Thermodynamic fluctuation relation for temperature and energy

TL;DR

This paper generalizes a fundamental thermodynamic fluctuation relation to include finite systems and negative heat capacities, providing new insights into thermodynamic states, and discusses implications for Monte Carlo methods and quantum analogies.

Contribution

It introduces a generalized fluctuation relation for finite systems in thermodynamic equilibrium with a generalized thermostat, extending the classical relation to negative heat capacities.

Findings

The relation accounts for negative heat capacities in thermodynamic states.

It imposes restrictions on determining the microcanonical caloric curve.

Implications for new Monte Carlo methods and quantum analogies are discussed.

Abstract

The present work extends the well-known thermodynamic relation $C=\beta ^{2}< \delta {E^{2}}>$ for the canonical ensemble. We start from the general situation of the thermodynamic equilibrium between a large but finite system of interest and a generalized thermostat, which we define in the course of the paper. The resulting identity $< \delta \beta \delta {E}> =1+< \delta {E^{2}}% > \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states with a negative heat capacity $C<0$; at the same time, it represents a thermodynamic fluctuation relation that imposes some restrictions on the determination of the microcanonical caloric curve $\beta (E) =\partial S(E) /\partial E$. Finally, we comment briefly on the implications of the present result for the development of new Monte Carlo methods and an apparent analogy with quantum mechanics.