Random Initial Phase Research Articles

Statistical limit in a completely integrable system with deterministic initial conditions

The asymptotic behavior of the solutions of the KdV equation in the classical limit with an oscil- lating nonperiodic initial function u 0 ( x ) prescribed on the entire x axis is investigated. For such an initial con- dition, nonlinear oscillations, which become stochastic in the asymptotic limit t × , develop in the system. The complete system of conservation laws is formulated in the integral form, and it is demonstrated that this system is equivalent to the spectral density of the discrete levels of the initial problem. The scattering problem is studied for the Schrodinger equation with the initial potential - u 0 ( x ), and it is shown that the scattering phase is a uniformly distributed random quantity. A modified method is developed for solving the inverse scattering problem by constructing the maximizer for an N -soliton solution with random initial phases. A one-to-one rela- tion is established between the spectrum of the discrete levels of the initial state of the system and the spectrum established in phase space. It is shown that when the system passes into the stochastic state, all KdV integral conservation laws are satisfied. The first three laws are satisfied exactly, while the remaining laws are satisfied in the WKB approximation, i.e., to within the square of a small dispersion parameter. The concept of a qua- sisoliton, playing in the stochastic state of the system the role of a standard soliton in the dynamical limit, is introduced. A method is developed for determining the probability density f ( u ), which is calculated for a spe- cific initial problem. Physically, the problem studied describes a developed one-dimensional turbulent state in dispersion hydrodynamics. © 2000 MAIK "Nauka/Interperiodica".

X-ray phase determination by the principle of minimum charge.

When the electron density in a crystal or a quasicrystal is reconstructed from its Fourier modes, the global minimum value of the density is sensitively dependent on the relative phases of the modes. The set of phases that maximizes the value of the global minimum corresponds, by positivity of the density, to the density having the minimum total charge that is consistent with the measured Fourier amplitudes. Phases that minimize the total electronic charge (i.e. the average electron density) have the additional property that the lowest minima of the electron density become exactly degenerate and proliferate within the unit cell. The large number of degenerate minima have the effect that density maxima are forced to occupy ever smaller regions of the unit cell. Thus, by minimization of the electronic charge, the atomicity of the electron density is enhanced as well. Charge minimization applied to simulated crystalline and quasicrystalline diffraction data successfully reproduces the correct phases starting from random initial phases.

Equivalence of Widrow's and Gray's approaches to uniform quantizers

In the paper we demonstrate the equivalence between Widrow's and Gray's approaches to the quantization error created in a uniform quantizer by the pure sine wave with a random initial phase and a deterministic bias. The exact finite-sum formulas are directly derived from the infinite-sum series resulting from the classical Quantization Theory. The analytical results lead to the simplification of data processing recommended by the IEEE Standards.

The crystal structure of 3-(9-anthracenyl)-1,5-diphenyl-verdazyl, a stable free radical

The crystal structure of the stable free radical 3-(9-anthracenyl)-1,5-diphenylverdazyl was determined. There are four independent molecules in the unit cell (space group P ̄ 1 ). The structure was solved by direct methods. A large number of sets of initial random phases were generated in space group P1 ( Z = 8, 256 non-hydrogen atoms) and the best solution gave the location of 250 atoms. The four symmetry-independent molecules have a variety of different conformations indicating considerable conformational freedom.

Periodically driven linear system with multiplicative colored noise

A periodically driven linear system subject to multiplicative correlated noise is considered. It has been argued recently by several authors that such a simple system exhibits stochastic resonance. By introducing a general type of composite stochastic process, bridging two previously considered limiting cases of dichotomous and Gaussian noise, it is proved that, indeed, the amplitude of the average of the driven linear process at long times shows a pronounced maximum both as a function of the noise strength and as a function of the autocorrelation time. However, this kind of stochastic resonant behavior can be experimentally observable only in a special case where the initial phase of the external forcing is somehow fixed. Additional averaging over the uniform distribution of the initial random phase, inherent in most physical systems, leads to that the periodic output vanishes identically at long times. Moreover, the system response is typically defined in terms of the power spectrum rather than the amplitude of the average. The output signal given by the spectral density corresponding to the frequency of the external forcing is calculated via the long-time phase-averaged correlation function. It appears that the output signal simply diverges upon approaching the second moment instability point with increasing noise strength. No stochastic resonance is observed for any parameter settings. Interestingly, the resonancelike behavior of the system response as a function of the autocorrelation time is retained.

Speckle-noise reduction on kinoform reconstruction using a phase-only spatial light modulator

Random-phase distributions that are statistically independent individually are used for computing kinoforms. These uncorrelated kinoforms are recorded and read out sequentially by a phase-only liquid-crystal spatial light modulator, and reconstructed images with well-developed speckles are added. The fidelity of the resultant image to an original is improved as the number of additions increases. The dependence of the speckle contrast on the initial random phase and the influence of the liquid-crystal spatial light modulator's display performance on the image quality are discussed.

Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions

Interactions among scales of motion within fully developed turbulence are given in the Fourier-space view by the sum of collections of triadic interactions among subsets of Fourier modes in three-dimensional Fourier space. To analyze general characteristics of interscale couplings in high Reynolds number turbulence, ‘‘chains’’ of roughly 20 000 interconnected triads are embedded within a model energy spectrum and intermodal energy exchange is examined within groups of triadic interactions based on triad geometry. Calculations of the inviscid momentum equation are carried out over a large number of realizations with random initial phase, and ensemble-averaged intermodal energy transfers are analyzed to extract general dynamical features associated with differences in relative scale and triad obtuseness. The studies demonstrate different dynamical effects within ‘‘local,’’ ‘‘nonlocal,’’ and ‘‘distant’’ triadic interactions in different spectral regions. The forward cascade within the inertial range, is found to be strongly dominated by local-to-nonlocal triadic interactions with scale separations as high as 10–15. A consequence is that for α decades of inertial subrange, Rλ=C×10(2/3)α, with C≊310–470, implying that Rλ must be roughly 1450–2200 for a decade of inertial range, and must exceed 300–400 for any natural inertial range to exist. ‘‘Backscatter’’ is primarily within triadic interactions in which energy moves both to larger and to smaller scales and highly local triadic interactions appear to be weakly inverse cascading, on average. Distant interactions are characterized by the asymptotic form of the triad equations whereby structural information is transferred from the largest to smallest scales, but not energy. It is found that direct energy transfer between the largest and smallest scales ceases when the ratio between scales is greater than 15–20, suggesting that a dominantly ‘‘distant’’ triadic characteristic requires scale separations greater than roughly 15. Energy transfer within the high wave-number modes in distant interactions, however, is nonenergy cascading, on average; energy tends to move laterally within thin spectral shells, but not from one shell to the next. Furthermore, as energy moves from large to small scales within progressively more local triadic interactions, the tendency is to distribute energy more and more uniformly at smaller and smaller scales, providing an increasingly strong isotropizing influence at successively smaller scales. By contrast, distant interactions are highly directional in three-dimensional Fourier space with characteristics that tend to transfer, over time, structural information (including anisotropy) directly from large to small scales. The various isotropizing and anisotropizing influences within local-to-nonlocal and distant scale interactions compete at the small scales. Which influences dominate over what periods of time depend on a number of interacting dynamical factors, including location within the spectrum, energy and temporal coherence within large-scale structure, the presence of direct forcing at different scales (as with mean shear), the stationary or nonstationary character of the turbulence, and time. It is argued that searches for deviations from local isotropy should be carried out (1) within systematically increasing scale separations, (2) in the limit of the smallest dynamical scales, and (3) as a function of systematic variations in large-scale anisotropy.

Cooperative radiation of acoustic waves by gas bubbles in a liquid.

The process of cooperative radiation of acoustic waves by radial bubble oscillations is investigated theoretically. In addition to instantaneous bubble interaction [E. A. Zabolotskaya, Sov. Phys. Acoust. 30, 365 (1984)], the interaction due to acoustic radiation has also been taken into account. Numerical results for the acoustic intensity produced by ten bubbles that pulsate in water with random initial phases are presented. Mutual bubble interaction causes synchronization of the bubble pulsation phases, which gives rise to collective radiation. The phenomenon is not as strong as in optics. In order to ensure superradiance by ten bubbles, the radial oscillations of the bubbles must have amplitudes of the order of their initial radii. [This work was supported by the David and Lucile Packard Foundation, and by the Office of Naval Research.] a)On leave from the Department of Physics, Moscow State University, 119899 Moscow, Russia. b)On leave from the General Physics Institute, Russian Academy of Sciences, 38 Vavilov Street, 117942 Moscow, Russia.

Optimal control of uncertain quantum systems.

The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions ${\mathit{H}}_{0}$(\ensuremath{\alpha}) indexed by the random variable \ensuremath{\alpha} taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions.The constrained optimization problem is reduced to an unconstrained optimization problem by the introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.

Statistical synthesis of algorithms for the reception of multicomponent discrete signals

Optimal algorithms have been derived for the joint nonlinear filtering of multicomponent discrete-continuous Markovian processes. Optimal reception algorithms have been synthesized for double phase-shift and frequency phase-shift telegraphy in the presence of a random initial phase of the received signal. The singularities of the resulting block diagrams have been discussed.

Influence of random phase modulation on the formation and propagation of a soliton

A study was made of the influence of random initial phase modulation on the characteristics of optical solitons formed in a fiber. A calculation is reported of a reduction in the amplitude of solitons and their velocity distribution for an arbitrary phase correlation function. Statistical characteristics are obtained for the optical noise which appears because of initial phase modulation.

Spontaneous generation of coherent optical beats.

We represent two groups of excited atoms which radiate at slightly different frequencies by means of a pair of inverted harmonic oscillators coupled to the radiation field. The radiation emitted spontaneously by these oscillators is amplified exponentially and the field generated by each exerts a strong dynamical influence on the other. For the regime in which the amplification rate is large compared to the frequency difference of the uncoupled oscillators, the indirect coupling via the field tends to lock the two systems in phase so that they radiate in unison. When the amplification rate is small compared to the frequency difference, however, each oscillator undergoes, in addition to its own spontaneous oscillation, a partially coherent oscillation forced at the frequency of the other oscillator. These mutually induced oscillations give rise to intensity beats in the radiated field. Such beats have predetermined phases which are independent of the random initial phases of the oscillator amplitudes. The beats can be present and detectable in statistical terms even before a single photon has been emitted on the average.

COMPUTATIONAL TRIALS FOR PHASE RETRIEVAL IN ELECTRON MICROSCOPY FROM IMAGE AND DIFFRACTION PATTERNS

The problem of phase retrieval from image and diffraction patterns is of importance in electron microscopy. In this paper the true phase can be extracted uniquely and accurately by Gerchberg-Saxton iterative algorithm for the wave function, its amplitude being unity or a linear function, while its phase being a parabolic, sinuous or exponential function.A computer program for retrieving the phase has been worked out. As an image wave function is given, its Fourier transform may be calculated by the discrete fast Fourier transform algorithm. So we can recover the phase with G-S algorithm, using a random initial phase between -π and +π. The numerical examples are given in detail. The phase retrieval solution is stable under fluctuation of input initial phase up to 1 rad.

The chaotic universe, friedmannian on the average: I

The chaotic Universe is considered, which is homogeneous and isotropic on the sealesL ≫L whereL is the scale of averaging. From the Einstein equations, the equations for the correlation functions have been obtained which describe the statistically chaotic model with the fluctuations of arbitrary amplitudeh. In the approximation of δ-correlated fluctuations andh≪1, the infinite set of coupled equations for the correlators closes and gives a self-consistent description of the inverse influence of the vortical and potential perturbations and gravitational waves with random initial phases on the Friedmann expansion of the Universe. For the state equationp=ne, the cosmological solutions have been foud.. They depend on the position of the maximum in the spectrum of the metric perturbations (λmax >ct or λmax 0.26 it goes below it. The effect of the finites ast → 0 long-wave fluctuations leads to an averaged quasi-isotropic solution. The contribution of the quantum and short-wave classical fluctuation to the expansion law is also considered. Their effect is equivalent to the contribution from an additional ultrarelativistic gas at corresponding energy density and pressure. In Paper II, the constraints on the degree of the chaos (the spectrum characteristics) of the Universe by the time of nucleosynthesis are obtained, which involve the observed helium abundance.

Toward explaining why events occur

The possibility is raised of adding a randomly fluctuating interaction term to the Schrodinger equation, so that the new equation “reduces” the state vector. The exact event that occurs is predicted by the equation and depends upon the precise time dependence of the interaction term. The uncertainty in nature is attributed to the random behavior of this term. A class of such terms is found. This class includes terms whose nonlinear dependence on the wave function is identical to that of terms introduced in a previous paper for a similar purpose. In the previous paper, the exact event predicted depends upon the initial phase factors in the superposition making up the state vector: the uncertainty in nature is attributed to random initial phase factors. Another derivation of the results in the previous paper is given in an appendix: the calculations in that paper and in this appendix are of second order in perturbation theory. On the other hand, the calculations in the present paper are exact. A possible answer is given to the question, raised in the previous paper, of the nature of the “observable” states to which the state vector reduces.

On the derivation of the Pauli and van Hove master equations

We discuss the derivation of the Pauli master equation, based on a repeated random phase assumption, and of van Hove's result, based on an initial random phase assumption. For the former we indicate a derivation which is closer to the general approach of stochastic theory than Pauli's original method. For the van Hove result, we show that the diagonal and nondiagonal parts of the evolution operator of the Schrödinger or von Neumann equation are readily obtained by Zwanzig's projection operator method.

Theory of Radiative and Nonradiative Electronic Processes in Ionic Crystals with Point Imperfections

After an outline of the quantum theory of decaying states on the basis of the resolvent method. a Hamiltonian is considered describing an ionic micro-crystal containing one electronic impurity center. The complete Hamiltonian of the coupled system micro-crystal plus radiation field can be decomposed into a diagonalized part and a “perturbation”, so that the perturbation leads to terms associated with nonradiative electronic transitions, anharmonic lattice transitions, radiative electronic transitions, and radiative lattice transitions. The state of the system micro-crystal plus radiation field is supposed to be given by a density operator in the representation where the unperturbed Hamiltonian is diagonal. By means of the assumption of initial random phases the quantum theory of decaying states yields an approximatively valid differential equation for the diagonal elements of the density operator describing the time-dependent radiative and nonradiative electronic processes in the micro-crystal. Under certain additional conditions the differential equation has the form of the Pauli-Master-Equation.

Informationstheoretische Beschreibung physikalischer Vorgänge. IV. Exakte Mastergleichung für gemittelte Verteilungsfunktionen

AbstractAn exact markovian master equation for the smoothed classical distribution function f̄ = Mf is derived using the existence of the operator [1 + M(−1 + exp (‐it L))]−1. It is shown that according to the information theory f̄0 = 0 (“initial random phase approximation”) should be taken. Then in the first order of a perturbation approach the master equation given by POMPE and VOSS can be derived in the long time approximation.

Weak-Coupling Rate Equations and Initial Conditions

Irreversible rate equations for the diagonal and offdiagonal elements of the density matrix are derived in the weak-coupling, long-time approximation for two types of crystal perturbations --- electron scattering from random impurities, and spin-spin interactions. No initial random phase assumption on the density matrix is necessary in the derivation using the random perturbation. For the spinspin interactions, a partial'' initial random phase assumption is introduced and used in a natural way. It is shown, alternatively, that if one takes into account the very extensive cancellation of terms for the vast majority of unperturbed energy states, no initial restrictions need be placed on the majority of the density matrix elements. In all three cases the equations obtained are the same, being Pauli equations for the diagonal density matrix elements, but simpler relaxation-oscillation equations for the offdiagonal elements. (auth)