Thermodynamic framework for compact q-Gaussian distributions

Recent works have associated systems of particles, characterized by short-range repulsive interactions and evolving under overdamped motion, to a nonlinear Fokker-Planck equation within the class of nonextensive statistical mechanics, with a nonlinear diffusion contribution whose exponent is given by ν=2−q. The particular case ν=2 applies to interacting vortices in type-II superconductors, whereas ν>2 covers systems of particles characterized by short-range power-law interactions, where correlations among particles are taken into account. In the former case, several studies presented a consistent thermodynamic framework based on the definition of an effective temperature θ (presenting experimental values much higher than typical room temperatures T, so that thermal noise could be neglected), conjugated to a generalized entropy sν (with ν=2). Herein, the whole thermodynamic scheme is revisited and extended to systems of particles interacting repulsively, through short-ranged potentials, described by an entropy sν, with ν>1, covering the ν=2 (vortices in type-II superconductors) and ν>2 (short-range power-law interactions) physical examples. One basic requirement concerns a cutoff in the equilibrium distribution Peq(x), approached due to a confining external harmonic potential, ϕ(x)=αx2∕2 (α>0). The main results achieved are: (a) The definition of an effective temperature θ conjugated to the entropy sν; (b) The construction of a Carnot cycle, whose efficiency is shown to be η=1−(θ2∕θ1), where θ1 and θ2 are the effective temperatures associated with two isothermal transformations, with θ1>θ2; (c) Thermodynamic potentials, Maxwell relations, and response functions. The present thermodynamic framework, for a system of interacting particles under the above-mentioned conditions, and associated to an entropy sν, with ν>1, certainly enlarges the possibility of experimental verifications.