Bitemporal Dynamics Sinai Divergence, An Energetic Analog to Boltzmann's Entropy?

TL;DR

This paper explores how Sinai chaos with smooth potentials and time dependence can lead to a classical entropy analog in real space, linking dynamical chaos to energetic divergence.

Contribution

It introduces a novel analogy between Sinai divergence and Boltzmann's entropy through energetic and dynamical analysis of smooth, time-dependent potentials.

Findings

Sinai divergence persists with smooth potentials

Energy decreases systematically in time-dependent scenarios

Classical entropy analog can be derived from dynamical chaos

Abstract

Sinai chaos is characterized by exponential divergence between neighboring trajectories of a point billiard. If the repulsive potential of the finite-diameter fixed particle in the middle of the table is made smooth, the Sinai divergence persists with finite measure. So it does if the smooth potential is made attractive. So it still does if the potential is in addition made time-dependent (periodic). Then a systematic decrease in energy of the moving particle can be predicted to occur in both time directions for a long time. If so, classical entropy acquires an analog in real space.