# 1. Anomalous Diffusion

Workshop organized by: S. Abe and J.P. Boon

*This workshop will bring together experts who have developed various theoretical approaches for the analysis of Anomalous Diffusion and experimentalists who have explored and measured diffusive phenomena and reaction-diffuusion processes in particular in biological systems.*

Diffusion is an ubiquitous phenomenon observed in physical, chemical, biological, social, algorithmic systems where "objects" (particles, molecules, cells, individuals, agents, ...) move in a seemingly random sequence of steps in such a way that their mean squared displacement increases linearly in time: <*r*^2>~*t*. However there are many instances where the "objects" do not move freely: obstacles, time delays, interactions can modify their trajectories such that the mean squared displacement deviates from the linear law. So more generally, one observes that the characteristic spatial scale varies like l~*t^(γ/2)* where for normal diffusion γ=1 while if γ<>1 one talks about anomalous diffusion: when 0<γ<1 the process is said to be sub-diffusive and when γ>1 it is super-diffusive. As a result, there has been considerable interest in developing stochastic models capable of generating such behavior. Diverse microscopic dynamics can give rise to "diffusion" at the macroscopic level, but the underlying mechanisms may be quite different; for instance the distinction should be made between tracer motion where experimentally one follows trajectories of distinguishable particles seeded in an active medium and molecular diffusion where the motion of tagged particles, while identical to the medium particles, is made observable by radioactive or fluorescent markers.

One class of diffusion models involves the use of memory. Non-Markovian dynamics is the mechanism behind the fractional Brownian motion and the use of correlated noise in the generalized Langevin equation leads to the fractional Fokker-Planck equation describing the phenomenology of anomalous diffusion in large ensembles and for single trajectories. On the other hand a Markovian random walk can also give rise to anomalous diffusion in a nonlinear formulation of the diffusion equation. A description of sub-diffusion based on non-Markovian models gives the fractional Fokker-Planck equation leading to a stretched-exponential distribution while the nonlinear theory gives algebraic power-law distributions. Both approaches have also been extended to the description of nonlinear reaction-diffusion systems. While in most studies the theories have been substantiated by numerical simulations, explicit comparisons between the theoretical results and experimental data are much scarcer. Therefore, new analytic developments in connection with experimental observations will be of considerable interest.