From particles scale to anomalous or classical convection-diffusion models with path integrals

Catherine Choquet (1) , Marie-Christine Néel (2)

1 MIA - Mathématiques, Image et Applications - EA 3165
2 EMMAH - Environnement Méditerranéen et Modélisation des Agro-Hydrosystèmes

The present paper is devoted to the rigorous upscaling of some particles displacement model with trapping events, to the continuum scale. It focuses especially on the transitions between sub-diffusive and diffusive models. The work gives emphasis to the following points: 1. The distribution of waiting times in passing to the continuum limit. In fact, the common idea is that the distributions with slowly decaying long tails produce anomalous diffusion while the classical diffusion model corresponds to distributions with short tails. This is shown to be not always true by introducing a simple model of geometrical heterogeneity leading to trapping events without characteristic time scale. 2. The extension of the Feynman-Kac theory to some non-Brownian settings without handling with double Laplace transforms. This yields to better numerical simulations. 3. Constructing a microscopic random walk model that, thought based on a MIM approach, gives both fMIM and FFPE at the mesoscopic limit.

Mots clés: fractal mobile-immobile model; fractional Fokker-Planck equa- tion; fractional Feynman-Kac equation; diffusion versus sub-diffusion; up- scaling; continuum limit

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Catherine Choquet, Marie-Christine Néel. From particles scale to anomalous or classical convection-diffusion models with path integrals. 2013. ⟨hal-00842002⟩