Projects per year
Abstract
The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $R^3$ according to a Poisson process with intensity $N$ and in velocity according some fattailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the BoltzmannGrad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. The convergence of the empiric particle dynamics to the Boltzmanntype dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{k}$.
Original language  English 

Publisher  arXiv 
Publication status  Published  29 May 2024 
Bibliographical note
38 pages. arXiv admin note: text overlap with arXiv:2405.04449Keywords
 math.AP
 mathph
 math.MP
 math.PR
Fingerprint
Dive into the research topics of 'Fractional diffusion as the limit of a short range potential Rayleigh gas'. Together they form a unique fingerprint.Projects
 1 Finished

Derivation of kinetic equation: From Newton to Boltzmann via trees
Matthies, K. (PI)
1/10/20 → 31/03/24
Project: UK charity