Calibração de parâmetros entre as escalas de voos difusivos anômalos: prescrição para corresponder simulação e modelos

Abstract

Anomalous diffusion is an ubiquitous phenomenon which have been studied by several approaches, including simulation, analytical methods and by experiments. Either fractional partial differential as non-linear partial differential equations were able to describe the phenomenon, in particular, the heavy tails observed in the PDFs. Besides, one can observe some lack in controlling anomalous diffusion in many attempts of simulation. Consequently, the relationship to adequate models becomes somewhat impaired. To perform a systematic calibration between simulations and models, in this work, we explore the relationship among the diffusion coefficients and diffusion exponents with the order of fractional derivatives, the q-Gaussian parameter and model diffusion constant by means of a systematic fitting procedure of the simulation data using linear and non-linear approaches. Using CTRW with a criterion to control mean waiting time and step length variance, a full range of well controlled cases from subdiffusion to superdiffusion were generated. Theoretical models are expressed by means of generalised diffusion equations with fractional derivatives in space, in time and by the non-linearity of the porous medium equation. To decide how to assess the diffusion constant, the order of fractional derivatives and the q-Gaussian parameter from the simulation 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 the solutions of the theoretical models to the histogram of positions. The relative accuracies of the models are also analysed for each regimen of anomalous diffusion. We highlight relations between the simulation parameters and model parameters. Among those, Tsallis-Buckman scaling law identity is verified. The study discusses methods to link model parameters to simulation parameters.