Generalization of the fractal Einstein law relating conduction and diffusion on networks.

We settle a long-standing controversy about the exactness of the fractal Einstein and Alexander-Orbach laws by showing that the properties of a class of fractal trees violate both laws. A new formula is derived which unifies the two classical results by showing that if one holds, then so must the other, and resolves a puzzling discrepancy in the properties of Eden trees and diffusion-limited aggregates. We also conjecture that the result holds for networks which have no fractal dimension. The failure of the classical laws is attributed to anisotropic exploration of the network by a random walker. The occurrence of this newly revealed behavior means that the conventional laws must be checked if they, or numerous results which depend on them, are to be applied accurately.

[1]  D. Jacobs,et al.  Diffusion on a DLA cluster in two and three dimensions , 1994 .

[2]  Shlomo Havlin,et al.  Topological properties of diffusion limited aggregation and cluster-cluster aggregation , 1984 .

[3]  G. Toulouse,et al.  Random walks on fractal structures and percolation clusters , 1983 .

[4]  S. Havlin,et al.  Diffusion in disordered media , 2002 .

[5]  Cassi,et al.  Spectral dimension of fractal trees. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Diffusion and spectral dimension on Eden tree , 1992, cond-mat/9211020.

[7]  S. Havlin,et al.  Scaling theory of transport in complex biological networks , 2007, Proceedings of the National Academy of Sciences.

[8]  L. Sander,et al.  Diffusion-limited aggregation, a kinetic critical phenomenon , 1981 .

[9]  Cassi,et al.  Random walks on bundled structures. , 1996, Physical review letters.

[10]  N. Sheibani,et al.  Paris , 1894, The Hospital.

[11]  András Telcs.  Spectra of graphs and fractal dimensions II , 1995 .

[12]  I. Procaccia,et al.  Analytical solutions for diffusion on fractal objects. , 1985, Physical review letters.

[13]  R. Orbach,et al.  Dynamics of Fractal Networks , 1986, Science.

[14]  M. Sahimi.  Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing , 1993 .

[15]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[16]  A. Roberts,et al.  Continuum diffusion on networks: trees with hyperbranched trunks and fractal branches. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  J. Klafter,et al.  First-passage times in complex scale-invariant media , 2007, Nature.

[18]  T. Hattori,et al.  Gaussian Field Theories on General Networks and the Spectral Dimensions , 1987 .

[19]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[20]  András Telcs,et al.  Random Walks on graphs, electric networks and fractals , 1989 .

[21]  Elmar Teufl,et al.  Asymptotics of the transition probabilities of the simple random walk on self-similar graphs , 2002 .

[22]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[23]  S. Alexander,et al.  Density of states on fractals : « fractons » , 1982 .

[24]  S. Havlin,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2000 .

[25]  W. Coffey,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2002 .

[26]  Paul D. Bourke.  Constrained diffusion-limited aggregation in 3 dimensions , 2006, Comput. Graph..