TY - JOUR
T1 - The random walk's guide to anomalous diffusion
T2 - A fractional dynamics approach
AU - Metzler, Ralf
AU - Klafter, Joseph
N1 - Funding Information:
We thank Eli Barkai for many lively and fruitful discussions as well as correspondence. We thank Gerd Baumann, Alexander Blumen, Albert Compte, Hans Fogedby, Paolo Grigolini, Rudolf Hilfer, Sune Jespersen, Philipp Maass, Theo Nonnenmacher, Michael Shlesinger, Bruce West, and Gerd Zumofen for helpful discussions, as well as Norbert Südland and Jürgen Dollinger for help in Mathematica problems and in programming details. Many thanks to the Abteilung für Mathematische Physik at the Universität Ulm for the hospitality and for the assistance in the reproduction of the historical figures. Financial assistance from the German–Israeli Foundation (GIF) and the Gordon Centre for Energy Studies is gratefully acknowledged. RM was supported in parts by a Feodor–Lynen fellowship from the Alexander von Humboldt Stiftung, Bonn am Rhein, Germany, and through an Amos-de Shalit fellowship from the Minerva foundation.
PY - 2000/12
Y1 - 2000/12
N2 - Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.
AB - Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.
KW - 02.50.Ey
KW - 05.40.-a
KW - 05.40.Fb
KW - Anomalous diffusion
KW - Anomalous relaxation
KW - Dynamics in complex systems
KW - Fractional Fokker-Planck equation
KW - Fractional diffusion equation
KW - Mittag-Leffler relaxation
UR - http://www.scopus.com/inward/record.url?scp=0002641421&partnerID=8YFLogxK
U2 - 10.1016/S0370-1573(00)00070-3
DO - 10.1016/S0370-1573(00)00070-3
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AN - SCOPUS:0002641421
VL - 339
SP - 1
EP - 77
JO - Physics Reports
JF - Physics Reports
SN - 0370-1573
IS - 1
ER -