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Is fractal dispersion subdiffusive or superdiffusive? A theoretical investigation
Author(s) -
Carlier E.
Publication year - 2007
Publication title -
hydrological processes
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.222
H-Index - 161
eISSN - 1099-1085
pISSN - 0885-6087
DOI - 10.1002/hyp.6645
Subject(s) - fractal , fractal dimension , fractal derivative , statistical physics , mathematics , dispersion (optics) , probability density function , anomalous diffusion , langevin equation , exponential function , distribution (mathematics) , mathematical analysis , fractal analysis , physics , statistics , optics , computer science , knowledge management , innovation diffusion
A probabilistic approach is used to simulate particle tracking along a fractal path. The particle tracking is modelled as the sum of elementary steps with independent random variables. An exponential distribution is obtained for each elementary step and a Gamma distribution or probability density function is then deduced. The relationship between fractal dispersivity and the elementary step is given. It is theoretically demonstrated that the fractal dispersion is subdiffusive and that the linear dispersion coefficient and the linear dispersivity decrease with the mean travel distance. A review of some fractal models showing an increase of the linear dispersivity is given and an explanation that shows why these models may be not correct is proposed. It is shown that the results presented are in agreement with other studies relating to the application of the fractional calculus to diffusion transport. Lastly, a relation between the fractal dimension and the order of the fractional Langevin equation is proposed. Copyright © 2007 John Wiley & Sons, Ltd.