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Infinite covariant density for diffusion in logarithmic potentials and optical lattices.
1Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel.
Physical Review Letters
|September 28, 2010
Summary
This study solves the Fokker-Planck equation for Brownian motion, revealing an infinite covariant density under critical conditions. This finding redefines the long-time behavior of anomalous diffusion systems, impacting atomic and charge dynamics.
Area of Science:
- Statistical Physics
- Non-equilibrium Systems
- Complex Fluids
Background:
- Brownian motion describes random particle movement.
- Logarithmic potentials are relevant in various physical systems.
- Anomalous diffusion deviates from standard Brownian motion.
Purpose of the Study:
- Solve the Fokker-Planck equation for Brownian motion in a logarithmic potential.
- Investigate the system's behavior below a critical diffusion constant.
- Determine the phase diagram of anomalous diffusion.
Main Methods:
- Analytical solution of the Fokker-Planck equation.
- Analysis of non-normalizable solutions.
- Phase diagram construction for anomalous diffusion.
Main Results:
- The solution approaches an infinite covariant density below a critical diffusion constant.
- A phase diagram for anomalous diffusion was obtained.
- Infinite covariant density, not Boltzmann equilibrium, describes the long-time limit.
Conclusions:
- The study clarifies the long-time behavior of anomalous diffusion in logarithmic potentials.
- Findings have implications for atoms in optical lattices and charges near polyelectrolytes.
- Highlights the importance of non-equilibrium statistical mechanics for certain systems.
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