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Anomalous diffusion in one dimension

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  • Balakrishnan, V.

Abstract

In view of the interest in the occurrence of anomalous diffusion (〈r2(t)〉 ∼ t2H, 0 < H < 12) in several physical circumstances, we study anomalous diffusion per se in terms of exactly solvable one-dimensional models. The basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion, and provide solvable analogues of more realistic physical situations. We first derive a general equation for a deterministic trajectory xε(t) that comprehensively characterizes the diffusive motion, by finding the ε-quantiles of the time-dependent probability distribution. The class of all diffusion processes (or, equivalently, symmetric random walks) for which xε(t) ∼ t12, and, subsequently, xε(t) ∼ tH, is identified. Explicit solutions are presented for families of such processes. Considering random walks whose step sequences in time are governed by renewal processes, and proceeding to the continuum limit, a true generalization of Brownian motion (the latter corresponds to the limiting value H = 12) is obtained explicitly: 〈x2(t)〉 ∼ t2H; the diffusive spread of the initial condition is given by xε(t) ∼ tH; and the first passage time from the origin to the point x has a stable Lévy distribution with an exponent equal to H.

Suggested Citation

  • Balakrishnan, V., 1985. "Anomalous diffusion in one dimension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 132(2), pages 569-580.
  • Handle: RePEc:eee:phsmap:v:132:y:1985:i:2:p:569-580
    DOI: 10.1016/0378-4371(85)90028-7
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    Cited by:

    1. Evgeny P. Kitsyuk & Renat T. Sibatov & Vyacheslav V. Svetukhin, 2020. "Memory Effect and Fractional Differential Dynamics in Planar Microsupercapacitors Based on Multiwalled Carbon Nanotube Arrays," Energies, MDPI, vol. 13(1), pages 1-14, January.
    2. Godoy, Salvador & Braun, E., 1994. "The approach to equilibrium in quantum random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 203(3), pages 414-424.
    3. Diaz-Diaz, Fernando & Estrada, Ernesto, 2022. "Time and space generalized diffusion equation on graph/networks," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    4. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.

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