Approach to equilibrium of diffusion in a logarithmic potential

Details Approach to equilibrium of diffusion in a logarithmic potential Approach to equilibrium of diffusion in a logarithmic potential

2011-06-02

arxiv.org/abs/1106.0458v2

Phys. Rev. E 84, 041111 (2011)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 3 figures; Published version

Scientific paper

10.1103/PhysRevE.84.041111

The late-time distribution function P(x,t) of a particle diffusing in a one-dimensional logarithmic potential is calculated for arbitrary initial conditions. We find a scaling solution with three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ t^(1/2)) and a subdiffusive (x ~ t^{\gamma} with a given {\gamma} < 1/2) length scale, respectively, (ii) the overall scaling function is selected by the initial condition, and (iii) depending on the tail of the initial condition, the scaling exponent which characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value.

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