arXiv subjects
Publications and source records attributed to Anatoly Zagorodny.
Yuri L'vovich Klimontovich (28.09.1924--26.10.2002) was an outstanding theoretical physicist who made major contributions to kinetic theory. On the occasion of his 100th birthday we recall his main scientific achievements.
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.
We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub- or super-diffusion) at longer times. Using the standard non-Markovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function (pdf) as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the pdf obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.
Diffusion of particles in velocity space undergoing turbulent field was extensively studied in the problem of warm beam relaxation. Under low field intensities the diffusion is described by the Fokker-Planck equation with the diffusion coefficient given by quasilinear theory. This diffusion coefficient is calculated on the free particle propagator and for weak fields its renormalization due to orbit diffusion is not necessary. To study effects which should be taken into account when the intensity of the turbulent field is increased a numerical simulation of particle motion in the external field of Langmuir waves with given k-spectrum and random phases is done. For strong fields meansquare velocity evolution shows that ballistic regime in the very beginning is changed for oscillatory one in the intermediate stage and later for diffusion. Asymptotically it behaves like fractional power of elapsed time with the exponent dependent on the particular field spectrum. Such evolution in the whole temporal interval of simulation is recovered from the numerical solution of generalized Fokker-Planck equation with time dependent diffusion coefficient obtained from the microscopic approach. The analytical approximation for this solution is also given.