Kolmogorov-Sinai entropy of many-body Hamiltonian systems

Abstract

The Kolmogorov-Sinai (KS) 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 KS 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 KS entropy. By way of a nontrivial illustration, the KS 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 KS entropy within the diagonal approximation is derived and its range of validity is also explored.