Langevin equation fractional dynamics

Fractional dynamics emerges as the macroscopic limit of the combination of the Langevin and the trapping processes. After straightforward calculations based on the continuous-time version of the Chapman-Kolmogorov equation [75, 114] which are valid in the long-time limit t max r, t, one obtains the fractional Klein-Kramers equation... 

However, neither of these models is adequate for describing multiffactal statistical processes as they stand. A number of investigators have recently developed multifractal random walk models to account for the multiffactal character of various physiological phenomena, and here we introduce a variant of those discussions based on the fractional calculus. The most recent generalization of the Langevin equation incorporates memory into the system s dynamics and has the simple form of Eq. (131) with the dissipation parameter set to zero ... 

Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Levy nestable probability density describing Levy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisfies the scaling relation [Eq. (35)] with the Levy index a such that 8 = 1 /a. The dynamics of a Levy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Levy noise is also Levy but with a change in parameters. This is a possible alternate model of the fluctuations in the interbeat intervals for the human heart shown to be Levy stable over a decade ago [25],... 

In this section, we formulate the dynamical description of Levy flights using both a stochastic differential (Langevin) equation and the deterministic fractional Fokker-Planck equation. For the latter, we also discuss the corresponding form in the domain of wavenumbers, which is a convenient form for certain analytical manipulations in later sections. 

We will first show how one can obtain the time-correlation function expression for the susceptibility in a classical statistical ensemble of particles which exhibits a linear response to an externally applied perturbation. This will be followed by an outline of the argument that leads to the generalized Langevin equation for the time-dependence of an arbitrary function of the molecular canonical coordinates. In both cases, derivations with minor modifications have been presented previously in numerous reviews, monographs, etc. However, the results are employed in a large fraction of current descriptions of dynamical processes in dense phases and thus, it seems worthwhile to again show the basic ideas underlying the formalism. 

This model for subdiffusion in the external force field F(x) — — ( ) provides a basis for fractional evolution equations, starting from Langevin dynamics that is combined with long-tailed trapping events possessing a... 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

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