Normal Random Walks Research Articles

Diffusive transport on networks with stochastic resetting to multiple nodes.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

Normal and anomalous random walks of 2-d solitons.

Solitons, which describe the propagation of concentrated beams of light through nonlinear media, can exhibit a variety of behaviors as a result of the intrinsic dissipation, diffraction, and the nonlinear effects. One of these phenomena, modeled by the complex Ginzburg-Landau equation, is chaotic explosions, transient enlargements of the soliton that may induce random transversal displacements, which in the long run lead to a random walk of the soliton center. As we show in this work, the transition from nonmoving to moving solitons is not a simple bifurcation but includes a sequence of normal and anomalous random walks. We analyze their statistics with the distribution of generalized diffusivities, a novel approach that has been used successfully for characterizing anomalous diffusion.

Fractional random walk lattice dynamics

We analyze time-discrete and time-continuous ‘fractional’ random walks on undirected regular networks with special focus on cubic periodic lattices in n = 1, 2, 3,.. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices where recovers the normal walk. First we demonstrate that the interval is admissible for the fractional random walk. We derive analytical expressions for the transition matrix of the fractional random walk and closely related the average return probabilities. We further obtain the fundamental matrix , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix relates fractional random walks with normal random walks. We show that the matrix elements of the transition matrix of the fractional random walk exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the transition matrix and return probabilities of the fractional random walk are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

High activity and Lévy searches: jellyfish can search the water column like fish

Over-fishing may lead to a decrease in fish abundance and a proliferation of jellyfish. Active movements and prey search might be thought to provide a competitive advantage for fish, but here we use data-loggers to show that the frequently occurring coastal jellyfish (Rhizostoma octopus) does not simply passively drift to encounter prey. Jellyfish (327 days of data from 25 jellyfish with depth collected every 1 min) showed very dynamic vertical movements, with their integrated vertical movement averaging 619.2 m d(-1), more than 60 times the water depth where they were tagged. The majority of movement patterns were best approximated by exponential models describing normal random walks. However, jellyfish also showed switching behaviour from exponential patterns to patterns best fitted by a truncated Lévy distribution with exponents (mean μ=1.96, range 1.2-2.9) close to the theoretical optimum for searching for sparse prey (μopt≈2.0). Complex movements in these 'simple' animals may help jellyfish to compete effectively with fish for plankton prey, which may enhance their ability to increase in dominance in perturbed ocean systems.

Discriminating between normal and anomalous random walks

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).

Optimal transport in a ratchet coupled to a modulated environment: The role of Levy walks

We consider a Brownian particle in a ratchet potential coupled to a modulated environment and subjected to an external oscillating force. The modulated environment is modelled by a finite number N of uncoupled harmonic oscillators. Superdiffusive motion and Levy walks (anomalous random walks) are observed for any N and for low values of the external amplitude F. The coexistence of left and right running states enhances the power α from the time dependence of the mean square displacement (MSD). It is shown that α is twice the average of the power of the separated left and right MSDs. Normal random walks are obtained by increasing F. We show that the maximal mobility of particles along the periodic structure occurs just before superdiffusive motion disappears and Levy walks are transformed into normal random walks.

A first passage time distribution for a discrete version of the Ornstein–Uhlenbeck process

The probability of the first entrance to the negative semi-axis for a one-dimensional discrete Ornstein–Uhlenbeck (O-U) process is studied in this work. The discrete O-U process is a simple generalization of the random walk and many of its statistics may be calculated using essentially the same formalism. In particular, the case in which Sparre-Andersen's theorem applies for normal random walks is considered, and it is shown that the universal features of the first passage probability do not extend to the discrete O-U process. Finally, an explicit expression for the generating function of the probability of first entrance to the negative real axis at step n is calculated and analysed for a particular choice of the step distribution.

Field-line transport in stochastic magnetic fields: Percolation, Lévy flights, and non-Gaussian dynamics.

The transport of magnetic field lines is studied numerically in the case where strong three-dimensional magnetic fluctuations are superimposed to a uniform average magnetic field. The magnetic percolation of field lines between magnetic islands is found, as well as a non-Gaussian regime where the field lines exhibit L\'evy random walks, changing from L\'evy flights to trapped motion. Anomalous diffusion laws 〈\ensuremath{\Delta}${\mathit{x}}_{\mathit{i}}^{2}$〉\ensuremath{\propto}${\mathit{s}}^{\mathrm{\ensuremath{\alpha}}}$ with \ensuremath{\alpha}g1 and \ensuremath{\alpha}1 are found for low fluctuation levels, while normal diffusion and Gaussian random walks are recovered for sufficiently high fluctuation levels.

Conditional Boundary Crossing Probabilities, with Applications to Change-Point Problems

For normal random walks $S_1, S_2,\ldots$, formed from independent identically distributed random variables $X_1, X_2,\ldots$, we determine the asymptotic behavior under regularity conditions of $P(S_n > mg(n/m) \text{for some} n < m\mid S_m = m\xi_0, U_m = m\lambda_0), \quad\xi_0 < g(1),$ where $U_m = X^2_1 + \cdots + X^2_m$. The result is applied to a normal change-point problem to approximate null distributions of test statistics and to obtain approximate confidence sets for the change-point.

Geometric Properties of Off-Lattice Self-Avoiding Random Walks

Simple geometric properties of self-avoiding random walks not restricted to lie on a lattice have been investigated. Both the mean-square end-to-end distance and the mean-square radius of gyration are found to deviate in their exponential dependence upon the number of steps in the walk from values established for lattice-restricted walks. As the allowed deviation of each step from its lattice-defined position increases, the exponent increases markedly from its lattice value and then attains a limiting value as free rotation is approached. For all degrees of deviation the exponent is significantly larger than the accepted value of 6/5. Although normal random walks have the same exponential dependence whether restricted to a lattice or not, self-avoiding random walks behave differently. Thus any generalization to off-lattice systems of properties calculated from self-avoiding walks constrained to a lattice may be difficult. Since all linear macromolecules have some rotational freedom, the lattice-constrained self-avoiding random walk may be an insufficient model of such systems, even though it is clearly better than the normal random-walk model.