Analogies between colored Lévy noise and random channel approach to disordered kinetics

Abstract

We point out some interesting analogies between colored Lévy noise and the random channel approach to disordered kinetics. These analogies are due to the fact that the probability density of the Lévy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics. Although the equations for the two approaches are not identical, the analogies can be used for deriving new, useful results for both problems. The random channel approach makes it possible to generalize the fractional Uhlenbeck–Ornstein processes (FUO) for space- and time-dependent colored noise. We describe the properties of colored noise in terms of characteristic functionals, which are evaluated by using a generalization of Huber’s approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We start out by investigating the properties of symmetrical white noise and then define the Lévy colored noise in terms of a Langevin equation with a Lévy white noise source. We derive exact analytical expressions for the various characteristic functionals, which characterize the noise, and a functional fractional Fokker–Planck equation for the probability density functional of the noise at a given moment in time. Second, by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics, we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics. These equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data. Special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics.