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

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.

5 figures, resubmitted to Phys. Rev. E

References in corpus (5)

• Determinants of Block Tridiagonal Matrices
• Fluctuations of finite-time stability exponents in the standard map and the detection of small islands
• Spectral duality and distribution of exponents for transfer matrices of block tridiagonal Hamiltonians
• Fluctuations in classical sum rules
• Chaoticity of the Wet Granular Gas

Cited by in corpus (5)

• Semiclassical roots of universality in many-body quantum chaos
• Chaos in high-dimensional dynamical systems
• Entanglement between two subsystems, the Wigner semicircle and extreme value statistics
• Spatio-temporal spread of perturbations in a driven dissipative Duffing chain: an OTOC approach
• Thermalization slowing down in multidimensional Josephson junction networks