Lyapunov exponent of many-particle systems: testing the stochastic approach

TL;DR

This paper tests the stochastic approach for calculating the largest Lyapunov exponent in many-particle XY-Hamiltonian systems, showing good agreement with simulations in disordered phases and highlighting the method's predictive power for correlation functions.

Contribution

It validates the stochastic approach for Lyapunov exponent estimation in mean-field XY-Hamiltonians and demonstrates its effectiveness in weakly chaotic regimes.

Findings

Quantitative agreement between theory and simulations in disordered phases

Successful prediction of correlation function shapes and times

Applicability to systems with attractive and repulsive interactions

Abstract

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.