Stochastic transport through complex comb structures
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Stochastic Transport through Complex Comb Structures V. Yu. Zaburdaeva, P. V. Popovb, A. S. Romanovb, and K. V. Chukbarb a
Institüt für Theoretische Physik, Technische Universität Berlin, D-10623 Berlin, Germany b Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia e-mail: [email protected] Received November 6, 2007
Abstract—A unified rigorous approach is used to derive fractional differential equations describing subdiffusive transport through comb structures of various geometrical complexity. A general nontrivial effect of the initial particle distribution on the subsequent evolution is exposed. Solutions having qualitative features of practical importance are given for joined structures with widely different fractional exponents. PACS numbers: 05.40.Fb, 05.45.Df DOI: 10.1134/S1063776108050178
1. INTRODUCTION In the past years, considerable attention has been given to anomalous stochastic transport described by fractional differential equations. Mathematical and physical aspects of this phenomenon have been discussed in numerous review papers (e.g., see [1–5]) and original publications. A very convenient and widely used proving ground for analyzing characteristics and laws of fractional transport are comb structures, which provide simple and graphic explanations for deviations from classical diffusion and are amenable to theoretical treatment. They were among the first physical systems for which transport equations were rigorously derived rather than inferred from scaling laws for the mean square displacement 〈x2〉 ∝ tα, where α ≠ 1 (see the pioneering study in [6] and derivation of a scaling law with α = 1/2 in [7]). Furthermore, permanent interest in these systems (e.g., see [8–10]) is stimulated by their importance in physics of heterogeneous materials with inclusions of arbitrary geometry. The purpose of this paper is twofold. First, several generalized comb structures are introduced in order to diversify their observed properties. Second, they are used as illustrative examples for analyzing important problems involving mutual influence between fractional and classical diffusion. As a starting point, we use a comb structure to expose one rarely discussed qualitative aspect of fractional differential models used in physical applications, which was pointed out in [11].
(branches) of cross sectional area S0 connected thereto with a spacing of l between them. Particle transport along each element of the structure is characterized by diffusion coefficient D; i.e., unusual behavior of the overall transport is entirely due to geometry rather than to microscopic particle dynamics. The particle concentrations on the backbone and the branch emanating from a point x are denoted by n1(x, t) and n0(x, y, t). Since we analyze macroscopic behavior of particles, particle concentration is hereinafter interpreted as the concentration along the horizontal axis averaged over scales much larger than l. We also assume that the distribution of particle concentration is suffi
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