Negative heat-capacity at phase-separations in microcanonical thermostatistics of macroscopic systems with either short or long-range interactions

TL;DR

This paper explores the microcanonical thermostatistics of macroscopic systems, revealing the fundamental role of convex entropy and negative heat capacity during phase separation, regardless of interaction range.

Contribution

It demonstrates the necessity of convex entropy and negative heat capacity in phase separation for both small and large systems, challenging traditional canonical approaches.

Findings

Negative heat capacity occurs at phase separation.

Convex entropy S(E) is essential in microcanonical thermodynamics.

Results apply to systems with short and long-range interactions.

Abstract

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck statistics. This was elaborated in the talk presented at this conference. It turns out that the Boltzmann-Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. This can be done to a far extend rigorously and analytically. The deep and essential difference between ``extensive'' and ``intensive'' control parameters, i.e. microcanonical and canonical statistics, was exemplified by rotating, self-gravitating systems. In the present paper the necessary appearance of a convex entropy $S(E)$ and the negative heat capacity at phase separation in small as well macroscopic systems independently of the range of the force is pointed out.