Scaling detection in time series: diffusion entropy analysis

The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and L\'{e}vy time series, as prototypes of ordinary and anomalus statistics, respectively, and we analyse them with the DEA and four ordinary methods of analysis, some of which are very popular. We show that the DEA determines the correct scaling exponent even when the statistical properties, as well as the dynamic properties, are anomalous. The other four methods produce correct results in the Gauss case but fail to detect the correct scaling in the case of L\'{e}vy statistics.Comment: 21 pages,10 figures, 1 tabl

Target-searching on the percolation

We study target-searching processes on a percolation, on which a hunter tracks a target by smelling odors it emits. The odor intensity is supposed to be inversely proportional to the distance it propagates. The Monte Carlo simulation is performed on a 2-dimensional bond-percolation above the threshold. Having no idea of the location of the target, the hunter determines its moves only by random attempts in each direction. For lager percolation connectivity $p\gtrsim 0.90$, it reveals a scaling law for the searching time versus the distance to the position of the target. The scaling exponent is dependent on the sensitivity of the hunter. For smaller $p$, the scaling law is broken and the probability of finding out the target significantly reduces. The hunter seems trapped in the cluster of the percolation and can hardly reach the goal.Comment: 5 figure

From deterministic dynamics to kinetic phenomena

We investigate a one-dimenisonal Hamiltonian system that describes a system of particles interacting through short-range repulsive potentials. Depending on the particle mean energy, $\epsilon$, the system demonstrates a spectrum of kinetic regimes, characterized by their transport properties ranging from ballistic motion to localized oscillations through anomalous diffusion regimes. We etsablish relationships between the observed kinetic regimes and the "thermodynamic" states of the system. The nature of heat conduction in the proposed model is discussed.Comment: 4 pages, 4 figure

Biased diffusion in a piecewise linear random potential

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.Comment: 11 pages, 3 figure

Truncated Levy Random Walks and Generalized Cauchy Processes

A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.Comment: 7 pages, 3 figure

Symbolic stochastic dynamical systems viewed as binary N-step Markov chains

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In the model, the conditional probability that the i-th symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker-Planck equation governing the distribution function of the L-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.Comment: 14 pages, 13 figure

Anomalous diffusion and Tsallis statistics in an optical lattice

We point out a connection between anomalous quantum transport in an optical lattice and Tsallis' generalized thermostatistics. Specifically, we show that the momentum equation for the semiclassical Wigner function that describes atomic motion in the optical potential, belongs to a class of transport equations recently studied by Borland [PLA 245, 67 (1998)]. The important property of these ordinary linear Fokker--Planck equations is that their stationary solutions are exactly given by Tsallis distributions. Dissipative optical lattices are therefore new systems in which Tsallis statistics can be experimentally studied.Comment: 4 pages, 1 figur

Number of distinct sites visited by N random walkers on a Euclidean lattice

The evaluation of the average number S_N(t) of distinct sites visited up to time t by N independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for large N, S_N(t) \approx \hat S_N(t) (1-\Delta), where \hat S_N(t) is the volume of a hypersphere of radius (4Dt \ln N)^{1/2}, \Delta={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N, d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the dimension and time. The first three terms of these series are calculated explicitly and the resulting expressions are compared with other approximations and with simulation results for dimensions 1, 2, and 3. Some implications of these results on the geometry of the set of visited sites are discussed.Comment: 15 pages (RevTex), 4 figures (eps); to appear in Phys. Rev.

Scaling-violation phenomena and fractality in the human posture control systems

By analyzing the movements of quiet standing persons by means of wavelet statistics, we observe multiple scaling regions in the underlying body dynamics. The use of the wavelet-variance function opens the possibility to relate scaling violations to different modes of posture control. We show that scaling behavior becomes close to perfect, when correctional movements are dominated by the vestibular system.Comment: 12 pages, 4 figures, to appear in Phys. Rev.

Does strange kinetics imply unusual thermodynamics?

We introduce a fractional Fokker-Planck equation (FFPE) for Levy flights in the presence of an external field. The equation is derived within the framework of the subordination of random processes which leads to Levy flights. It is shown that the coexistence of anomalous transport and a potential displays a regular exponential relaxation towards the Boltzmann equilibrium distribution. The properties of the Levy-flight FFPE derived here are compared with earlier findings for subdiffusive FFPE. The latter is characterized by a non-exponential Mittag-Leffler relaxation to the Boltzmann distribution. In both cases, which describe strange kinetics, the Boltzmann equilibrium is reached and modifications of the Boltzmann thermodynamics are not required