One-dimensional numerical simulations of random sound impulses

Abstract

We perform one-dimensional numerical simulations of small-amplitude acoustic pulses in space- and time-dependent random mass density and time-dependent velocity fields. Numerical results reveal that: (a) random fields affect the speeds, amplitudes and, consequently, shapes of sound pulses; (b) for weak random fields and short propagation times the numerical data converge with the analytical results of the mean field theory which says that a space-dependent (time-dependent) random field leads to wave attenuation (amplification) and all random fields speed up sound pulses; (c) for sufficiently strong random fields and long propagation times numerical simulations reveal pulse splitting into smaller components, parts of which propagate much slower than a wave pulse in a non-random medium. These slow waves build an initial stage of a wave localization phenomenon. However, this effect can be very weak in a real three-dimensional medium.     

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.     

双模随机晶场对纳米管上Blume-Capel模型磁化强度和相变的影响

近年来, 磁性纳米管的物理性质和相关应用得到了人们的广泛关注. 利用有效场理论研究了纳米管上双模随机晶场中Blume-Capel模型的磁化强度和相变性质, 得到了系统的磁化强度与温度和随机晶场的关系及其相图. 结果表明: 系统在稀释晶场、交错晶场和同向晶场中会表现出不同的磁学性质和相变行为; 稀释晶场和交错晶场会抑制系统的磁化强度, 导致其基态饱和值小于1, 而同向晶场则不会; 随着随机晶场参量的变化, 系统存在多个相变温度, 并呈现出三临界现象和重入现象.     

On the probabilistic treatment of fields

Some basic problems of the probabilistic treatment of fields are considered, proceeding from the fundamentals of the complete probability theory. Two essentially equivalent definitions of random fields related to continuous objects are suggested. It is explained why the conventional classical probabilistic treatment generally is inapplicable to fields in principle. Two types of finite-dimensional random variables created by random fields are compared. Some general regularities related to Lagrangian and Hamiltonian partial equations, obtainable proceeding from the corresponding sets of ordinary differential equations, are revealed by using the functional derivative defined anew. It is shown that Hamiltonian random fields give rise to two types of Hamiltonian random variables, variables of the second type being those considered in the author's previous paper and immediately suited to the quantum approach. The results obtained are illustrated by some general examples. Critical remarks concerning second quantization are made, demonstrating the artificiality of this method. It is emphasized that the given probabilistic consideration of fields cannot be directly applied to, for instance, the electromagnetic field, which needs a special treatment.     

The three-dimensional three-state Potts ferromagnet exposed to random fields: evidence for a second order transition

Using large scale Monte Carlo simulations, the ordering of the three-dimensional three state Potts ferromagnet exposed to random fields is investigated. Studies of the order parameter probability distribution and of various of its moments suggest that the order of the transition depends on the strength of the random field: i.e., the first order transition of the pure ferromagnetic model persists for weak random fields, but turns into a second order transition for a range of random fields of medium strength. For large random fields the transition seems to be first order again. In this range large domains of strongly aligned Potts spins occur already in the disordered phase and the associated slow relaxation hampers significantly the Monte Carlo study of thermodynamic equilibrium phenomena. These results are discussed in the light of current theoretical concepts. Possible applications to experiments on diluted anisotropic molecular crystals and orientational glasses are briefly mentioned.     

The three-dimensional three-state Potts ferromagnet exposed to random fields: evidence for a second order transition

Using large scale Monte Carlo simulations, the ordering of the three-dimensional three state Potts ferromagnet exposed to random fields is investigated. Studies of the order parameter probability distribution and of various of its moments suggest that the order of the transition depends on the strength of the random field: i.e., the first order transition of the pure ferromagnetic model persists for weak random fields, but turns into a second order transition for a range of random fields of medium strength. For large random fields the transition seems to be first order again. In this range large domains of strongly aligned Potts spins occur already in the disordered phase and the associated slow relaxation hampers significantly the Monte Carlo study of thermodynamic equilibrium phenomena. These results are discussed in the light of current theoretical concepts. Possible applications to experiments on diluted anisotropic molecular crystals and orientational glasses are briefly mentioned.     

Tricritical Points and Reentry in the Quantum Hopfield Neural-Network Model

We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.     

Critical behavior of an Ising metamagnet in uniform and random fields

We use an effective field method to analyze the thermodynamic behavior of an Ising metamagnet in uniform and random magnetic fields. Depending on the strength of the random fields, we show that the phase diagram displays a tricritical point and reentrant second-order transition lines. This reentrance is due to the competition between uniform and random interactions.     

Irreversible effects of random modulation on coherence properties of partially polarized light

The effect of the multiplication of two vector electric fields by random Jones matrices on their coherence properties is investigated by analyzing their interference fringes. It is shown that these random modulations cannot increase the maximal visibility obtained when the polarization states of the two analyzed electric fields are optimized. This new result generalizes a standard property of perfectly polarized light to partially polarized light and is representative of the irreversible effects of random modulations.     

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.     

On Pfaffian Random Point Fields

We study Pfaffian random point fields by using the Moore-Dyson quaternion determinants. First, we give sufficient conditions that ensure that a self-dual quaternion kernel defines a valid random point field, and then we prove a CLT for Pfaffian point fields. The proofs are based on a new quaternion extension of the Cauchy-Binet determinantal identity. In addition, we derive the Fredholm determinantal formulas for the Pfaffian point fields which use the quaternion determinant.     

Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions)

A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio--Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case). Received: 25 April 1996 / Accepted: 29 July 1996     

Potentials for almost Markovian random fields

A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.     

Universality in three dimensional random-field ground states

We investigate the critical behavior of three-dimensional random-field Ising systems with both Gauss and bimodal distribution of random fields and additional the three-dimensional diluted Ising antiferromagnet in an external field. These models are expected to be in the same universality class. We use exact ground-state calculations with an integer optimization algorithm and by a finite-size scaling analysis we calculate the critical exponents , , and . While the random-field model with Gauss distribution of random fields and the diluted antiferromagnet appear to be in same universality class, the critical exponents of the random-field model with bimodal distribution of random fields seem to be significantly different. Received: 9 July 1998 / Received in final form: 15 July 1998 / Accepted: 20 July 1998     

Non-ergodic instability and the quadrupolar glass state in K(CN) x Br1−x and (NaCN) x (KCN)1−x

Random strain fields, due to substitutional disorder, couple to translational and orientational dynamic modes in plastic crystals. One distinguishes the cases of weak and strong random fields. In weak fields, the crystal comes close to a structural phase transition. Then elastic restoring forces are weak while internal friction dominates. The resulting non ergodic instability marks the onset of a structural glass state. The evolution of the glass state is discussed within a selfconsistent theory. The results of an extensive numerical study are compared with experiment. In presence of strong random fields no instability occurs, and orientational freezing is a continuous process.     

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.     

Pairs of opposite decaying evanescent waves in random media and tunneling radiative transfer

A tunneling mechanism of radiative transfer through a dielectric random medium is revealed applying technique of Dyson and Bethe-Salpeter equations for electromagnetic wave multiple scattering by medium inhomogeneities (scatterers) with near fields effects in scattered fields. The mechanism consists in existing inside of a random inhomogeneity a pair of virtually opposite decaying evanescent waves whose interference results in energy flux.     

Higher-Order Spectral Densities of Fractional Random Fields

This paper presents the second- and higher-order spectral densities of stationary (in space) random fields arising as approximations of rescaled solutions of the heat and fractional heat equations with singular initial conditions. The development is based on the diagram formalism and the Riesz composition formula. Our results are the first step to full parametrization of higher-order spectra of some classes of fractional random fields.     

Induced random fields in the LiHoxY1-xF4 quantum Ising magnet in a transverse magnetic field

The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.