Microscopic dynamics of nonlinear Fokker-Planck equations

We propose an approach to describe the effective microscopic dynamics of (power-law) nonlinear Fokker-Planck equations. Our formalism is based on a nonextensive generalization of the Wiener process. This allows us to obtain, in addition to significant physical insights, several analytical results with great simplicity. Indeed, we obtain analytical solutions for a nonextensive version of the Brownian free-particle and Ornstein-Uhlenbeck processes, and we explain anomalous diffusive behaviors in terms of memory effects in a nonextensive generalization of Gaussian white noise. Finally, we apply the developed formalism to model thermal noise in electric circuits.

Figure 1

Temporal correlations with $q>1$ [Eq. (45)].

Figure 2

RLC circuit where the resistor $R$ is coupled to a thermal bath at a temperature $T$.

Figure 3

Thermal fluctuations in an RLC circuit in series with $C=1.0$ mF, $L=50.0$ mH, and $R=1.15\mathrm{m}\mathrm{\Omega }$.