Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem.
Strong anomalous diffusion, where ⟨|x(t)|(q)⟩ ∼ tqν(q) with a nonlinear spectrum ν(q) ≠ const, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomenon is related to infinite covariant densities; i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multifractal anomalous diffusion, as it is complementary to the central limit theorem.
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel.
Adi
A
Rebenshtok
Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine and Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany and Department for Bioinformatics, Lobachevsky State University, Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia.
Sergey
S
Denisov
Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany.
Peter
P
Hänggi
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel.
Eli
E
Barkai