Proceedings of the RHSAS

PROCEEDINGS OF THE RUSSIAN HIGHER SCHOOL
ACADEMY OF SCIENCES

Print ISSN: 1727-2769    Online ISSN: 2658-3747
English | Русский

Recent issue
№2(67) April - June 2025

On the master equation approach to modeling anomalous diffusion

Issue No 2 (31) April-June 2016
Authors:

Arkashov Nikolay Sergeevich,
Seleznev Vadim Alexandrovich
DOI: http://dx.doi.org/10.17212/1727-2769-2016-2-7-15
Abstract
A large amount of experimental data on various processes of the so-called anomalous diffusion for which variance varies nonlinearly over time has currently been accumulated. Various methods of modeling anomalous diffusion are associated with such properties of the corresponding processes as “a strong form” of increment dependence and increment nonstationarity (see, e.g., [1]–[4].). The well-known examples of such processes are continuous time random walk (CTRW) models and the fractional Brownian motion (see e. g, [4], [5]). Today, apparently, there are no modeling formats (see. [3]) covering all of these properties, similar to the Wiener process which is a classical format of the Brownian motion. Questions of modeling transport processes in singular phase spaces were raised in [1]–[4] etc., where the modeling of transport processes in continuous media with a fractal structure was studied. These processes were considered as a subset of the zero Lebesgue and some non-zero Hausdorff measures. A technique of fractional integro-differential calculus was used as a modeling tool in these studies. In this paper we depart from the paradigm that transfer processes are modeled in continuous media with a fractal structure. We construct a master equation that makes it possible to model anomalous diffusion processes in such a way as to take into account both the aftereffect fractal structure and correlation properties of the process. This master equation allows obtaining the Wiener process and the fractional Brownian motion as limiting cases. This present paper is a natural continuation of a series of papers [6], [7], [8], [9] in which an anomalous character of mass, energy, and momentum transport was closely linked with the introduction of values singular relative to the Lebesgue measure.
Keywords: Cantor set; fractional Brownian motion; moving averages; anomalous diffusion; self-similarity.

References
  1. Feder J. Fractals. New York, Plenum Press, 1988. 254 p. (Russ. ed.: Feder E. Fraktaly. Moscow, Mir Publ., 1991. 254 p.).
  2. Zelenyi L.M., Milovanov A.V. Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics. Physics-Uspekhi, 2004, vol. 47, no. 8, pp. 749–788. Translated from Uspekhi fizicheskikh nauk, 2004, vol. 174, no. 8, pp. 809–852.
  3. Zaslavsky G.M. Hamiltonian chaos and fractional dynamics. Oxford, Oxford University Press, 2008. 448 p. (Russ. ed.: Zaslavskii G.M. Gamil'tonov khaos i fraktal'naya dinamika. Moscow, Institut komp'yuternykh issledovanii Publ., Izhevsk, Regulyarnaya i khaotiche­skaya dinamika Publ., 2010. 472 p.).
  4. Uchaikin V.V. Self-similar anomalous diffusion and Lévy-stable laws. Physics-Uspekhi, 2003, vol. 46, no. 8, pp. 821–849. Translated from Uspekhi fizicheskikh nauk, 2003, vol. 173, no. 8, pp. 847–876.
  5. Mandelbrot B.,Ness J. van. Fractional Brownian motions, fractional noise and applications. SIAM Review, 1968, vol. 10, pp. 422–437.
  6. Arkashov N.S. Ergodic properties of a transformation of a self-similar space with a Hausdorff measure. Mathematical Notes, 2015, vol. 97, iss. 1–2, pp. 155–163. doi: 10.1134/S0001434615010186. Translated from Matematicheskie zametki, 2015, vol. 97, no. 2, pp. 163–173.
  7. Arkashov N.S., Seleznev V.A. On a random walk model on sets with self-similar structure. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 968–983. Translated from Sibirskii matematicheskii zhurnal, 2013, vol. 54, no. 6, pp. 1216–1236.
  8. Arkashov N.S., Seleznev V.A. O modeli sub- i superdiffuzii na topologicheskikh prostranstvakh s samopodobnoi strukturoi [On one model of sub- and superdiffusion on topological spaces with a self-similar structure]. Teoriya veroyatnostei i ee primeneniya – Theory of Probability and its Applications, 2015, vol. 60, no. 2, pp. 209–226. (In Russian)
  9. Seleznev V.A., Arkashov N.S. Ob usloviyakh formirovaniya protsessov sub- i su-perdiffuzii na samopodobnykh mnozhestvakh [On conditions of forming processes of sub- and superdiffusion on sets with self-similar structures]. Doklady Akademii nauk vysshei shkoly Rossiiskoi Federatsii – Proceedings of the Russian higher school Academy of sciences, 2014, no. 4 (25), pp. 33–38.
  10. Arkashov N.S., Borisov I.S. Gaussian approximation to the partial sum processes of moving averages. Siberian Mathematical Journal, 2014, vol. 45, iss. 6, pp. 1000–1030. Translated from Sibirskii matematicheskii zhurnal, 2004, vol. 45, no. 6, pp. 1221–1255.
  11. Arkashov N.S., Borisov I.S., Mogul'skii A.A. Large deviation principle for partial sum processes of moving averages. Theory of Probability and its Applications, 2008, vol. 52, iss. 2, pp. 181–208. doi: 10.1137/s0040585x97982955. Translated from Teoriya veroyatnostei i ee primeneniya, 2007, vol. 52, no. 2, pp. 209–239.
  12. Gorin E.A., Kukushkin B.N. Integrals related to the Cantor ladder. St. Petersburg Mathema­tical Journal, 2004, vol. 15, iss. 3, pp. 449–468. Translated from Algebra i analiz, 2003, vol. 15, no. 3, pp. 188–220.
  13. Shiryaev A.N. Veroyatnost' [Probability]. Moscow, Nauka Publ., 1980. 574 p.
  14. Edgar G. Measure, topology, and fractal geometry. New York, Springer, 2008. 268 p.
Views: 2966