Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

TL;DR

This paper investigates the statistical behavior of sums of chaotic variables in multi-dimensional Hamiltonian systems, revealing long-lasting q-Gaussian distributions in weak chaos regimes and their transition to Gaussian distributions over time.

Contribution

It demonstrates the emergence and evolution of q-Gaussian distributions in weakly chaotic regimes of FPU chains and microplasma systems, highlighting finite-size effects and energy-dependent chaos regimes.

Findings

q-Gaussian distributions approximate sums in weak chaos for long times

Distributions tend to Gaussian as orbits explore larger phase space regions

Identification of weak chaos regimes in microplasma via q-index variations

Abstract

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$\beta$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian ($1<q<3$) distribution and tend to a Gaussian ($q=1$) for longer times, as the orbits eventually enter into "large size" chaotic domains. However, in agreement with other studies, we find in certain cases that the $q$-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called "crossover" function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these $q$-Gaussian distributions to identify two energy regimes of "weak chaos"-one where the system melts and one where it transforms from liquid to a gas state-by observing where the $q$-index of the distribution increases significantly above the $q=1$ value of strong chaos.