Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise

A recent proposal of an effective temperature θ, conjugated to a generalized entropy s_{q}, typical of nonextensive statistical mechanics, has led to a consistent thermodynamic framework in the case q=2. The proposal was explored for repulsively interacting vortices, currently used for modeling type-II superconductors. In these systems, the variable θ presents values much higher than those of typical room temperatures T, so that the thermal noise can be neglected (T/θ≃0). The whole procedure was developed for an equilibrium state obtained after a sufficiently long-time evolution, associated with a nonlinear Fokker-Planck equation and approached due to a confining external harmonic potential, ϕ(x)=αx^{2}/2 (α>0). Herein, the thermodynamic framework is extended to a quite general confining potential, namely ϕ(x)=α|x|^{z}/z (z>1). It is shown that the main results of the previous analyses hold for any z>1: (i) The definition of the effective temperature θ conjugated to the entropy s_{2}. (ii) The construction of a Carnot cycle, whose efficiency is shown to be η=1-(θ_{2}/θ_{1}), where θ_{1} and θ_{2} are the effective temperatures associated with two isothermal transformations, with θ_{1}>θ_{2}. The special character of the Carnot cycle is indicated by analyzing another cycle that presents an efficiency depending on z. (iii) Applying Legendre transformations for a distinct pair of variables, different thermodynamic potentials are obtained, and furthermore, Maxwell relations and response functions are derived. The present approach shows a consistent thermodynamic framework, suggesting that these results should hold for a general confining potential ϕ(x), increasing the possibility of experimental verifications.