Multiplicative processes and power laws

Details Multiplicative processes and power laws Multiplicative processes and power laws

1997-08-29

arxiv.org/abs/cond-mat/9708231v2

Phys. Rev. E 57 N4, 4811-4813 (1998)

Physics

Condensed Matter

Statistical Mechanics

Extended version (7 pages). Phys. Rev. E (to appear April 1998)

Scientific paper

10.1103/PhysRevE.57.4811

[Takayasu et al., Phys. Rev.Lett. 79, 966 (1997)] revisited the question of stochastic processes with multiplicative noise, which have been studied in several different contexts over the past decades. We focus on the regime, found for a generic set of control parameters, in which stochastic processes with multiplicative noise produce intermittency of a special kind, characterized by a power law probability density distribution. We briefly explain the physical mechanism leading to a power law pdf and provide a list of references for these results dating back from a quarter of century. We explain how the formulation in terms of the characteristic function developed by Takayasu et al. can be extended to exponents $\mu >2$, which explains the ``reason of the lucky coincidence''. The multidimensional generalization of (\ref{eq1}) and the available results are briefly summarized. The discovery of stretched exponential tails in the presence of the cut-off introduced in \cite{Taka} is explained theoretically. We end by briefly listing applications.

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