Classical thermodynamics from quasi-probabilities

Abstract

The basic idea of a microscopic understanding of Thermodynamics is to derive its main features from a microscopic probability distribution. In such a vein, we investigate the thermal statistics of quasi-probabilities's semi-classical analogs in phase space for the important case of quadratic Hamiltonians, focusing attention in the three more important instances, i.e., those of Wigner, P -, and Husimi distributions. Introduction of an effective temperature permits one to obtain a unified thermodynamic description that encompasses and unifies the three different quasi-probability distributions. This unified description turns out to be classical.

The Fisher Thermodynamics of Quasi-Probabilities

“…In Section 2, we revisit some key ideas developed in [4,9] but reexpressing them in a phase space language. In Section 3, we recapitulate the quasi-probabilities treatment of Lee [23], and finally, in Section 4, we discuss the effective temperature approach for quasi-probabilities developed in [24]. The subsequent three Sections convey our new materials.…”

“…The subsequent three Sections convey our new materials. Section 5 relates Fisher's measure to the effective temperature of [24]. Section 6 discusses that special kind of Fisher information indicator (cf.…”

“…To end our recapitulation process, we review here now ideas of [24] that will be later generalized à la Fisher. Let us define the classical expectation value of the arbitrary function A(x, p) as…”

“…The pertinent effective, classical HO-Hamiltonian reads [24] H(x, p) =hω|α| 2 (29) where x and p are phase space variables, |α| 2 = x 2 /4σ 2 x + p 2 /σ 2 p , and σ 2 x =h/2mω and σ 2 p =hmω/2 [3]. Let us discuss in some more detail the concept of effective temperature [24] with reference to the mean value U * of the effective Hamiltonian for a given free parameter n that mimics the mean phonon number n , defined as n = e −βhω 1 − e −βhω (30) so that…”

“…This entails τ > 0. Taking into account the definition Equation (31), we recast Equation (24) in the following manner…”