Second Law of Thermodynamics, Macroscopic Observables within Boltzmann's Principle but without Thermodynamic Limit

TL;DR

This paper extends Boltzmann's principle to small and non-equilibrium systems without relying on the thermodynamic limit, using ensemble averaging and box-counting volume to derive the Second Law and thermodynamics.

Contribution

It introduces a generalized framework for thermodynamics and the Second Law applicable to small and fractal systems without the thermodynamic limit, using ensemble averages and box-counting volume.

Findings

Derives the Second Law without the thermodynamic limit.

Generalizes Boltzmann's principle to non-equilibrium systems with fractal phase space distributions.

Shows thermodynamics can be deduced from ensemble averages and coarse-graining based on box-counting volume.

Abstract

Boltzmann's principle S=k ln W allows to extend equilibrium thermo-statistics to ``Small'' systems without invoking the thermodynamic limit. The clue is to base statistical probability on ensemble averaging and not on time averaging. It is argued that due to the incomplete information obtained by macroscopic measurements thermodynamics handles ensembles or finite-sized sub-manifolds in phase space and not single time-dependent trajectories. Therefore, ensemble averages are the natural objects of statistical probabilities. This is the physical origin of coarse-graining which is not anymore a mathematical ad hoc assumption. From this concept all equilibrium thermodynamics can be deduced quite naturally including the most sophisticated phenomena of phase transitions for ``Small'' systems. Boltzmann's principle is generalized to non-equilibrium Hamiltonian systems with possibly fractal distributions ${\cal{M}}$ in 6N-dim. phase space by replacing the conventional Riemann integral for the volume in phase space by its corresponding box-counting volume. This is equal to the volume of the closure $\bar{\cal{M}}$. With this extension 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 by the volume of its closure $\bar{\cal{M}}$. The physical reason for this replacement is that macroscopic measurements cannot distinguish ${\cal{M}}$ from $\bar{\cal{M}}$. Whereas the former is not changing in time due to Liouville's theorem, the volume of the closure can be larger. In contrast to conventional coarse graining the box-counting volume is defined in the limit of infinite resolution. I.e. there is no artificial loss of information.