Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-in-time random noise with the spectrum $\overline{|f(k)|^2}\propto k^{-1}$ exhibit a biscaling behavior: All moments of velocity differences $S_{n\le 3}(r)=\overline{|u(x+r)-u(x)|^n}\equiv\overline{|\Delta u|^n}\propto r^{n/3}$, while $S_{n>3}\propto r^{\zeta_n}$ with $\zeta_n\approx 1$ for real $n>0$ (Chekhlov and Yakhot, Phys. Rev. E {\bf 51}, R2739, 1995). The probability density function, which is dominated by coherent shocks in the interval $\Delta u<0$, is ${\cal P}(\Delta u,r)\propto (\Delta u)^{-q}$ with $q\approx 4$.Comment: 12 pages, psfig macro, 4 figs in Postscript, accepted to Phys. Rev. E as a Brief Communicatio

Phenomenology for the decay of energy-containing eddies in homogeneous MHD turbulence

We evaluate a number of simple, one‐point phenomenological models for the decay of energy‐containing eddies in magnetohydrodynamic(MHD) and hydrodynamicturbulence. The MHDmodels include effects of cross helicity and Alfvénic couplings associated with a constant mean magnetic field, based on physical effects well‐described in the literature. The analytic structure of three separate MHDmodels is discussed. The single hydrodynamic model and several MHDmodels are compared against results from spectral‐method simulations. The hydrodynamic model phenomenology has been previously verified against experiments in wind tunnels, and certain experimentally determined parameters in the model are satisfactorily reproduced by the present simulation. This agreement supports the suitability of our numerical calculations for examining MHDturbulence, where practical difficulties make it more difficult to study physical examples. When the triple‐decorrelation time and effects of spectral anisotropy are properly taken into account, particular MHDmodels give decay rates that remain correct to within a factor of 2 for several energy‐halving times. A simple model of this type is likely to be useful in a number of applications in space physics, astrophysics, and laboratory plasma physics where the approximate effects of turbulence need to be included

A travelling wave model to interpret a wound healing migration assay for human peritoneal mesothelial cells

The critical determinants of the speed of an invading cell front are not well known. We performed a "wound-healing" experiment that quantifies the migration of human peritoneal mesothelial cells over components of the extracellular matrix. Results were interpreted in terms of Fisher's equation, which includes terms for the modeling of random cell motility (diffusion) and proliferation. The model predicts that, after a short transient, the invading cell front will move as a traveling wave at constant speed. This is consistent with the experimental findings. Using the model, a relationship between the rate of cell proliferation and the diffusion coefficient was obtained. We used the model to deduce the cell diffusion coefficients under control conditions and in the presence of collagen IV and compared these with other published data. The model may be useful in analyzing the invasive capacity of cancer cells as well in predicting the efficacy of growth factors in tissue reconstruction, including the development of monolayer sheets of cells in skin engineering or the repair of injured corneas using grafts of cultured cells

The Effect of Coherent Structures on Stochastic Acceleration in MHD Turbulence

We investigate the influence of coherent structures on particle acceleration in the strongly turbulent solar corona. By randomizing the Fourier phases of a pseudo-spectral simulation of isotropic MHD turbulence (Re $\sim 300$), and tracing collisionless test protons in both the exact-MHD and phase-randomized fields, it is found that the phase correlations enhance the acceleration efficiency during the first adiabatic stage of the acceleration process. The underlying physical mechanism is identified as the dynamical MHD alignment of the magnetic field with the electric current, which favours parallel (resistive) electric fields responsible for initial injection. Conversely, the alignment of the magnetic field with the bulk velocity weakens the acceleration by convective electric fields - \bfu \times \bfb at a non-adiabatic stage of the acceleration process. We point out that non-physical parallel electric fields in random-phase turbulence proxies lead to artificial acceleration, and that the dynamical MHD alignment can be taken into account on the level of the joint two-point function of the magnetic and electric fields, and is therefore amenable to Fokker-Planck descriptions of stochastic acceleration.Comment: accepted for publication in Ap

Analysis of unbounded operators and random motion

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure

Front propagation in A+B -> 2A reaction under subdiffusion

We consider an irreversible autocatalytic conversion reaction A+B -> 2A under subdiffusion described by continuous time random walks. The reactants' transformations take place independently on 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 (FKPP) 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

Linear representations of probabilistic transformations induced by context transitions

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of ${\bf C}$-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse superposition principle in this framework.Comment: Added a physical discussion and new reference

A generalized empirical interpolation method : application of reduced basis techniques to data assimilation

In an effort to extend the classical lagrangian interpolation tools, new interpolating methods that use general interpolating functions are explored. The method analyzed in this paper, called Generalized Empirical Interpolation Method (GEIM), belongs to this class of new techniques. It generalizes the plain Empirical Interpolation Method by replacing the evaluation at interpolating points by application of a class of interpolating linear functions. The paper is divided into two parts: first, the most basic properties of GEIM (such as the well-posedness of the generalized interpolation problem that is derived) will be analyzed. On a second part, a numerical example will illustrate how GEIM, if considered from a reduced basis point of view, can be used for the real-time reconstruction of experiments by coupling data assimilation with numerical simulations in a domain decomposition framework

Propagation and Extinction in Branching Annihilating Random Walks

We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite density wave, or extinction may occur, in which the number of particles vanishes in the long-time limit. The number parity conserving case where 2-offspring are produced in each branching event can be solved exactly for unit reaction probability, from which qualitative features of the transition between propagation and extinction, as well as intriguing parity-specific effects are elucidated. An approximate analysis is developed to treat this transition for general BAW processes. A scaling description suggests that the critical exponents which describe the vanishing of the particle density at the transition are unrelated to those of conventional models, such as Reggeon Field Theory. P. A. C. S. Numbers: 02.50.+s, 05.40.+j, 82.20.-wComment: 12 pages, plain Te

Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed