TY - JOUR

T1 - Generalized fractional Fokker-Planck equation for anomalous diffusion

AU - Veksler, Alex

AU - Granek, Rony

PY - 2009/3/1

Y1 - 2009/3/1

N2 - The problem of anomalous diffusion is important for a variety of
systems, such as fluids, glasses, polymers, proteins etc. It is
characterized by a mean square displacement evolving in time as a
power-law = 2 D0t^α. However, a
Fokker-Planck-like equation which could describe a stationary Gaussian
process with anomalous-diffusion behavior, such as the one described by
the Generalized Langevin equation, is still missing. We propose a
generalization for constant force to the fractional Fokker-Planck
equation (fFP) [Metzler, R. and Klafter, J., Phys. Rep. 339 (2000),
1-77], based on a series expansion in spatial and fractional time
derivatives and powers of the Fokker-Planck operator. The proposed
equation, GfFP, recovers the generalized Einstein relation and leads to
Gaussian distribution, in particular, for free particle diffusion. We
apply GfFP to 1-D first passage time problem. The long-time asymptote of
the probability distribution behaves like (-t^α). This contrasts
with the power-law behavior of the corresponding solutions of the fFP.
We further propose to generalize GfFP for treating other outstanding
problems, such as the anomalous diffusion under an harmonic potential
and the Kramers` escape problem.

AB - The problem of anomalous diffusion is important for a variety of
systems, such as fluids, glasses, polymers, proteins etc. It is
characterized by a mean square displacement evolving in time as a
power-law = 2 D0t^α. However, a
Fokker-Planck-like equation which could describe a stationary Gaussian
process with anomalous-diffusion behavior, such as the one described by
the Generalized Langevin equation, is still missing. We propose a
generalization for constant force to the fractional Fokker-Planck
equation (fFP) [Metzler, R. and Klafter, J., Phys. Rep. 339 (2000),
1-77], based on a series expansion in spatial and fractional time
derivatives and powers of the Fokker-Planck operator. The proposed
equation, GfFP, recovers the generalized Einstein relation and leads to
Gaussian distribution, in particular, for free particle diffusion. We
apply GfFP to 1-D first passage time problem. The long-time asymptote of
the probability distribution behaves like (-t^α). This contrasts
with the power-law behavior of the corresponding solutions of the fFP.
We further propose to generalize GfFP for treating other outstanding
problems, such as the anomalous diffusion under an harmonic potential
and the Kramers` escape problem.

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JO - American Physical Society, 2009 APS March Meeting, March 16-20, 2009

JF - American Physical Society, 2009 APS March Meeting, March 16-20, 2009

ER -