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V. I. Klyatskin - One of the best experts on this subject based on the ideXlab platform.
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dynamic stochastic systems Typical Realization Curve and lyapunov s exponents
Izvestiya Atmospheric and Oceanic Physics, 2008Co-Authors: V. I. KlyatskinAbstract:The relationship between a statistical description of dynamic stochastic systems on the basis of the ideas of statistical topography and the conventional analysis of Lyapunov stability of dynamic systems with the aid of Lyapunov’s exponents is discussed.
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Dynamic stochastic systems, Typical Realization Curve, and Lyapunov’s exponents
Izvestiya Atmospheric and Oceanic Physics, 2008Co-Authors: V. I. KlyatskinAbstract:The relationship between a statistical description of dynamic stochastic systems on the basis of the ideas of statistical topography and the conventional analysis of Lyapunov stability of dynamic systems with the aid of Lyapunov’s exponents is discussed.
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Dynamics of Stochastic Systems - Chapter 4 – Random quantities, processes and fields
Dynamics of Stochastic Systems, 2005Co-Authors: V. I. KlyatskinAbstract:This chapter discusses the basic concepts of the theory of random quantities, processes, and fields. It discusses the concept of Typical Realization Curve of random process, which concerns the fundamental features of the behavior of a separate process Realization as a whole for temporal intervals of arbitrary duration. Consideration of specific random processes allows the obtaining an additional information concerning the Realization's spikes relative to the Typical Realization Curve. The one-point probability density of random process is a result of averaging the singular indicator function over an ensemble of Realizations of this process. This function is concentrated at points at which the process crosses the line. The chapter starts the discussion with continuous processes, namely, with the Gaussian random process with zero-valued mean and correlation function. The chapter also considers the random processes whose points of discontinuity form Poisson streams of points. Currently, three types of such processes–the Poisson process, telegrapher's process, and generalized telegrapher's process–are mainly used in model problems of physics.
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Stochastic Equations through the Eye of the Physicist - Chapter 3 – Random quantities, processes, and fields
Stochastic Equations through the Eye of the Physicist, 2005Co-Authors: V. I. KlyatskinAbstract:This chapter discusses the basic properties of random quantities, processes, and fields that are widely used in analyzing dynamic systems with fluctuating parameters. It is found that if one deals with random function, then all this function statistical characteristics at any fixed instant are exhaustively described in terms of the one-point probability density. The Fourier transform of the correlation function with respect to the spatial variable defines the spatial spectral function. The Typical Realization Curve of random process concerns the fundamental features of the behavior of a separate process Realization as a whole for temporal intervals of arbitrary duration. The one-point probability density of random process is a result of averaging the singular indicator function over an ensemble of Realizations of this process. The discontinuous processes are the random functions that change their time-dependent behavior at discrete instants given statistically. The characteristic functional that describes all statistical characteristics of random process is analyzed. The one-dimensional discrete-continuous Markovian process is also elaborated.
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Dynamics of Stochastic Systems - Chapter 11 – Passive tracer clustering and diffusion in random hydrodynamic flows
Dynamics of Stochastic Systems, 2005Co-Authors: V. I. KlyatskinAbstract:This chapter focuses on passive tracer clustering and diffusion in random hydrodynamic flows. In the Lagrangian representation, the behavior of passive tracer is described in terms of three ordinary differential equations, from which one can easily pass to the linear Liouville equation in the corresponding phase space. With this goal in view, an indicator function is introduced, which explicitly emphasizes the fact that the solution to the initial dynamic equations depends on the Lagrangian coordinates r 0 . The chapter also demonstrates that Lagrangian statistical properties of a particle in flows containing the potential random component are qualitatively different from the statistical properties of a particle in divergence-free flows where j( t |r 0 ) = 1 and particle concentration remains invariant in the vicinity of a fixed particle ρ ( t | r o ) = ρ 0 (r 0 ) = const. These statistical estimates are indicative of the fact that the statistics of random processes j( t |r 0 ) and p ( t\r 0 ) is formed by the Realization spikes relative Typical Realization Curve. In the case of the delta-correlated random velocity field, linear equation in the absence of mean flow allows a relatively simple passage to the closed equations for both buoyant tracer average concentration and its higher multipoint correlation functions. To describe the local behavior of tracer Realizations in random velocity field, the probability distribution of tracer concentration is required, which can be obtained only in the absence of molecular diffusion.
