Diffusion, what else?

Diffusion essentially refers to a motion which leads to the spreading of some quantity: density of particles, energy, and so on. It owes its universality to the central limit theorem, according to which che combination of independent, random events has general features not related to the details of the single events. For example, in a standard, diffusive motion the spreading quantity does not vary proportionally to the time, but its variation may double only if time quadruples!

If however the hypotheses of the central limit theorem break down, spreading can be faster or slower depending on the underlying physical mechanisms of transport. This is usually refereed to as "anomalous diffusion" and applies to processes as diverse as light scattering in disordered materials, porous media, tracer motion in turbulent fluids and plasmas. Interdisciplinary applications of non-Brownian processes also arised recently in diverse fields as animal movement and even social and cognitive phenomena. Within this general context, one of the issues that attracted a certain interest in the last decade is the problem of anomalous heat conduction in low-dimensional lattice models where energy cand spread superdiffusively and description by standard heat equation no longer holds.

Diffusion can also be the mediation mechanism allowing the exchange of matter, energy or momentum. This is typical when dynamics conserves these quantities, for example matter in phase separation processes (see the figure) or energy and momentum in the propagation of nonlinear waves. In all these cases it is a diffusion process which leads the system towards equilibrium or allows relaxation.

Contact persons: Stefano Lepri and Paolo Politi