Theoretical estimates for the largest Lyapunov exponent of many-particle systems

TL;DR

This paper develops a theoretical approach to estimate the largest Lyapunov exponent in many-particle Hamiltonian systems by modeling the tangent vector evolution as a stochastic process and simplifying the calculation through cumulant expansion.

Contribution

It introduces a novel method to estimate the Lyapunov exponent using a small matrix derived from the cumulant expansion of the stochastic process.

Findings

Lyapunov exponent can be obtained from a 3x3 matrix in some cases

The method relates the Lyapunov exponent to correlation functions of the potential's Hessian

Connection established between stochastic approach and geometric method

Abstract

The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N-particle Hamiltonian system, with a smooth Hamiltonian of the type p^2 + v(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cummulant expansion. We show that if cummulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a small-dimension matrix, which, in some cases, can be as small as 3x3. In all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.