A thermodynamic fluctuation relation for temperature and energy

Abstract

The present work extends the well-known thermodynamic relation C = β 2 δE 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 δβδE = 1 + δE 2 ∂ 2 S (E) /∂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) = ∂S (E) /∂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.

Extending canonical Monte Carlo methods

“…This fact avoids the incidence of the so-called super-critical slowing down, a dynamical anomaly that significantly affects the efficiency of large-scale canonical MC simulations [7]. Our proposal follows as a direct application of the recently obtained fluctuationdissipation relation [8,9]:…”

“…Our analysis starts from the consideration of the following generic energy distribution function [8]:…”

Complementarity in Quantum Mechanics and Classical Statistical Mechanics

“…Our discussion does not only demonstrate the existence of complementary relations involving thermodynamic variables (7)(8)(9), but also the existence of a remarkable analogy between the conceptual features of quantum mechanics and classical statistical mechanics. This chapter is organized as follows.…”

Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a reweighting technique

“…Section II is devoted to discuss som e im portant antecedents o f this study. F or the sake o f self-consistency o f the paper, we start review ing som e generalized fluctuation relations d e rived by Velazquez and C urilef and their relevance in MC sim ulations [23][24][25][26][27][28]. A fterw ards, w e discuss the m ain ideas associated w ith extension o f canonical M C m ethods [1,2,9] as well as connections w ith other M C m ethods that perform m icrocanonical calculations [12], Section III is devoted to discuss the application o f the m ultihistogram s m ethod to im prove these type o f m icrocanonical M C calculations.…”

“…As shown earlier by Boltzmann and Gibbs [29], the canonical ensemble (2) describes a system of interest that is put in thermal contact with an environment of constant temperature or, equivalently, a thermal bath of infinite heat capacity. In full analogy with the known relation [30] C = p2(SU2) (3) of classical fluctuation theory is employed in any MC study based on canonical ensemble (2) to obtain the heat capacity C from the energy fluctuations, the more general fluctuation relation (1) can be employed with the same purpose in any MC study where the environmental inverse temperature experiences thermal fluctuations that are coupled with thermal fluctuations of the system energy U [23][24][25].…”