On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems

TL;DR

This paper develops a transfer matrix formalism to compute the Kolmogorov-Sinai entropy in many-body Hamiltonian systems, introducing approximations and applying them to coupled kicked rotors to analyze chaos complexity.

Contribution

It introduces a novel transfer matrix approach and a hierarchy of approximations for calculating K-S entropy in many-body systems, including an analytic formula within the diagonal approximation.

Findings

The transfer matrix formalism reveals a duality analogous to Anderson models.

The diagonal approximation effectively estimates K-S entropy using instantaneous Hessians.

Analytic expressions for K-S entropy are validated for coupled kicked rotors.

Abstract

The Kolmogorov-Sinai (K-S) entropy is a central measure of complexity and chaos. Its calculation for many-body systems is an interesting and important challenge. In this paper, the evaluation is formulated by considering $N$-dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem. This approach makes explicit a duality relation that is exactly analogous to one found in a generalized Anderson tight-binding model, and leads to a formally exact expression for the finite-time K-S entropy. Within this formalism there is a hierarchy of approximations, the final one being a diagonal approximation that only makes use of instantaneous Hessians of the potential to find the K-S entropy. By way of a non-trivial illustration, the K-S entropy of $N$ identically coupled kicked rotors (standard maps) is investigated. The validity of the various approximations with kicking strength, particle number, and time are elucidated. An analytic formula for the K-S entropy within the diagonal approximation is derived and its range of validity is also explored.