Abstract
We address the problem of subdiusion or normal diusion to perform a calibration between the parameters used in simulation and the parameters of a subdifusive model. The theoretical model is written as a generalized diusion equation with fractional derivatives in time. The data is generated by simulations consisting of continuous-time random walks with controlled mean waiting time and jump length variance to provide a full range of cases between subdiusion andnormal diusion. From the simulations, we compare the accuracy of two methods to obtain the diusion constant, the order of fractional derivatives: the analysis of the dispersion of the variance in time and the optimization tting of theoretical model solutions to histogram of positions. We highlight the connection between the parameters of the simulations the parameters of the theoretical models.
Highlights
Anomalous diffusion has been attracting the attention of the scientific community due the large plethora of natural systems which display large deviations from normal diffusion, as plasma diffusion [3], fluid flow in porous media [7, 22], diffusion in fractal structures [23], turbulence [8] etc
To the normal diffusion, associated to local, short range correlations, the presence of long range correlations lead to non-Gaussian probability density functions (PDF)
In the class of problems generically called anomalous diffusionsome or anomalous Brownian motions, the microscopic kinetics is characterized by increasing probabilities to observe large steps, causing non-Gaussian PDF with heavy tales, or to observe large waiting times elapsed between steps causing non-Gaussian PDF, causing non-Gaussian PDF with light tales
Summary
Anomalous diffusion has been attracting the attention of the scientific community due the large plethora of natural systems which display large deviations from normal diffusion, as plasma diffusion [3], fluid flow in porous media [7, 22], diffusion in fractal structures [23], turbulence [8] etc. The description of the macroscopic behavior of u(x, t ) for anomalous diffusion process can be modeled by means of generalized diffusion equations with fractional derivatives in time as well as in space [1, 2, 13, 14, 15]. Another class of generalizations of diffusion equations is obtained by the inclusion of nonlinearity in the partial differential equation [4, 13, 25]. Section (4) comprises the aspects of the simulations which is followed by the Section (5) of results and Section (6) for the conclusions
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