A system of interacting vortices under overdamped motion, which has been commonly used in the literature to model flux-front penetration in disordered type-II superconductors, was recently related to a nonlinear Fokker-Planck equation, characteristic of nonextensive statistical mechanics, through an analysis of its stationary state. Herein, this connection is extended by means of a thorough analysis of the time evolution of this system. Numerical data from molecular-dynamics simulations are presented for both position and velocity probability distributions P(x,t) and P(vx,t), respectively; both distributions are well fitted by similar q-Gaussian distributions, with the same index q=0, for all times considered. Particularly, the evolution of the system occurs in such a way that P(x,t) presents a time behavior for its width, normalization, and second moment, in full agreement with the analytic solution of the nonlinear Fokker-Planck equation. The present results provide further evidence that this system is deeply associated with nonextensive statistical mechanics.

Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved;for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.

Under the assumption that the physically appropriate entropy of generic complex systems satisfies thermodynamic extensivity, we investigate the recently introduced entropy $S_{\delta }$ (which recovers the usual Boltzmann-Gibbs form for $\delta =1$) and establish the microcanonical and canonical extremizing distributions. Using a generalized version of the H theorem, we find the nonlinear Fokker-Planck equation associated with that entropic functional and calculate the stationary-state probability distributions. We demonstrate that both approaches yield one and the same equation, which in turn uniquely determines the probability distribution. We show that the equilibrium distributions asymptotically behave like stretched exponentials, and that, in appropriate probability-energy variables, an interesting return occurs at $\delta =4/3$. As a mathematically simple illustration, we consider the one-dimensional harmonic oscillator and calculate the generalized chemical potential for different values of $\delta$.

A model of superconducting vortices under overdamped motion is currently used for describing type-II superconductors. Recently, this model has been identified to a nonlinear Fokker-Planck equation and associated to an entropic form characteristic of nonextensive statistical mechanics, ${S}_{2}(t)$\equiv${}{S}_{q=2}(t)$. In the present work, we consider a system of superconducting vortices under overdamped motion, following an irreversible process, so that by using the corresponding nonlinear Fokker-Planck equation, the entropy time rate $[d{S}_{2}(t)/dt]$ is investigated. Both entropy production and entropy flux from the system to its surroundings are analyzed. Molecular dynamics simulations are carried for this process, showing a good agreement between the numerical and analytical results. It is shown that the second law holds within the present framework, and we exhibit the increase of ${S}_{2}(t)$ with time, up to its stationary-state value.

A recent proposal of an effective temperature $\theta$, 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 $\theta$ presents values much higher than those of typical room temperatures $T$, so that the thermal noise can be neglected ($T/\theta \simeq 0$). 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, $\varphi \left(x\right)=\alpha x^2/2$ ($\alpha >0$). Herein, the thermodynamic framework is extended to a quite general confining potential, namely $\varphi \left(x\right)={\alpha \left|x\right|}^z/z$ ($z>1$). It is shown that the main results of the previous analyses hold for any $z>1$: (i) The definition of the effective temperature $\theta$ conjugated to the entropy $s_2$. (ii) The construction of a Carnot cycle, whose efficiency is shown to be $\eta =1-({\theta }_2/{\theta }_1)$, where ${\theta }_1$ and ${\theta }_2$ are the effective temperatures associated with two isothermal transformations, with ${\theta }_1>{\theta }_2$. 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 $\varphi \left(x\right)$, increasing the possibility of experimental verifications.