2011/06/30 by Ori Hirschberg, David Mukamel, Gunter M. Schütz · 47 citations
Mathematics · Physics and Astronomy · #Diffusion #Distribution (mathematics) #Distribution function #Exponent #Fractional Differential Equations Solutions #Function (biology) #Geometry #Initial value problem #Logarithm #Mathematical analysis #Mathematical physics #Mathematics #Physics #Scaling #Statistical Mechanics and Entropy #Statistical physics #Theoretical and Computational Physics #Thermodynamics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.84.041111
published in Physical Review E 84(4), 041111 (American Physical Society) · 4 pages, 3 figures; Published version
openalex publication_date 2011/10/10 · arxiv created 2011/12/14 · arxiv updated 2011/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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(γ) with a given γ<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 that characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value.