https://doi.org/10.1007/s100510170333
Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion
Department of Physics and School of Chemical Sciences, University of
Illinois at Urbana-Champaign, 600 S. Mathews MC-712 Box 24-6, Urbana, IL 61801, USA
Corresponding author: a metz@mit.edu
Received:
30
June
2000
Revised:
12
November
2000
Published online: 15 January 2001
A generalised random walk scheme for random walks in an arbitrary external
potential field is investigated. From this concept which accounts
for the symmetry breaking of homogeneity through the external field, a
generalised master equation is constructed. For long-tailed transfer
distance or waiting time distributions we show that this generalised
master equation is the genesis of apparently different fractional
Fokker-Planck equations discussed in literature. On this basis, we
introduce a generalisation of the Kramers-Moyal expansion for broad
jump length distributions that combines multiples of both ordinary and
fractional spatial derivatives. However, it is shown that the nature of the
drift term is not changed through the existence of anomalous transport
statistics, and thus to first order, an external potential
feeds back on the probability density function W through the classical
term
, i.e., even for Lévy
flights, there exists a linear infinitesimal generator that accounts for
the response to an external field.
PACS: 05.10.Gg – Stochastic analysis methods (Fokker-Planck, Langevin, etc.) / 05.40.Fb – Random walks and Lévy flights / 05.60.-k – Transport processes / 02.50.Ey – Stochastic processes
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2001