Balakrishnan, V.
(1985)
*Anomalous diffusion in one dimension*
Physica A: Statistical Mechanics and its Applications, 132
(2-3).
pp. 569-580.
ISSN 0378-4371

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/037843...

Related URL: http://dx.doi.org/10.1016/0378-4371(85)90028-7

## Abstract

In view of the interest in the occurrence of anomalous diffusion (<r^{2}(t)>~t^{2H}, 0 < H < ½) 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 x_{ε}(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 x_{ε}(t) ~ t^{½} , and, subsequently, x_{ε}(t) ~ t^{H} , 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= ½ ) is obtained explicitly: <x^{2}(t)> ~ t^{2H}; the diffusive spread of the initial condition is given by x_{ε}(t) ~ t^{H}; and the first passage time from the origin to the point x has a stable Levy distribution with an exponent equal to H.

Item Type: | Article |
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Source: | Copyright of this article belongs to European Physical Society. |

ID Code: | 1092 |

Deposited On: | 27 Sep 2010 04:31 |

Last Modified: | 11 May 2011 12:22 |

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