也谈概率密度和概率流密度算符

针对<大学物理>2008年第8期上关于粒子概率密度算符和概率流密度算符的讨论一文进行了补充,对一般空间(如动量空间)中的概率和概率流进行了较系统的阐述,并纠正了该文中的有关错误论断.     

On the kinetic theory of energy losses in a randomly heterogeneous medium

The equation describing the distribution of energy losses of a particle propagating in a fractal medium with quenched and dynamic heterogeneities has been derived. It has been shown that in the case of the medium with fractal dimension 2 < D < 3, the losses Δ are characterized by the sublinear anomalous dependence Δ ∼ x ^α with a power-law dependence on the distance x from the surface and exponent α = D − 2.     

Permittivity of a randomly inhomogeneous medium

A theory of permittivity of suspension-type systems is developed that allows one to calculate such optical parameters of inhomogeneous systems as the length of scattering and the transport length. It is shown that, in the Born approximation, which takes into account two-particle correlations in the arrangement of scattering particles, the theoretical and experimental data are in agreement only to within tens of percent. The contribution of three-particle correlations to the permittivity of a system of solid spheres is determined. It is shown that, in describing the optical properties of suspensions with a large difference between the refractive indices of the medium and the particles, it does not suffice to replace the Rayleigh-Gans form factor by the Mie form factor, even under a restriction to two-particle correlations.     

On the low-frequency sound scattering in a moving microinhomogeneous medium

The Rayleigh law that governs low-frequency sound attenuation due to the scattering by inhomogeneities in a microinhomogeneous medium is generalized to the case of particles moving in a flow or falling under gravity. Corrections to the scattering’s cross section that adjust the Rayleigh law to the case of a potential flow around inhomogeneities are calculated. It is shown that, when microinhomogeneities are moving in a viscous medium, the characteristics of discrete scatterers may considerably deviate from the Rayleigh law. Based on the data on the velocity and size distribution of falling drops of water in air, refinements are proposed for the laws of low-frequency sound scattering by rain.     

运动介质的电磁性质方程

应用电磁场张量、电磁感应张量和它们的对偶张量讨论运动介质的电磁性质方程,指出D=εE和B=μH在运动介质中不成立,给出运动介质电磁性质方程的协变形式、向量形式.     

Average number and probability distribution of caustics in randomly inhomogeneous medium and behind a random phase screen

Formulas are derived for the average cross-sectional caustic density and the probability density of the distances to caustics behind a random phase screen. The constancy of the average number of caustics at great distances behind the screen as the probability density approaches zero is explained.Nizhny Novgorod Architecture and Construction Academy. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 4, pp. 471–478, April, 1994.     

On the statistics of ray-tube divergence and wave multirayness in a randomly inhomogeneous medium

A solution of the problem of ray diffusion in a randomly inhomogeneous medium is used to find the statistical properties of multiray fields. The longitudinal-coordinate dependence for the average ray number and the Lagrangian mean-square ray-tube divergence, which is related to the Eulerian mean of a quantity which is inversely proportional to intensity, is obtained.Nizhnii Novgorod Academy of Architecture and Civil Engineering. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 3, pp. 323–329, March, 1994.     

On the scintillations of radiation from an extended source in a randomly inhomogeneous medium

Using the method of refractive scattering of radio waves (RSRW), we solve the problem of space correlation of scintillations of radiation from an extended source in a randomly inhomogeneous medium. General expressions are obtained for the index and radius of the space correlation of fluctuations of intensity of radiation from a source with finite angular dimensions as it propagates though a multilayered medium with refractive index fluctuations. The RSRW method is similar to the diffraction calculation of space correlation of scintillations of an extended source in a thick layer with inhomogeneities. We note that under certain conditions an increase in the number of inhomogeneous layers on the radio wave propagation path leads to a pronounced decrease in the scintillation index and an increase in the space correlation of fluctuations of intensity of received radiation from an extended source. The results obtained have a simple geometric-optical interpretation. We indicate a specific feature in determining the angular dimensions of extended radio sources using the known method of scintillations. Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 5, pp. 586–593, May, 1997.     

On the stability of acoustic waves propagating in a nonstationary moving medium

We analyze the stability of acoustic waves in a medium moving with a time-variable velocity. An instability criterion for these waves is obtained and analyzed for the weak modulation of medium velocity. The harmonics that emerge during the propagation under parametric resonance are shown to have an additional frequency shift.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 9, pp. 1087–1090, September, 1996.This work was supported by the Russian Foundation for Fundamental Research under Projects 95-02-05001 (G. I. Grigor'ev and V. V. Tamoikin) and 96-05-64277 (O. N. Savina).     

Statistical properties of an effective index of refraction in a flat-layered randomly inhomogeneous medium

The notion of the effective index of refraction in a one-dimensional randomly inhomogeneous medium is introduced based on the generalized eikonal equation. This physical quantity furnishes an unambiguous definition of the wave field propagating under prescribed boundary conditions in the medium considered. Within the framework of a Markovian diffusion approximation, the one-point probability density of the averaged effective index of refraction is found as well as a number of its statistical properties, namely, the mean value and its fluctuation variance. The applicability range of the results obtained is discussed for random inhomogeneities with a finite correlation radius.Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 9, pp. 914–927, September, 1993.     

Beam propagation in a randomly inhomogeneous medium

An integrodifferential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the “3/2 law.”     

On the probability distribution for rays in a medium having random inhomogeneities

Radiophysics and Quantum Electronics -     

On the probability density of the envelope of a sum of random vectors

A most efficient way to calculate the probability density for the projection of the sum of n sinusoidal waves with differing amplitudes and phases that are uniform in (0, 2π) is to expand it in a Fourier series. The corresponding series for the envelope of such a series is known to be a Fourier-Bessel series. It is difficult, however, to calculate moments of the envelope from this series. It is shown that by relating the probability density of the envelope to that of the projection one can find moments of the envelope in an easily computable form. It is also possible to calculate an approximate form for the density function for the envelope amplitude near its maximum value. A generalized expansion is developed, valid when the probability density function depends on both angle and radial distance.     

Relation between the probability density and other properties of a stationary random process

We consider the Pope-Ching differential equation [Phys. Fluids A 5, 1529 (1993)] connecting the probability density p(x)(x) of a stationary, homogeneous stochastic process x(t) and the conditional moments of its squared velocity and acceleration. We show that the solution of the Pope-Ching equation can be expressed as n(x), where n(x) is the mean number of crossings of the x level per unit time and is the mean inverse velocity of crossing. This result shows that the probability density at x is fully determined by a one-point measurement of crossing velocities, and does not imply knowledge of the x(t) behavior outside of the infinitesimally narrow window near x.     

Anisotropic distribution of quantum‐vacuum momentum density in a moving electromagnetic medium

An isotropic electromagnetic medium becomes gyrotropically anisotropic when it moves, and an anisotropic electromagnetic environment can then be created in this motion‐induced anisotropic medium. One of the most remarkable features is that the quantum vacuum in the anisotropic electromagnetic environment exhibits a nonzero electromagnetic momentum density, since the universal symmetry of the vacuum fluctuation field is broken, and the anisotropic quantum vacuum mode structure is produced because of the symmetry breaking. This would give rise to a noncompensation effect among the four vacuum eigenmodes (i.e., the forward and backward propagating modes as well as their respective mutually perpendicular polarized components), and leads to an anisotropic correction to the vacuum momentum in the moving medium. The physical significance and the potential applications of the anisotropic quantum vacuum are discussed. This quantum‐vacuum effect may be used to develop sensitive sensor techniques and to design new quantum optical and photonic devices.