Turbulent plasma diffusion in velocity space

The concept of turbulent diffusion in positional space which has been used by the author and others for the study of turbulent plasma, has been generalized to include the effects of diffusion in velocity space.     

Structural ion-sound plasma turbulence as a self-similar random process

It is shown that the experimentally investigated structural ion-sound plasma turbulence is a self-similar stationary random process. The self-similarity parameter is determined by two temporal laws: the nonrandom character of the appearance of nonlinear structures (nonlinear ion-sound solitons) in the plasma, and the nonlinear interaction between them. As the distance from the threshold of the ion-sound current instability increases, the self-similar random process approaches a Gaussian random process, but this limit has not been attained experimentally. The possibility of recording superlong time series of the fluctuations of the signal of the plasma process and processing of the time series by the R/S analysis method has made it possible to prove self-similarity of the plasma structural turbulence. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 3, 203–208 (10 August 1999)     

Quantum transport in a time-dependent random potential with a finite correlation time

Using an earlier density matrix formalism in momentum space we study the motion of a particle in a time-dependent random potential with a finite correlation time τ, for 0 < t ? τ. Within this domain we consider two subdomains bounded by kinetic time scales (t _c ² = 2m? ^-1 c ², c ² = σ ², ξ ², σξ, with 2σ the width of an initial wavepacket and the correlation length of the gaussian potential fluctuations), where we obtain power law scaling laws for the effect of the random potential in the mean squared displacement 〈x ²〉 and in the mean kinetic energy 〈E _kin〉. At short times, ? min (t _σ ², 1/2t _ξ ²), 〈x ²〉 and 〈E _kin〉 scale classically as t ⁴ and t ², respectively. At intermediate times, t _σξ ? t ? 2t _σ ² and 1/2t _ξ ² ? t ? t _σξ, these quantities scale quantum mechanically as t ^3/2 and as √t, respectively. These results lie in the perspective of recent studies of the existence of (fractional) power law behavior of 〈x ²〉 and 〈E _kin〉 at intermediate times. We also briefly discuss the scaling laws for 〈x ²〉 and 〈E _kin〉 at short times in the case of spatially uncorrelated potential.     

Spartan random processes in time series modeling

A Spartan random process (SRP) is used to estimate the correlation structure of time series and to predict (interpolate and extrapolate) the data values. SRPs are motivated from statistical physics, and they can be viewed as Ginzburg-Landau models. The temporal correlations of the SRP are modeled in terms of ‘interactions’ between the field values. Model parameter inference employs the computationally fast modified method of moments, which is based on matching sample energy moments with the respective stochastic constraints. The parameters thus inferred are then compared with those obtained by means of the maximum likelihood method. The performance of the Spartan predictor (SP) is investigated using real time series of the quarterly S&P 500 index. SP prediction errors are compared with those of the Kolmogorov-Wiener predictor. Two predictors, one of which is explicit, are derived and used for extrapolation. The performance of the predictors is similarly evaluated.     

Nonlinear wave processes in a plasma with stationary random irregularities. Part II

We analyze the influence of large-scale random irregularities of a medium on nonlinear wave interactions. First, we study three-wave interactions in an equilibrium medium, and then we consider the problems of stabilizing the explosive instability in the presence of random irregularities. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 1, pp. 54–71, January 2007     

Nonlinear wave processes in a plasma with stationary random irregularities. Part I

In this review comprising two papers, we discuss the results of studying the influence of random irregularities of a medium on nonlinear wave interactions. In this paper representing the first part of the review, we summarize the results of studying the mean fields of interacting waves in the presence of three-dimensional random irregularities and discuss the basic approximations within the framework of which these results are obtained. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 11, pp. 977–1001, November 2006.     

Turbulent diffusion in a compressible medium

This paper examines the diffusion of impurity particles in a compressible turbulent medium and compares it to diffusion in an incompressible medium. The turbulent diffusion coefficients are calculated using exact formulas expressed in terms of the Green’s function describing impurity transport in an infinite homogeneous, isotropic, stationary turbulent medium. To obtain an approximate expression for the Green’s function, numerical solutions of the nonlinear DIA (direct interaction approximation) equation (which in this paper are obtained for the first time for the case of compressible turbulence) are employed. Two types of turbulence are examined, acoustic and a mixture of shock waves. These are described by different generalized spectra. Finally, it is shown that compressibility significantly enhances the diffusion coefficient in the case of acoustic turbulence and reduces it in the second case. Zh. éksp. Teor. Fiz. 114, 930–945 (September 1998)     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

Nondiffusive transport in plasma turbulence: a fractional diffusion approach

Numerical evidence of nondiffusive transport in three-dimensional, resistive pressure-gradient-driven plasma turbulence is presented. It is shown that the probability density function (pdf) of tracer particles' radial displacements is strongly non-Gaussian and exhibits algebraic decaying tails. To model these results we propose a macroscopic transport model for the pdf based on the use of fractional derivatives in space and time that incorporate in a unified way space-time nonlocality (non-Fickian transport), non-Gaussianity, and nondiffusive scaling. The fractional diffusion model reproduces the shape and space-time scaling of the non-Gaussian pdf of turbulent transport calculations. The model also reproduces the observed superdiffusive scaling.     

Turbulent transport measurements with a laser Doppler velocimeter

The power spectrum of phototube current from a laser Doppler velocimeter operating in the heterodyne mode has been computed. The spectrum is obtained in terms of the space time correlation function, G(r,), of the fluid. The spectral width and shape predicted by the theory is in agreement with experiment. For normal operating parameters the time average spectrum contains information only for times shorter than the Lagrangian integral time scale of the turbulence. To examine the long time behaviour one must use either extremely small scattering angles, much longer wave-length radiation or a different mode of signal analysis, e.g. F.M. detection.     

Subordinated diffusion and continuous time random walk asymptotics

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Le?vy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.     

Quantitative model of price diffusion and market friction based on trading as a mechanistic random process

We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.     

Volatility estimators and the inverse range process in a random volatility random walk and Wiener processes

The purpose of this paper is to study the mean, the variance, the probability distribution and the hazard rate of the inverse range process of an a-priori unknown volatility random walk. Motivation for this process arises when it is necessary to obtain statistics that pertain to a process volatility in addition to the usual variance statistics. As a result, range process statistics are indicated as an additional source of information in the study of processes’ volatility. Examples and applications are considered.     

Quantum mechanics as a classical diffusion process

A direct construction is provided showing that the classical expectation value for the energy of a particle executing a classical diffusion process is equivalent to the quantum mechanical form with a Hamiltonian structured like that for a charged particle in an electromagnetic field.     

Influence of photon transport on diffusion processes in stripe heterolasers

The influence of photon transport on the diffusion of nonequilibrium charge carriers in heterolasers with stripe contact and in structures with a transverse p-n junction arrangement is taken into account. Results are presented of a machine computation of the nonequilibrium charge carrier (NCC) concentration profile and the radiation density in the active layer of AlGaAs lasers in the prethreshold pumping mode.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 12–17, June, 1984.     

Optimal realization time in the experimental investigation of random processes

Radiophysics and Quantum Electronics -     

Influence of a random telegraph process on the transport through a point contact

We describe the transport properties of a point contact under the influence of a classical two-level fluctuator. We employ a transfer matrix formalism allowing us to calculate arbitrary correlation functions of the stochastic process by mapping them on matrix products. The result is used to obtain the generating function of the full counting statistics of a classical point contact subject to a classical fluctuator, including extensions to a pair of two-level fluctuators as well as to a quantum point contact. We show that the noise in the quantum point contact is a sum of the (quantum) partitioning noise and the (classical) noise due to the two-level fluctuator. As a side result, we obtain the full counting statistics of a quantum point contact with time-dependent transmission probabilities.