de Leeuw Y, Cohen D. Diffusion in sparse networks: linear to semilinear crossover.
PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012;
86:051120. [PMID:
23214751 DOI:
10.1103/physreve.86.051120]
[Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2012] [Revised: 09/27/2012] [Indexed: 06/01/2023]
Abstract
We consider random networks whose dynamics is described by a rate equation, with transition rates w(nm) that form a symmetric matrix. The long time evolution of the system is characterized by a diffusion coefficient D. In one dimension it is well known that D can display an abrupt percolation-like transition from diffusion (D>0) to subdiffusion (D = 0). A question arises whether such a transition happens in higher dimensions. Numerically D can be evaluated using a resistor network calculation, or optionally it can be deduced from the spectral properties of the system. Contrary to a recent expectation that is based on a renormalization-group analysis, we deduce that D is finite, suggest an "effective-range-hopping" procedure to evaluate it, and contrast the results with the linear estimate. The same approach is useful in the analysis of networks that are described by quasi-one-dimensional sparse banded matrices.
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