A New Thermodynamics from Nuclei to Stars III

TL;DR

This paper presents a fundamental, geometrical approach to equilibrium thermodynamics based on microcanonical ensemble, applicable to both extensive and non-extensive systems, and capable of describing phase transitions in small systems without relying on the thermodynamic limit.

Contribution

It introduces a geometrical, information-theory-independent definition of equilibrium statistics valid for all system sizes, including nuclei and stars, and clarifies the nature of phase transitions and the second law.

Findings

Microcanonical ensemble accurately describes all systems without thermodynamic limit.

Phase transitions are sharply distinguishable even in small systems.

The second law naturally emerges from the statistical framework.

Abstract

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the $N-$body phase space with the given total energy. Due to Boltzmann's principle, $e^S=tr(\delta(E-H))$, its geometrical size is related to the entropy $S(E,N,...)$. This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption, as are needed in conventional (canonical) thermo-statistics. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the {\em fundamental} definition of any classical equilibrium statistics. It can address nuclei and astrophysical objects as well. All kind of phase transitions can be distinguished sharply and uniquely for even small systems. It is further shown that the second law is a natural consequence of the statistical nature of thermodynamics which describes all systems with the same -- redundant -- set of few control parameters simultaneously. It has nothing to do with the thermodynamic limit. It even works in systems which are by far {\em larger} than any thermodynamic "limit".