The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, $T_{eq}$, follows a universal law; i.e., $T_{eq}\sim \epsilon^{-2}$, where the perturbation strength, $\epsilon$, characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.