Abstract
Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schrödinger equation.
Highlights
We consider realistic models of non-Markovian transport that takes place in a comb geometry
A comb model is a simple caricature of various types of natural branched structures, see Figure 1, where random walks on comb structures provide a geometrical explanation of anomalous diffusion
If we add additional degrees of freedom y ∈ Rn to form a comb, the Markov process in fingers or branches leads to a memory effect in the x-space, which destroys the Markov property (46)
Summary
We consider realistic models of non-Markovian transport that takes place in a comb geometry. Comb-like models are widely employed to describe various experimental applications These models have proven useful for describing the transport along spiny dendrites [4,5,6], diffusion of drugs in the circulatory system [7], anomalous diffusion in cold atoms [8,9], and diffusion in crowded media [10,11] and many other interdisciplinary applications. Another example is the occupation time statistics for random walkers on combs where the branches can be regarded as independent complex structures, namely fractal or other ramified branches [12]. We obtain that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schrödinger equation
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