Chaos, the Critical Phenomenon in Phase Space: Feigenbaum Constants and Critical Exponents

TL;DR

This paper explores chaos as a critical phenomenon in phase space, linking Feigenbaum constants to critical exponents, and discusses implications for entropy and thermodynamics in conservative systems.

Contribution

It establishes a universal connection between Feigenbaum constants and critical exponents in chaos, and analyzes entropy increase in conservative systems through phase space correlations.

Findings

Feigenbaum constants correspond to universal critical exponents.

Chaos in conservative systems leads to phase space correlations.

Entropy increases in isolated systems due to phase space information loss.

Abstract

Chaos in both dissipative systems and conservative systems is investigated on the approach of renormalization group. It is found that the chaos is regarded as the critical phenomenon of equilibrium statistics in phase space. The two Feigenbaum constants in the period-doubling bifurcation systems correspond to two independent critical exponents, which are universal and can be adopted to distinguish the classes of chaos. For the conservative systems, due to the critical nature of the chaos, the isolated systems with different parameters are correlated in the phase space, and therefore the isolated system is no longer isolated in the phase space. The information of conservative systems is irreversibly lost over time, which leads to the increase entropy in an isolated system, and the contradiction between the second law of thermodynamics and the reversibility of isolated systems can be resolved.