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

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.Comment: PRL, in pres

Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

We find a general formula for the distribution of time-averaged observables for systems modeled according to the sub-diffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly non-ergodic statistical mechanical framework is constructed, which is based on L\'evy's generalized central limit theorem. As an example we calculate the distribution of $\bar{X}$: the time average of the position of the particle, for unbiased and uniformly biased particles, and show that $\bar{X}$ exhibits large fluctuations compared with the ensemble average $$.Comment: 5 pages, 2 figure

On distributions of functionals of anomalous diffusion paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.Comment: 25 pages, 4 figure

Weakly non-ergodic Statistical Physics

We find a general formula for the distribution of time averaged observables for weakly non-ergodic systems. Such type of ergodicity breaking is known to describe certain systems which exhibit anomalous fluctuations, e.g. blinking quantum dots and the sub-diffusive continuous time random walk model. When the fluctuations become normal we recover usual ergodic statistical mechanics. Examples of a particle undergoing fractional dynamics in a binding force field are worked out in detail. We briefly discuss possible physical applications in single particle experiments

Jamming and correlation patterns in traffic of information on sparse modular networks

89.75.Hc Networks and genealogical trees, 05.40.Ca Noise, 02.70.-c Computational techniques; simulations,