Thermodynamic fluctuation relation for temperature and energy
arXiv:cond-mat/0702031 · doi:10.1088/1751-8113/42/9/095006
Abstract
The present work extends the well-known thermodynamic relation $C=β^{2}< δ{E^{2}}>$ 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 $< δβδ{E}> =1+< δ{E^{2}}% > \partial ^{2}S(E) /\partial {E^{2}}$ 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 $β(E) =\partial S(E) /\partial E$. 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.
Version accepted for publication in J. Phys. A: Math and Theo