Generalized R\'enyi Entropy and Structure Detection of Complex Dynamical Systems

TL;DR

This paper introduces a method using generalized Rénnyi entropy to detect the structure of complex dynamical systems with Hamiltonian components, based solely on probability density functions at specific times.

Contribution

It provides necessary and sufficient conditions to determine when the Hamiltonian subsystem's influence on the rest of the system is negligible, without knowing explicit evolution laws.

Findings

Conditions for negligible back reaction of Hamiltonian subsystem.

Method applicable to measure-preserving subsystems.

Framework based on generalized Rénnyi entropy.

Abstract

We study the problem of detecting the structure of a complex dynamical system described by a set of deterministic differential equation that contains a Hamiltonian subsystem, without any information on the explicit form of evolution laws. We suppose that initial conditions are random and the initial conditions of the Hamiltonian subsystem are independent from the initial conditions of the rest of the system. The single numerical information is the probability density function of the system at one or several, finite number of time instants. In the framework of the formalism of the generalized R\'{e}nyi entropy we find necessary and sufficient conditions that the back reaction of the Hamiltonian subsystem to the rest of the system is negligible.The results can be easily generalized to the case of general, measure preserving subsystem.