  The equation which describes a particle diffusing in a logarithmic potential
arises in diverse physical problems such as momentum diffusion of atoms in
optical traps, condensation processes, and denaturation of DNA molecules. A
detailed study of the approach of such systems to equilibrium via a scaling
analysis is carried out, revealing three surprising features: (i) the solution
is given by two distinct scaling forms, corresponding to a diffusive (x ~
\sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii)
the scaling exponents and scaling functions corresponding to both regimes are
selected by the initial condition; and (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 selection mechanism which is found has many similarities
to the marginal stability mechanism which has been widely studied in the
context of fronts propagating into unstable states. The general scaling forms
are presented and their practical and theoretical applications are discussed.
