Diffusion in a logarithmic potential: some unexpected surprises

 Abstract:

 The diffusion equation of a particle in an external logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. I will describe a scaling analysis of this seemingly simple equation which reveals several surprising features: 
 

(i) the solution is given by two distinct scaling forms, corresponding to a diffusive and a subdiffusive  length scale; 

 (ii) the overall scaling function is selected by the initial condition. This selection mechanism has many similarities to the marginal stability mechanism which has been widely studied in the context of fronts propagating into unstable states.
 

(iii) this dependence on the initial condition manifests a ``phase transition'' from a regime in which the scaling solution depends on the initial condition to a regime in which it is independent of it.
 

The derivation of these features will be briefly outlined, and their implications will be discussed.