Abstract
The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non-Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times, these solutions agree with the predictions of the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial.
Highlights
Particle diffusion processes have been studied for about two centuries [1], there are still some subtle issues that need to be faced at the present time
The strategy followed in this paper is to construct a time-nonlocal Fokker-Planck equation which reproduces the time dependence of the mean square displacement of an underlying process throughout the time domain
It should be stressed that the same mean square displacement may correspond, in general, to different models for the time-dependence of the probability distribution function (PDF)
Summary
Particle diffusion processes have been studied for about two centuries [1], there are still some subtle issues that need to be faced at the present time. The correct theory for the PDF should satisfy the two asymptotic limits given by equation (5) and, in general, by equation (1) simultaneously The cure for both the infinite speed problem and for the ballistic regime in 1 dimension was proposed by Davydov [12], who introduced an explicit time interval tc of the mean free path. In the limit tc → 0, the time evolution of the PDF is possible only if D 0, e.g., the mean particle velocity c → ∞ In this case, equation (7) is reduced to the classical diffusion equation (2) and the solution of this equation with the initial condition (3) is equation (4). The asymptotic behavior of these functions is defined by Iν(z)z→∞ ∼ exp(z)/ 2πz, in the long time limit τ ≫ 1, equation (11) is reduced to the solution of the classical diffusion equation given by equation (4).
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