Stochastic transport in random wave fields

An analysis is made of particle diffusion and the field of a passive impurity in random wave fields. A characteristic of this problem is that the statistical transport coefficients (diffusion coefficients) vanish in the approximations normally used (delta-correlated random field or diffusion) giving the Fokker-Planck equation. In this study perturbation theory is used in the first nonvanishing order of smallness which allows these transport coefficients to be calculated for waves of various types.     

Phase conjugation with random fields and with deterministic and random scatterers

The theory of distortion correction by phase conjugation, developed since the discovery of this phenomenon many years ago, applies to situations when the field that is conjugated is monochromatic and the medium with which it interacts is deterministic. In this Letter a generalization of the theory is presented that applies to phase conjugation of partially coherent waves interacting with either deterministic or random weakly scattering nonabsorbing media.     

Phase coherence of RVB wave function

We establish the phase coherence of a variational resonating valence bond (RVB) wave function in the case of nearest neighbor singlets on a square lattice in the fermion representation. The result applies both in the half filled band case and in the doped case.     

Construction of non-Gaussian self-similar random fields with hierarchical structure

In the present work we construct non-Gaussian self-similar random fields with hierarchical structure. The construction is based on non-Gaussian solutions of the main nonlinear equation of the hierarchical models theory. The existence of such solutions was proved originally by Sinai and the author and later by another method by Collet and Eckmann. Next we establish the uniqueness of a Gibbs state for the constructed self-similar field. Finally for a class of hierarchical models we prove the convergence of renormalization transformations of a random field at the critical point to the self-similar field.     

Wigner function for highly convergent three-dimensional wave fields

The angle-impact Wigner function for highly convergent three-dimensional scalar wave fields is derived directly by use of the three-dimensional generalized optical transfer function rather than from a six-dimensional Wigner function. The angle-impact Wigner function is a real four-dimensional function from which the intensity at any point in space is readily determined.     

Gaussian random fields

Two results on Gaussian random fields are presented. The first characterizes the unit Gaussian random field by a strong independence property and the second determines Gaussian random fields that are generated by stochastic processes.     

Phase diagram and tricritical behavior of an metamagnet in uniform and random fields

A two-sublattice Ising metamagnet in both uniform and random fields is studied within the mean-field approach based on Bogoliubov’s inequality for the Gibbs free energy. We show that the qualitative features of the phase diagrams are dependent on the parameters of the model and the uniform field values. The tricritical point and reentrant phenomenon can be observed on the phase diagram. The reentrance is due to the competition between uniform and random interactions.     

On random transformations of the wave function of a two-level system

The process of random diffusion variation of the wave function of a system with two states is analyzed. A method is developed for calculating the evolution operator and the damping increment of the probability distribution function of the state of the system on the basis of quaternion apparatus. It is proved analytically that the second moments formed from the wave function play the major role since all other statistical characteristics tend to equilibrium at a higher rate. For more general models of a random action, the result remains asymptotically the same, but the relative orders of increments may be different. Exceptional cases of incomplete statistical equilibrium are singled out. The possible role of the given model problem in the actual problem of state splitting in the transition from the microworld to macroworld is discussed. It is shown that, in spite of the views expressed in modern literature, the distribution of finite probabilities in the white noise model does not allow the well-known Schrödinger’s Cat paradox to be resolved.     

Experimental investigation of local properties and statistics of optical vortices in random wave fields

We present the first direct experimental evidence of the local properties of optical vortices in a random laser speckle field. We have observed the Berry anisotropy ellipse describing the anisotropic squeezing of phase lines close to vortex cores and quantitatively verified the Dennis angular momentum rule for its phase. Some statistics associated with vortices, such as density, anisotropy ellipse eccentricity, and its relation to zero crossings of real and imaginary parts of the random field, are also investigated by experiments.     

Wave function structure in two-body random matrix ensembles

We study the structure of eigenstates in two-body interaction random matrix ensembles and find significant deviations from random matrix theory expectations. The deviations are most prominent in the tails of the spectral density and indicate localization of the eigenstates in Fock space. Using ideas related to scar theory we derive an analytical formula that relates fluctuations in wave function intensities to fluctuations of the two-body interaction matrix elements. Numerical results for many-body fermion systems agree well with the theoretical predictions.     

Phase transitions in diluted magnets: Critical behavior,percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn⁺⁺ or Mg⁺⁺ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe⁺⁺ and Co⁺⁺ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb₂Mn_0.5Ni_0.5F₄ and Fe_cZn_1–cF₂ which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb₂Mn_cMg_l–cF₄, Rb₂Co_cMg_l–cF₄, KMn_cZ_l-cF₃, and Mn_cZn_l–cF₂ which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.     

Smooth decomposition of random fields

The Smooth Decomposition (SD) method was introduced to analyze discrete-time signals and generalized to continuous-time vector-valued random processes. The SD is obtained solving a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process which defines the decomposition basis. This paper presents a new extension of the SD to continuous-time and continuous-space vector-valued random processes, classically named random fields. This generalization is a major step since one now deals with operators in infinite-dimensional spaces and not matrices. It is shown that in this new context the main properties of the SD are preserved. Applied to the responses of randomly excited continuous mechanical systems, the SD can be considered as an output-only analysis tool. Moreover, two natural orderings are defined to classify the decomposition terms which permit to interpret the SD in terms of modal analysis or in terms of Karhunen–Loève analysis.     

Optical polarimetry of random fields

The state of polarization of an optical field provides detailed information concerning both the radiation emission processes and the intricate interaction between light and matter. We report here a novel approach for characterizing the polarization properties of electromagnetic fields for which the electric field vector at a point may fluctuate in three dimensions. Using probes which couple all three components of the field, we were able to extract the polarized and unpolarized components of such fields. Our results constitute the proof of concept for what could be called three-dimensional optical polarimetry.     

Martin-Dynkin boundaries of random fields

Analogous of exit spaces of Dynkin [4] for Markov processes are constructed for random fields introduced by Dobrushin [2].     

Transport in random correlated fields

A review is given of recent developments in the diffusion properties of particles in the presence of local random fields as well as the conductivity of the analog random resistor network. The effect of long-range ferro- and antiferro-type correlations between the local fields on the diffusion and conductivity properties is considered. A physical realization for such spatial correlations is diffusion on linear polymers in the presence of external uniform bias field. For this case a universal diffusion law was found independent of the fractal dimension of the polymer chain or the Euclidean dimension in which the polymer is embedded. Recent results for diffusion in two dimensions in the presence of a special case of correlated local fields are also reviewed.     

Acoustic waves in random fields

The effect of space- and time-dependent random mass density, velocity, and pressure fields on frequencies and amplitudes of acoustic waves is considered by means of the analytical perturbative method. The analytical results, which are valid for weak fluctuations and long wavelength sound waves, reveal frequency and amplitude alteration, the effect of which depends on the type of random field. In particular, the effect of a random mass density field is to increase wave frequencies. Space-dependent random velocity and pressure fields reduce wave frequencies. While space-dependent random fields attenuate wave amplitudes, their time-dependent counterparts lead to wave amplification. In another example, sound waves that are trapped in the vertical direction but are free to propagate horizontally are affected by a space-dependent random mass density field. This effect depends on the direction along which the field is varying. A random field, which varies along the horizontal direction, does not couple vertically standing modes but increases their frequencies and attenuates amplitudes. These modes are coupled by a random field which depends on the vertical coordinate, but the dispersion relation remains the same as in the case of the deterministic medium.     

Phase diagram of random heteropolymers

We propose a new analytic approach to study the phase diagram of random heteropolymers, based on the cavity method. For copolymers we analyze the nature and phenomenology of the glass transition as a function of sequence correlations. Depending on these correlations, we find that two different scenarios for the glass transition can occur. We show that, beside the much studied possibility of an abrupt freezing transition at low temperature, the system can exhibit, upon cooling, a first transition to a soft glass phase with fully broken replica symmetry and a continuously growing degree of freezing as the temperature is lowered.