Approach to equilibrium of diffusion in a logarithmic potential

Area of Science:

• Statistical Physics
• Physical Chemistry
• Computational Physics

Background:

• Particle diffusion is fundamental in various physical and chemical processes.
• Logarithmic potentials present unique challenges in modeling diffusion dynamics.
• Understanding late-time behavior is crucial for long-term predictions.

Purpose of the Study:

• To calculate the late-time distribution function for a particle in a 1D logarithmic potential.
• To analyze the scaling solution for arbitrary initial conditions.
• To identify novel features in the diffusion process.

Main Methods:

• Analytical calculation of the late-time distribution function.
• Investigation of scaling solutions.
• Analysis of the influence of initial conditions on scaling behavior.

Main Results:

• A scaling solution with two distinct forms: diffusive (x~t(1/2)) and subdiffusive (x~t(γ), γ<1/2).
• The initial condition uniquely selects the overall scaling function.
• A transition in the scaling exponent is observed, dependent on the initial condition's tail.

Conclusions:

• The diffusion in a logarithmic potential displays complex scaling properties.
• Initial conditions play a critical role in determining the diffusion's long-term behavior.
• The identified scaling transitions offer new insights into anomalous diffusion phenomena.