Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Abstract

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

Correlation properties of the backscattered field in a nonstationary statistical problem

The statistical problem of the scattering of wideband pulses by a random layered medium at normal incidence is considered in the framework of the wave approach in the space—time domain. Simulated correlation functions and power spectral densities of the backscattered field are presented. They extend the earlier findings concerning the backscattered field formation and also confirm and refine a number of conclusions drawn earlier from the behavior of the field’s statistical moments. The simulation technique is free from approximations commonly used in the statistical analysis of the propagation problems and can be used to study the statistical properties of the scattered field in a wide range of time intervals, as well as to find the limits of applicability of the approximate methods.     

Numerical modeling of one-dimensional problem of self-action of a wave in a stochastic nonlinear medium

Pacific Oceanology Institute, Far East Scientific Center, Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 31, No. 1, pp. 53–60, January, 1988.     

Statistical mechanics of the one-dimensional plane rotator model in an external field

A method of calculating the partition function and the pair correlation function of the one-dimensional plane rotator model in an external field is proposed. In this method these quantities are expressed in terms of the modified Bessel function. These explicit expressions are given to fourth and second order, respectively.     

Statistical properties of one-dimensional inversion for the wave field diffracted by a two-dimensional phase screen

We study theoretically the statistical properties of one-dimensional wave-field inversions. We show that the real and imaginary parts of the logarithm of the normalized coherence function are the invariants of the inverted field if the field is measured on the statistical symmetry axis. Using these invariants, one can easily reconstruct two-point statistical moments of the phase distribution on the screen. We derive equations for the reconstruction of phase-distribution moments in the general case. Numerical simulations show that these equations can be solved by an iterative technique. The convergence range of the iteration method with variation in the parameters is studied. A. M. Obukhov Institute for Physics of the Atmosphere of the Russian Academy of Sciences, Moscow, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 3, pp. 234–242, March., 2000.     

Statistical mechanics of a one-dimensional lattice gas

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be aK-system.     

Statistical mechanics of a quasiperiodic one-dimensional magnetic chain

In this paper we study the thermodynamic properties of the Ising model on a linear chain in which the sites are generated according to the Fibonacci sequence. We calculate the partition function, the specific heat and the q-dependent magnetic susceptibility.     

Identification of the structure parameters using short-time non-stationary stochastic excitation

In this paper, we propose an approach to the flexural stiffness or eigenvalue frequency identification of a linear structure using a non-stationary stochastic excitation process. The idea of the proposed approach lies within time domain input-output methods. The proposed method is based on transforming the dynamical problem into a static one by integrating the input and the output signals. The output signal is the structure reaction, i.e. structure displacements due to the short-time, irregular load of random type. The systems with single and multiple degrees of freedom, as well as continuous systems are considered.     

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.Supported by a grant from the National Research Council of Canada (to RCD) and by the Sherman Fairchild Foundation (to RZ).Also Sherman Fairchild Distinguished Scholar, 1974–75, at the California Institute of Technology, where the early part of this research was done.     

On the one-dimensional Coulomb problem

We analyse the one-dimensional Coulomb problem (1DCP) pointing out some mistaken beliefs on it. We show that no eigenstates of even or odd parity can represent states of the system. The 1DCP exhibits a sort of spontaneous breaking of parity. We also show that a superselection rule operates in the system. Such rule explains some of its peculiarities. We build the superpotential associated to the 1DCP.     

直线加速动态黑洞Dirac场的熵

采用薄层模型brickwall方法,计算出了直线加速动态黑洞视界面上Dirac场的熵以及Rindler视界面上Dirac场的熵密度,通过适当选择时间依赖的截断因子ε和薄层厚度δ,仍可得出熵与面积成正比的结论 关键词： 熵 加速动态黑洞 薄层模型 Dirac场 Dirac方程     

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.     

Statistical properties of chaos demonstrated in a class of one-dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.     

Anomalous fall to the center in a one-dimensional singular field

The fall to the center in a singular, one-dimensional, even field is investigated. It is shown that the fall of the ground level in a field with a weak singularity , 0 < 2) is caused by a -well arising from the nonphysical choice of a self-adjoint broadening of the Hamiltonian, unconnected with the increased symmetry of the Hamiltonian and easily eliminated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 75–79, October, 1989.     

First passage time for a class of one-dimensional stochastic systems

We consider the time evolution of a class of stochastic systems of finite size with polynomial nearest neighbor transition rates. We obtain analytical expressions for the first passage time (FPT) and its moments. We show that the mean FPT, averaged over a uniform initial distribution, shows a simple asymptotoc behavior with the system size and the parameters of the transition rates.