Effective-temperature concept: A physical application for nonextensive statistical mechanics

The H-theorem [$(df/dt)\le 0$] for a free-energy functional, $f=u-\theta s$ (with $u$ and $s$ representing, respectively, the internal energy and a generalized entropy of a given physical system), has been proven previously by making use of a nonlinear Fokker-Planck equation. Herein we focus on a nonlinear Fokker-Planck equation derived by means of a coarse-graining procedure on the equations of motion of a system of interacting vortices, under overdamped motion, in the absence of thermal noise ($T=0$). In this case, we show that the parameter $\theta$ is directly related to the density as well as to the interactions among vortices. Generalized quantities such as entropy, internal energy, free energy, and heat capacity are analyzed for varying $\theta$: important relations and physical behavior analogous to those of standard thermodynamics are found, showing that $\theta$ plays the role of an effective temperature. Estimates of $\theta$ in typical physical situations of different type-II superconductors are presented; in addition to this, possible experimental procedures for varying $\theta$ are proposed.

Figure 1

The dimensionless equilibrium distribution $\lambda P_{\mathrm{st}}\left(x\right)$ is represented vs the position $x$ (in units of $\lambda$) for typical choices of $\tau$, namely $\tau =1,2,4$ (from top to bottom). In the inset, we present the same distribution in the special value ${\tau }^{*}=0.072$, for which the entropy becomes zero, in a different scale.

Figure 2

The internal energy $u$ (in units of $\alpha {\lambda }^2$) and the entropy $s$ (in units of $k$) are represented vs the dimensionless parameter $\tau$. The entropy becomes negative only for very small values of $\tau$, namely for $\tau <0.072$.