Fisher information and the thermodynamics of scale-invariant systems

Abstract

We present a thermodynamic formulation for scale-invariant systems based on the minimization with constraints of Fisher's information measure. In such a way a clear analogy between these systems's thermal properties and those of gases and fluids is seen to emerge in natural fashion. We focus attention on the non-interacting scenario, speaking thus of scale-free ideal gases (SFIGs) and present some empirical evidences regarding such disparate systems as electoral results, city populations and total citations in Physics journals, that seem to indicate that SFIGs do exist. We also illustrate the way in which Zipf's law can be understood in a thermodynamical context as the surface of a finite system. Finally, we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature.

Extreme Fisher Information, Non-Equilibrium Thermodynamics and Reciprocity Relations

“…As a didactic example we will here discuss the Fisher treatment of the ideal gas, by following the considerations expounded in [23]. We look for the density distribution, in configuration space, of the (translational invariant) ideal gas (IG) that describes non-interacting classical particles of mass m with coordinates q = (r, p), where mdr/dt = p. The translational invariance is described by the translational family of distributions F (r, p|θ r , θ p ) = F (r , p ) whose form does not change under the transformations r = r − θ r and p = p − θ p .…”

“…This assumption is introduced into the Fisher information measure (FIM) (11), with 2D dimensions (phase space). For the sake of dimensional balance we introduce in (11) two appropriate dimensional constants, namely, c r for space coordinates and c p for momentum coordinates [23]. The phase space probability density F (r, p) can obviously be factorized in the fashion F (r, p) = ρ(r)η(p), and then it follows from the additivity of the information measure [5] that I = I r + I p , i.e., FIM becomes the sum of a coordinate-FIM and a momentum-one.…”

“…In extremizing FIM we constrain [23] the normalization of ρ(r) and η(p) to the total number of particles N and to 1, respectively, i.e.,…”

“…In addition, we must penalize infinite values for the particle momentum (infinite energies are un-physical) with a constraint on the variance of η(p) to force it to be finite [23], namely,…”

Interplay of information quantifiers and the modified Jaynes-Cummings model

“…and is encountered in many physical applications (see, for instance, [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], and references therein). The FIM associated to Husimi distributions Q F (X 1 X 2 ) is defined as [49] …”