Parameter calibration between models and simulations: Connecting linear and non-linear descriptions of anomalous diffusion

Abstract

Anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches, including simulation, analytical methods and experiments. Partial differential equations, either fractional linear ones as non-linear, are able to describe the phenomenon, in particular, the heavy tails observed in the probability distribution functions. Besides, one can observe some difficulties to tame the anomalous diffusion parameters in many simulations studies. Consequently, the relationship between simulations results to the corresponding model becomes somewhat impaired. In this work, we present a systematic calibration between simulations and models, exploring the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives, q-Gaussian parameter and model diffusion constant. By means of a statistical fitting procedure of the simulations data, we establish a connection between linear and non-linear approaches. Simulations considered a CTRW with a criterion to control mean waiting time and step length variance, and a full range of well controlled cases ranging from subdiffusion to superdiffusion regimes were generated. Theoretical models were expressed by means of generalised diffusion equations with fractional derivatives in space and time and by the non-linear porous medium equation. To assess the diffusion constant, the order of fractional derivatives and the q-Gaussian parameter of simulations data in each case of anomalous diffusion, we compare the accuracy of two methods: (1) by analysis of the dispersion of the variance over time and, (2) by the optimisation of solutions of the theoretical models to the histogram of positions. The relative accuracies of the models were also analysed for each regimen of anomalous diffusion. We highlight relations between simulations parameters and model parameters and discuss methods to link them. Among those, Tsallis–Buckman scaling law was verified.