The spatial correlations in the velocities arising from a random distribution of point vortices

This paper is devoted to a statistical analysis of the velocity fluctuations arising from a random distribution of point vortices in two-dimensional turbulence. Exact results are derived for the correlations in the velocities occurring at two points separated by an arbitrary distance. We find that the spatial correlation function decays extremely slowly with the distance. We discuss the analogy with the statistics of the gravitational field in stellar systems.Comment: 37 pages in RevTeX format (no figure); submitted to Physics of Fluid

Velocity field distributions due to ideal line vortices

We evaluate numerically the velocity field distributions produced by a bounded, two-dimensional fluid model consisting of a collection of parallel ideal line vortices. We sample at many spatial points inside a rigid circular boundary. We focus on ``nearest neighbor'' contributions that result from vortices that fall (randomly) very close to the spatial points where the velocity is being sampled. We confirm that these events lead to a non-Gaussian high-velocity ``tail'' on an otherwise Gaussian distribution function for the Eulerian velocity field. We also investigate the behavior of distributions that do not have equilibrium mean-field probability distributions that are uniform inside the circle, but instead correspond to both higher and lower mean-field energies than those associated with the uniform vorticity distribution. We find substantial differences between these and the uniform case.Comment: 21 pages, 9 figures. To be published in Physical Review E (http://pre.aps.org/) in May 200

Statistical mechanics of two-dimensional vortices and stellar systems

The formation of large-scale vortices is an intriguing phenomenon in two-dimensional turbulence. Such organization is observed in large-scale oceanic or atmospheric flows, and can be reproduced in laboratory experiments and numerical simulations. A general explanation of this organization was first proposed by Onsager (1949) by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics. Similarly, the structure and the organization of stellar systems (globular clusters, elliptical galaxies,...) in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by Chandrasekhar (1942). These statistical mechanics turn out to be relatively similar and present the same difficulties due to the unshielded long-range nature of the interaction. This analogy concerns not only the equilibrium states, i.e. the formation of large-scale structures, but also the relaxation towards equilibrium and the statistics of fluctuations. We will discuss these analogies in detail and also point out the specificities of each system.Comment: Chapter of the forthcoming "Lecture Notes in Physics" volume: ``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002

Memory effects in stochastic transport

The memory effects in stochastic transport, namely, the dependence of the form of transport equations on the macroscopic time are considered. Equations explicitly taking into account the microscopic aspect of the problem, without which the transfer processes cannot be adequately described, are derived; the methods of their solution are suggested; and the asymptotic properties of the latter are analyzed. (C) 2003 MAIK "Nauka/Interperiodica"

Enhanced superdiffusion and finite velocity of Levy flights

A fractional differential equation is derived that describes the transformation of a stochastic transport from fast spreading ((x) over bar proportional to t (alpha), alpha > 1) to a pseudowave regime (alpha = 1) due to the finiteness of the velocities of individual particles. Qualitative features of the new regime are discussed. (C) 2002 MAIK "Nauka / Interperiodica"

Enhanced superdiffusion and finite velocity of Levy flights

A fractional differential equation is derived that describes the transformation of a stochastic transport from fast spreading (x) over bar proportional to t(alpha), alpha > 1 to a pseudowave regime alpha = 1 due to the finiteness of the velocities of individual particles. Qualitative features of new regime are discussed

Comment on "Towards deterministic equations for Levy walks: The fractional material derivative"

The connection of problems considered in the paper by Sokolov and Metzler with stochastic transport in usual space and uniform medium is discussed

Subdiffusion in random compressible flows

In this work, we study the diffusion of admixture particles in a one-dimensional velocity field given by a gradient of a random potential. This refers us to the case of random compressible flows, where previously only scaling estimates were available. We develop a general approach which allows to solve this problem analytically. With its help we derive the macroscopic transport equation and rigorously show in which cases transport can be subdiffusive. We find the Fourier-Laplace transform of the Green's function of this equation and prove that for some potential distributions it satisfies the subdiffusive equation with fractional derivative with respect to time

Nonlinear dynamics of electron vortex lattices

Weak and strong nonlinearities that determine the evolution of regular ensembles of electron vortices in a magnetized plasma are analyzed. Qualitative differences in behavior between such a medium and standard nonlinear media are revealed. (C) 2004 MAIK "Nauka/Interpetiodica"

Translated from Pis'ma v Zhurnal Éksperimental'no oe i

In recent years, growing interest has been shown in the processes of stochastic transport because of the spatial and temporal nonlocalities inherent in this phenomenon There are many physical reasons that are responsible for the above-mentioned nonlocalities (fractional derivatives) in the transport equations (see discussion in Evidently, one should expect that evolution is continuous for any physical process satisfying the causality principle: if the solution to the equations is functionally related to the initial state by the Green's function G t , i.e., if n ( x , t ) = G t * n ( t = 0) , then transport with a fractional time derivative, including a recent excellent review [4], strictly speaking, do not possess this property. This unpleasant fact has in no way been discussed in the literature, though it is precisely the point that is expected to be helpful in the recognition of a hidden defect of the above-mentioned description, namely, of the incompleteness in the description of a particle cloud only in terms of its macroscopic concentration n ( x , t ). Interestingly, similar problems arise for strongly coupled coulombic systems in the quantum kinetic theory, where the solutions show a strong dependence on the initial correlations (1)), the defect is often "hidden under the rug." In reality, the time for approaching the microscopic evolution regime strongly depends on the initial condition and can be much longer than the microscopic time 〈τ〉 characterizing the random walk of individual particles (see below). This is especially characteristic of the subdiffusion time operators. Therefore, the memory effects considered in this work consist not in the familiar temporal nonlocality (fractional derivative) in the effective transport equation but in the fact that the form of this equation depends on the macroscopic time t (see below). When deriving the transport equations, we will use, as in Memory Effects in Stochastic Transpor