Abstract
One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.
Highlights
In a review [6], a detailed analysis of Einstein’s derivation of diffusion equation was done and revealed an important circumstance taking place in the case with the telegraph equation as well: random positions of scatterers acting on the tracer are considered to be statistically independent
The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis
Several particular cases have been considered when the convolutions can be expressed in explicit form
Summary
In a review [6], a detailed analysis of Einstein’s derivation of diffusion equation was done and revealed an important circumstance taking place in the case with the telegraph equation as well: random positions of scatterers acting on the tracer are considered to be statistically independent In other words, these equations model passing a tracer through an ideal gas, molecules of which do not interact with each other, so there are no correlations between consecutive collisions. A similar situation takes place in the case of charged particle diffusion in random magnetic fields, correlations of which are produced by a random system of magnetic force lines Some of these cases allow the broken-line approximation of certain parts of the trajectory, but the lengths of the random segments have distributions that differ from the classical exponential form.
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