Fractional dynamics in comb-like structures

"Random walks often provide the underlying mesoscopic mechanism for transport phenomena in physics, chemistry and biology. In particular, anomalous transport in branched structures has attracted considerable attention. Combs are simple caricatures of various types of natural branched structures that belong to the category of loopless graphs. The comb model was introduced to understand anomalous transport in percolation clusters. Comb-like models have been widely adopted to describe kinetic processes in various experimental applications in medical physics and biophysics, chemistry of polymers, semiconductors, and many other interdisciplinary applications. The authors present a random walk description of the transport in specific comb geometries, ranging from simple random walks on comb structures, which provide a geometrical explanation of anomalous diffusion, to more complex types of random walks, such as non-Markovian continuous-time random walks. The simplicity of comb models allows to perform a rigorous analysis and to obtain exact analytical results for various types of random walks and reaction-transport processes."--

2.3 Fox H-functions2.4 Fractional integration; 2.4.1 Properties of fractional integration with integer derivatives; 2.5 Fractional derivative; 2.6 Laplace transform; 2.7 Weyl and Riesz fractional derivatives; 3. Markov processes; 3.1 Stochastic processes; 3.1.1 Averaged characteristics of stochastic processes; 3.2 Markov processes; 3.2.1 Chapman-Kolmogorov equation; 3.2.2 Solutions of the Chapman-Kolmogorov equation; 3.2.3 Fokker-Planck equation; 3.3 Stochastic differential equations; 3.4 Markov chains; 3.4.1 The simple random walk; 3.4.2 The first passage problem.

5.2.1.1 Subdiffusion in fingers5.2.1.2 Levy flights in fingers; 5.2.1.3 Diffusion in finite-length fingers; 5.3 Langevin dynamics along the branches: Characteristic functions; 5.3.1 Gaussian colored noise: Correlation function; 5.3.2 Gaussian colored noise: MSD; 5.3.3 Non-Gaussian white noise; 6. Inhomogeneous advection and superdiffusion; 6.1 Introduction; 6.2 Superdiffusion on a comb structure; 6.2.1 Diffusion vs subdiffusion; 6.2.2 Superdiffusion as turbulent diffusion; 6.2.3 Exponential diffusion and log-normal distribution; 6.3 Liouville-Green approximation.

6.3.1 The Fox function solution6.4 Classification of so…