Stochastic processes crossing from ballistic to fractional diffusion with memory: exact results

• Дата: 2010
• Автори: Ilyin, V., Procaccia, I., Zagorodny, A.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут фізики конденсованих систем НАН України 2010
• Назва видання: Condensed Matter Physics
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/32087
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Stochastic processes crossing from ballistic to fractional diffusion with memory: exact results / V. Ilyin, I. Procaccia, A. Zagorodny // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23001: 1-8. — Бібліогр.: 19 назв. — англ.

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 as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the probability distribution function 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.