The similarity group and anomalous diffusion equations
- 28 July 2000
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 33 (31), 5501-5511
- https://doi.org/10.1088/0305-4470/33/31/305
Abstract
A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.Keywords
This publication has 25 references indexed in Scilit:
- Anomalous diffusion of water in biological tissuesBiophysical Journal, 1996
- Submonolayer Growth with Repulsive Impurities: Island Density Scaling with Anomalous DiffusionPhysical Review Letters, 1995
- Anomalous Diffusion at Liquid SurfacesPhysical Review Letters, 1995
- Fractional diffusion equation for transport phenomena in random mediaPhysica A: Statistical Mechanics and its Applications, 1992
- Fractional diffusion equation on fractals: three-dimensional case and scattering functionJournal of Physics A: General Physics, 1992
- Anomalous Diffusion of Hydrogen in Amorphous MetalsEurophysics Letters, 1990
- Diffusion in disordered mediaAdvances in Physics, 1987
- Diffusion on fractalsPhysical Review A, 1985
- Analytical Solutions for Diffusion on Fractal ObjectsPhysical Review Letters, 1985
- A higher-order model for inelastic electron-atom scatteringJournal of Physics B: Atomic and Molecular Physics, 1983