Anomalous Diffusion in Random-Media of Any Dimensionality

Physica A: Statistical Mechanics and its Applications, 1990

This paper investigates whether spontaneous, stationary velocity fluctuations can lead to deviations from the regular Fickian diffusion. A kinematic analysis reveals that anomalous diffusion, both fast and slow, arises from long-tailed velocity auto-correlation functions (VACF). This infinite span of interdependence of the random velocity leads to the breakdown of the central limit theorem for particle displacements. A generalized Langevin equation, which features a retarded friction, has been used to describe the particle dynamics in the long-time limit. The analysis reveals that simple power-law decay models for the friction kernel are adequate to yield the pathological VACFs which imply anomalous diffusion. The fluctuation dissipation theorem is invoked to infer that a fractional noise gives rise to anomalous diffusion. Such a Langevin equation represents a mean-field description of disorder effects and the friction kernel then becomes a constitutive property of the medium.

Journal of Statistical Mechanics: Theory and Experiment, 2011

We study how the Einstein relation between spontaneous fluctuations and the response to an external perturbation holds in the absence of currents, for the comb model and the elastic single-file, which are examples of systems with subdiffusive transport properties. The relevance of nonequilibrium conditions is investigated: when a stationary current (in the form of a drift or an energy flux) is present, the Einstein relation breaks down, as it is known to happen in systems with standard diffusion. In the case of the comb model, a general relation -appeared in the recent literaturebetween response function and an unperturbed suitable correlation function, allows us to explain the observed results. This suggests that a relevant ingredient in breaking the Einstein formula, for stationary regimes, is not the anomalous diffusion but the presence of currents driving the system out of equilibrium.

Subdiffusive transport in tilted washboard potentials is studied within the fractional Fokker-Planck equation approach, using the associated continuous time random walk (CTRW) framework. The scaled subvelocity is shown to obey a universal law, assuming the form of a stationary Lévystable distribution. The latter is defined by the index of subdiffusion α and the mean subvelocity only, but interestingly depends neither on the bias strength nor on the specific form of the potential. These scaled, universal subvelocity fluctuations emerge due to the weak ergodicity breaking and are vanishing in the limit of normal diffusion. The results of the analytical heuristic theory are corroborated by Monte Carlo simulations of the underlying CTRW.

Acta Physica Polonica B, 2013

Diffusion regimes most frequently found in nature are described in terms of asymptotic behaviors. In this work, we use a generalization of the finalvalue theorem for Laplace transform in order to investigate the anomalous diffusion phenomenon for asymptotic times. We generalize the concept of the diffusion exponent, including a wide variety of asymptotic behaviors than the power law. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor, λ. We obtain as well an exact expression for λ, which makes possible to describe all diffusive regimes featuring a universal parameter determined by the diffusion exponent. We show the existence of two kinds of ballistic diffusion, ergodic and non-ergodic. The method is general and may be applied to many types of stochastic problem.

Physical review. E, 2017

Logarithmic or Sinai-type subdiffusion is usually associated with random force disorder and nonstationary potential fluctuations whose root-mean-squared amplitude grows with distance. We show here that extremely persistent, macroscopic logarithmic diffusion also universally emerges at sufficiently low temperatures in stationary Gaussian random potentials with spatially decaying correlations, known to exist in a broad range of physical systems. Combining results from extensive simulations with a scaling approach we elucidate the physical mechanism of this unusual subdiffusion. In particular, we explain why with growing temperature and/or time a first crossover occurs to standard, power-law subdiffusion, with a time-dependent power-law exponent, and then a second crossover occurs to normal diffusion with a disorder-renormalized diffusion coefficient. Interestingly, the initial, nominally ultraslow diffusion turns out to be much faster than the universal de Gennes-Bässler-Zwanzig limit...

Physical Review E, 2000

The time dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a non-linear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347 (1995)]. The scope of the present paper is twofold. Firstly we show that this distribution can be obtained also as solution of the non-linear porous media equation. Secondly we prove that the time dependent Tsallis distribution can be obtained also as solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the velocity, that describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in presence of multiplicative noise. PACS number(s): 05.10.Gg , 05.20.-y

Journal de Physique, 1990

2014 Anomalous diffusion in presence of a (fractal) boundary is investigated. Asymptotic behaviour of the survival probability and current through an absorbing boundary are exactly calculated in specific examples. They agree with recent findings related with anomalous Warburg impedance experiments. The autocorrelation function for sites near the boundary is carefully discussed ; in presence of reflecting boundary conditions it can be different from the (spatial) average of the autocorrelation at variance with the normal diffusion case. Interesting issues from a methodological point of view are discussed in the renormalization groups analysis. Scaling arguments are given relating various exponents.

Journal de Physique, 1988

Physical Review E, 2010

We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The long-time behavior of the particle position is studied in the frame of a continuous-time random walk on a semi-infinite one-dimensional lattice. We formulate the conditions for anomalous diffusion, derive the diffusion laws and analyze their dependence on the particle mass and the distribution of the random force.

Physical Review E, 1996