Second Law of Thermodynamics and Macroscopic Observables within Boltzmann's principle, an attempt

TL;DR

This paper extends Boltzmann's principle to non-equilibrium systems with fractal phase space distributions, deriving the Second Law without thermodynamic limits by using box-counting volumes and phase space closures.

Contribution

It introduces a generalized phase-space volume measure for non-equilibrium systems, deriving irreversibility without coarse graining or thermodynamic limits.

Findings

Derivation of the Second Law from fractal phase space distributions.

Introduction of box-counting volume as a limit of infinite resolution.

Explanation of irreversibility via phase space closure volumes.

Abstract

Boltzmann's principleS=k*ln W is generalized to non-equilibrium Hamiltonian systems with possibly fractal distributions in phase space by the box-counting volume. The probabilities P(M) of macroscopic observables M are given by the ratio P(M)=W(M)/W of these volumes of the sub-manifold {M} of the microcanonical ensemble with the constraint M to the one without. With this extension of the phase-space integral the Second Law is derived without invoking the thermodynamic limit. The irreversibility in this approach is due to the replacement of the phase space volume of the possibly fractal sub-manifold {M} by the volume of the closure of {M}. In contrast to conventional coarse graining the box-counting volume is defined by the limit of infinite resolution.