Equilibrium fluctuation theorems compatible with anomalous response

Abstract

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.

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.…”

“…(76) is the famous Cramer-Rao theorem of inference theory (11), which puts a lower bound to the efficiency of any unbiased estimatorsθ α . As clearly shown in Table 1, fluctuation theory and inference theory can be regarded as dual counterpart statistical approaches (9). In fact, there exists a direct correspondence among their respective definitions and theorems.…”

Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory

“…One can perform an indirect but complete characterization of the abstract statistical manifolds M and P studying the behavior of a complete set of random quantities ξ. Specifically, the behavior of these random quantities is fully determined by the knowledge of the family of continuous distributions: dp ξ (x|θ) = ρ ξ (x|θ)dx, (1) where the nonnegative function ρ ξ (x|θ) is the probability density, while dx denotes the ordinary volume element (Lebesgue measure of the n-dimensional real space R n ). Denoting by S ⊂ R x , the integral:…”

“…Here, each point θ ∈ R θ is associated with only one member of the distributions family (1). The set of continuous variables θ = (θ 1 , θ 2 , .…”

“…The analysis of such statistical methods is outside the interest of the present work. On the contrary, the main interest here concerns to the information about the abstract statistical manifolds M and P that is obtained from the knowledge of the family of continuous distributions (1).…”