Understanding deterministic diffusion by correlated random walks

Low-dimensional periodic arrays of scatterers with a moving point particle are ideal models for studying deterministic diffusion. For such systems the diffusion coefficient is typically an irregular function under variation of a control parameter. Here we propose a systematic scheme of how to approximate deterministic diffusion coefficients of this kind in terms of correlated random walks. We apply this approach to two simple examples which are a one-dimensional map on the line and the periodic Lorentz gas. Starting from suitable Green-Kubo formulas we evaluate hierarchies of approximations for their parameter-dependent diffusion coefficients. These approximations converge exactly yielding a straightforward interpretation of the structure of these irregular diffusion coeficients in terms of dynamical correlations.Comment: 13 pages (revtex) with 5 figures (postscript

Comparison of pure and combined search strategies for single and multiple targets

We address the generic problem of random search for a point-like target on a line. Using the measures of search reliability and efficiency to quantify the random search quality, we compare Brownian search with L\'evy search based on long-tailed jump length distributions. We then compare these results with a search process combined of two different long-tailed jump length distributions. Moreover, we study the case of multiple targets located by a L\'evy searcher.Comment: 16 pages, 12 figure

Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals

We show that the generalized diffusion coefficient of a subdiffusive intermittent map is a fractal function of control parameters. A modified continuous time random walk theory yields its coarse functional form and correctly describes a dynamical phase transition from normal to anomalous diffusion marked by strong suppression of diffusion. Similarly, the probability density of moving particles is governed by a time-fractional diffusion equation on coarse scales while exhibiting a specific fine structure. Approximations beyond stochastic theory are derived from a generalized Taylor-Green-Kubo formula.Comment: 4 pages, 3 eps figure

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Galilean invariance is a cornerstone of classical mechanics. It states that for closed systems the equations of motion of the microscopic degrees of freedom do not change under Galilean transformations to different inertial frames. However, the description of real world systems usually requires coarse-grained models integrating complex microscopic interactions indistinguishably as friction and stochastic forces, which intrinsically violate Galilean invariance. By studying the coarse-graining procedure in different frames, we show that alternative rules -- denoted as "weak Galilean invariance" -- need to be satisfied by these stochastic models. Our results highlight that diffusive models in general can not be chosen arbitrarily based on the agreement with data alone but have to satisfy weak Galilean invariance for physical consistency

Extended Poisson-Kac Theory: A Unifying Framework for Stochastic Processes

Stochastic processes play a key role for modeling a huge variety of transport problems out of equilibrium, with manifold applications throughout the natural and social sciences. To formulate models of stochastic dynamics the conventional approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or L\'evy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless \textit{unbounded}, meaning that arbitrarily large fluctuations can be obtained with finite probability. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing physically realistic finite propagation velocity. Our approach is motivated by the theory of L\'evy walks, which we embed into an extension of conventional Poisson-Kac processes. The resulting extended theory employs generalised transition rates to model subtle microscopic dynamics, which reproduces non-trivial spatio-temporal correlations on macroscopic scales. It thus enables the modelling of many different kinds of dynamical features, as we demonstrate by three physically and biologically motivated examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking `Brownian yet non Gaussian' diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory can therefore be used to model a wide range of finite velocity dynamical phenomena that are observed experimentally.Comment: 26 pages, 5 figure

Метод экспертных оценок в лингводидактике

Рассматривается использование метода экспертных оценок как в общей системе научно-педагогической экспертной деятельности, так и в рамках ее лингводидактического аспекта. Дается характеристика процедуры проведения экспертизы, а также оценивается перспективность обращения к интеллектуальным компьютерным системам как инструментам экспертного анализа

Anomalous dynamics of cell migration

Cell movement, for example during embryogenesis or tumor metastasis, is a complex dynamical process resulting from an intricate interplay of multiple components of the cellular migration machinery. At first sight, the paths of migrating cells resemble those of thermally driven Brownian particles. However, cell migration is an active biological process putting a characterization in terms of normal Brownian motion into question. By analyzing the trajectories of wildtype and mutated epithelial (MDCK-F) cells we show experimentally that anomalous dynamics characterizes cell migration. A superdiffusive increase of the mean squared displacement, non-Gaussian spatial probability distributions, and power-law decays of the velocity autocorrelations are the basis for this interpretation. Almost all results can be explained with a fractional Klein- Kramers equation allowing the quantitative classification of cell migration by a few parameters. Thereby it discloses the influence and relative importance of individual components of the cellular migration apparatus to the behavior of the cell as a whole.Comment: 20 pages, 3 figures, 1 tabl

Logarithmic oscillators: ideal Hamiltonian thermostats

A logarithmic oscillator (in short, log-oscillator) behaves like an ideal thermostat because of its infinite heat capacity: when it weakly couples to another system, time averages of the system observables agree with ensemble averages from a Gibbs distribution with a temperature T that is given by the strength of the logarithmic potential. The resulting equations of motion are Hamiltonian and may be implemented not only in a computer but also with real-world experiments, e.g., with cold atoms.Comment: 5 pages, 3 figures. v4: version accepted in Phys. Rev. Let

Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity.

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson-Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson-Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables