Periodic orbit theory of strongly anomalous transport

We establish a deterministic technique to investigate transport moments of arbitrary order. The theory is applied to the analysis of different kinds of intermittent one-dimensional maps and the Lorentz gas with infinite horizon: the typical appearance of phase transitions in the spectrum of transport exponents is explained.Comment: 22 pages, 10 figures, revised versio

Follow the fugitive: an application of the method of images to open dynamical systems

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open dynamical systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in more general dynamical systems.Comment: 9 pages, 1 figur

Temporal-varying failures of nodes in networks

We consider networks in which random walkers are removed because of the failure of specific nodes. We interpret the rate of loss as a measure of the importance of nodes, a notion we denote as failure-centrality. We show that the degree of the node is not sufficient to determine this measure and that, in a first approximation, the shortest loops through the node have to be taken into account. We propose approximations of the failure-centrality which are valid for temporal-varying failures and we dwell on the possibility of externally changing the relative importance of nodes in a given network, by exploiting the interference between the loops of a node and the cycles of the temporal pattern of failures. In the limit of long failure cycles we show analytically that the escape in a node is larger than the one estimated from a stochastic failure with the same failure probability. We test our general formalism in two real-world networks (air-transportation and e-mail users) and show how communities lead to deviations from predictions for failures in hubs.Comment: 7 pages, 3 figure

L\'evy walks on lattices as multi-state processes

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of L\'evy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.Comment: 27 pages, 4 figure

On universality of algebraic decays in Hamiltonian systems

Hamiltonian systems with a mixed phase space typically exhibit an algebraic decay of correlations and of Poincare' recurrences, with numerical experiments over finite times showing system-dependent power-law exponents. We conjecture the existence of a universal asymptotic decay based on results for a Markov tree model with random scaling factors for the transition probabilities. Numerical simulations for different Hamiltonian systems support this conjecture and permit the determination of the universal exponent.Comment: 4 pages, 3 figure

Random walks in a one-dimensional L\'evy random environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as L\'evy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.Comment: Final version to be published in J. Stat. Phys. 23 pages. (Changes from v1: Theorem 2.4 and Corollary 2.6 have been removed.

Transport properties of L\'evy walks: an analysis in terms of multistate processes

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such L\'evy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.Comment: 5 pages, 2 figures. New references adde

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of super-diffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time-range accessible to numerical simulations. We compare our simulations to the known analytical results for the variance of the anomalously-rescaled limiting normal distributions.Comment: 10 pages, 4 figure