Thermostatistics in the neighborhood of the $\pi$-mode solution for the Fermi-Pasta-Ulam $\beta$ system: from weak to strong chaos

TL;DR

This paper investigates the transition from weak to strong chaos in the Fermi-Pasta-Ulam β system near the $pi$-mode, revealing a connection to Tsallis thermostatistics during the weak chaos regime.

Contribution

It introduces a novel analysis of the $pi$-mode stability and chaos transition, linking stochastic indicators to symmetry breaking and thermostatistics in the FPU system.

Findings

Transition from weak to strong chaos linked to symmetry breaking.

In weak chaos, thermostatistics follows Tsallis distribution.

Numerical evidence of stochasticity indicator $ ho$.

Abstract

We consider a $\pi$-mode solution of the Fermi-Pasta-Ulam $\beta$ system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio $\rho$ (when is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.