The Lévy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time averaged mean squared displacement δ 2 often used to analyze single particle tracking experiments. The ballistic phase of the motion is non-ergodic and we obtain analytical expressions for the fluctuations of δ 2 . For enhanced sub-ballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto Phys. Rev. Lett. 108, 164101 (2012) deviations of temporal averages δ 2 from the ensemble average x 2 depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time averaged response to a bias is considered and the resultant generalized Einstein relations are discussed. PACS numbers: 02.50.-r, 05.40.Fb

We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard non-ballistic random walks.

We investigate a Lévy-Walk alternating between velocities ±v0 with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is x 2 ∝ t 2 , the latter to enhanced diffusion with x 2 ∝ t ν , 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.PACS. 0 5.40.Fb, 02.50.-r

We consider stationary profiles of reactants' concentrations and of reaction zones of an A + B --> 0 reaction in a flat subdiffusive medium fed by reactants of both types on both sides. The structures formed under such conditions differ strikingly from those in simple diffusion and exhibit accumulation and depletion zones close to the boundaries and nonmonotonic behavior of the reaction intensity with respect to the reactants' concentrations at the boundaries. These findings are connected to an effectively nonlinear character of transport in subdiffusive systems under reactions.

We consider an irreversible autocatalytic conversion reaction A + B → 2A under subdiffusion described by continuous-time random walks. The reactants' transformations take place independently of their motion and are described by constant rates. The analog of this reaction in the case of normal diffusion is described by the Fisher-Kolmogorov-Petrovskii-Piskunov equation leading to the existence of a nonzero minimal front propagation velocity, which is really attained by the front in its stable motion. We show that for subdiffusion, this minimal propagation velocity is zero, which suggests propagation failure.

We study numerically the autocatalytic irreversible reaction A+B-->2A on a one-dimensional lattice for the case of subdiffusive reactants performing symmetric continuous-time random walks with the power-law waiting time density function psi(t) proportional, t(-1-alpha) with 0

We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for ergodic (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The Lévy walk model is used to exemplify the no-go theorem.

We discuss the front propagation in the A + B → 2A reaction under subdiffusion which is described by continuous time random walks with a heavy-tailed power law waiting time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front