Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Abstract

Strong anomalous diffusion, where $\langle |x(t)|^q \rangle \sim t^{q \nu(q)}$ with a nonlinear spectrum $\nu(q) \neq \mbox{const}$, 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 phenomena 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 multi-fractal anomalous diffusion, as it is complementary to the central limit theorem.