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.

Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations C = β2⟨δU2⟩, which is able to describe the existence of macrostates with negative heat capacities C < 0. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the fundamental and the complementary fluctuation theorems, which represent the generalization of two fluctuation identities already obtained in previous works, and the associated fluctuation theorem, a result that has no counterpart in the framework of Boltzmann–Gibbs distributions. These results are applied to study the anomalous susceptibility of a ferromagnetic system, in particular, the case of the 2D Ising model.

In this work, we discuss the implications of a recently obtained equilibrium fluctuation-dissipation relation on the extension of the available Monte Carlo methods based on the consideration of the Gibbs canonical ensemble to account for the existence of an anomalous regime with negative heat capacities C < 0. The resulting framework appears as a suitable generalization of the methodology associated with the so-called dynamical ensemble, which is applied to the extension of two well-known Monte Carlo methods: the Metropolis importance sample and the Swendsen-Wang clusters algorithm. These Monte Carlo algorithms are employed to study the anomalous thermodynamic behavior of the Potts models with many spin states q defined on a d-dimensional hypercubic lattice with periodic boundary conditions, which successfully reduce the exponential divergence of decorrelation time τ with the increase of the system size N to a weak power-law divergence τ ∝ N α with α ≈ 0.2 for the particular case of the 2D 10-state Potts model.

Recently, we have presented some simple arguments supporting the existence of a certain complementarity between thermodynamic quantities of temperature and energy, an idea suggested by Bohr and Heinsenberg in the early days of Quantum Mechanics. Such a complementarity is expressed as the impossibility to perform an exact simultaneous determination of the system energy and temperature by using an experimental procedure based on the thermal equilibrium with other system regarded as a measuring apparatus (thermometer). In this work, we provide a simple generalization of the last approach with the consideration of a thermodynamic situation with several control parameters.