Thermodynamics and fractional Fokker-Planck equations

The relaxation to equilibrium in many systems that show strange kinetics is described by fractional Fokker-Planck equations (FFPEs). These can be considered as phenomenological equations of linear nonequilibrium theory. We show that the FFPEs describe a system whose noise in equilibrium fulfills the Nyquist theorem. Moreover, we show that for subdiffusive dynamics, the solutions of the corresponding FFPEs are probability densities for all cases in which the solutions of the normal Fokker-Planck equation (with the same Fokker-Planck operator and with the same initial and boundary conditions) exist. The solutions of the FFPEs for superdiffusive dynamics are not always probability densities. This fact means only that the corresponding kinetic coefficients are incompatible with each other and with the initial conditions.