Contribution of statistical processes to elastic scattering

The sensitivity of the statistical theory to the choice of the distance-dependent transmittivity is investigated within the optical model. It is shown that for exponential density distributions, obtained in field theory, statistical scattering is negligibly small compared to diffraction. The alternating series appearing in the theory are summed rigorously. Some difficulties are noticed in comparing theory with experiment.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 6, pp. 17–21, June, 1971.We are grateful to V. M. Chudakov for discussing several problems of the statistical model.     

Standard deviations of chemical reaction rates and validity ranges of the statistical theory

Validity ranges of the statistical theory of chemical reactions are discussed. On the basis of the quantum mechanical scattering theory, the rate of the chemical reaction of the complex-formation mode is formulated. The interaction matrix elements between resonance states and scattering continua are regarded as stochastic variables. The expectation value of the reaction rate is shown to coincide with the prediction of the conventional statistical theory if the condition of overlapping resonance is fulfilled. The standard deviation is found to be inversely proportional to the number of resonance states involved in the width of the collision energy. High density of vibrational energy levels of the collision complex serves to suppress the standard deviation and to make the statistical theory accurate. The condition for the conventional statistical theory to hit the correct value with probabilities more than 99% is obtained as a relation between the number of vibrational modes and the depth of the potential energy well of the collision complex.     

Mathematical modeling of randomness and fuzziness phenomena in scientific studies. II. Applications

This paper considers some elements of the optimal fuzzy decision theory that are similar to the optimal statistical decision theory, in particular, the theory of optimal fuzzy identification and optimal fuzzy hypothesis testing, such as Neyman–Pearson statistical hypothesis testing and optimal fuzzy estimation along with a sequential fuzzy identification algorithm similar to the Wald sequential statistical criterion. Some elements of the fuzzy measuring and computing transducer theory and its applications in the problems of the analysis and interpretation of measurement experiment data are given.     

Theorems on estimating perturbative coefficients in quantum field theory and statistical physics

We present rigorous proofs for several theorems on using Padé approximants to estimate coefficients in perturbative quantum field theory and statistical physics. As a result, we find new trigonometric and other identities where the estimates based on this approach are exact. We discuss hypergeometric functions, as well as series from both perturbative quantum field theory and statistical physics.     

Response function theory for far-from-equilibrium statistical systems

A formalism to determine the response function of a sample in conditions far from thermal equilibrium is presented. It consists in a generalization of scattering theory coupled to the statistical theory of irreversible processes, the nonequilibrium statistical operator method, developed by Zubarev. The scattering cross section is expressed in terms of double-time correlation functions, which are related to appropriate nonequilibrium thermodynamic Green's functions. The latter are also used to treat generalized transport equations, and, as an illustration, the method is applied to the study of the time-resolved Raman spectroscopy of a photoexcited semiconductor plasma.     

微波器件微放电阈值计算的快速单粒子蒙特卡罗方法

为了计算微波器件的微放电阈值,提出了一种快速单粒子蒙特卡罗方法.该方法对二次电子出射能量、出射角度和相位等参数进行随机处理,结合四阶龙格库塔法和Furman模型模拟了电子运动和二次电子发射系数,并以多次连续碰撞的二次电子发射系数的算数平均值作为微放电效应发生的判据.以平板传输线横电磁模式为研究对象,分别采用快速单粒子蒙特卡罗方法、统计模型、传统蒙特卡罗方法以及粒子模拟方法计算其微放电阈值和敏感区域.计算结果表明,该方法不仅具有与统计模型和粒子模拟方法相当的计算精度,而且比统计模型方法的适应性更强,比传统蒙特卡罗方法的稳定性更好,比粒子模拟方法的计算效率高几十倍以上.     

Fatigue life of metal treated by magnetic field

This paper investigates theoretically the influence of magnetization on fatigue life by using non-equilibrium statistical theory of fatigue fracture for metals. The fatigue microcrack growth rate is obtained from the dynamic equation of microcrack growth, where the influence of magnetization is described by an additional term in the potential energy of microcrack. The statistical value of fatigue life of metal under magnetic field is derived, which is expressed in terms of magnetic field and macrophysical as well as microphysical quantities. The fatigue life of AISI 4140 steel in static magnetic field from this theory is basically consistent with the experimental data.     

Quantum statistical dynamics: Statistics origin,measurement, and irreversibility

It is shown that in the quantum theory of systems with a finite number of degrees of freedom which employs a set of algebraic states, a statistical element introduced by averaging the mean values of operators over the distribution of continuous quantities (a spectrum point of a canonical operator and time) is conserved for the limiting transition to the distribution. On that basis, quantum statistical dynamics, i.e., a theory in which dynamics (time evolution) includes a statistical element, is advanced. The theory is equivalent to orthodox quantum mechanics as regards the orthodox states, but is essentially different with respect to the coherence properties in a continuous spectrum. The measurement-process theory, including the statistical interpretation of quantum mechanics, and the irreversibility theory are constructed, and the law of increasing chaos, which is a strengthening of the law of entropy increase, is obtained. In our theory, mechanics and statistics are organically connected, whereby the fundamental nature of probabilities in quantum physics manifests itself.     

Critical behavior in earthquake energy dissipation

We explore bursty multiscale energy dissipation from earthquakes flanked by latitudes 29° S and 35.5° S, and longitudes 69.501° W and 73.944° W (in the Chilean central zone). Our work compares the predictions of a theory of nonequilibrium phase transitions with nonstandard statistical signatures of earthquake complex scaling behaviors. For temporal scales less than 84 hours, time development of earthquake radiated energy activity follows an algebraic arrangement consistent with estimates from the theory of nonequilibrium phase transitions. There are no characteristic scales for probability distributions of sizes and lifetimes of the activity bursts in the scaling region. The power-law exponents describing the probability distributions suggest that the main energy dissipation takes place due to largest bursts of activity, such as major earthquakes, as opposed to smaller activations which contribute less significantly though they have greater relative occurrence. The results obtained provide statistical evidence that earthquake energy dissipation mechanisms are essentially “scale-free”, displaying statistical and dynamical self-similarity. Our results provide some evidence that earthquake radiated energy and directed percolation belong to a similar universality class.     

RADIATIVE CAPTURE IN THE INELASTIC CHANNEL OF THE COMPOUND NUCLEUS

In this paper, the radiative capture in the inelastic channel of the compound nucleus is discussed.
Starting from the S-matrix theory, the calculation formular for this situation is derived. Then connected with the optical model and Hauser-Feshbach's statistical theory with the width fluctuation correction, a formular convenient for calculations is given.
Making use of the formular obtained above, the numerical calculations are made for the average cross-section of ⁵¹V and other nuclei, and the comparison with the statistical (n, γ) and (n, n′) cross sections is made.
The results show that if the incident energies of the neutron are large enough to open the inelastic channel, the cross sections of this process will have the same order of magnitude as the statistical cross sections. Hence it is an important mechanism in the (n, γ) reactions.     

Quantum mechanics as relativistic statistics. I: The two-particle case

It is shown that non-relativistic quantum mechanics can be treated as a kind of relativistic statistical theory, which describes the indeterministic motion of classical particles. The theory is relativistic in the sense that the relativistic notion of the state and two-time equations of motion are used. The principles and relations of quantum mechanics are obtained from the principles of statistics and those of classical mechanics.     

Normal and Abnormal Grain Growth as Transient Phenomena

For the case of uniform grain boundaries the basic equations of the statistical theory of grain growth (GG) of the present authors are shortly reviewed and compared to the classical Hillert model. On the basis of this present theory normal and abnormal 3-D GG is simulated and particularly the effect of the initial grain size distribution function (SDF) on the GG behaviour which may lead to normal or abnormal GG is demonstrated. Finally results of simulations of normal 2-D GG by the statistical method and by a direct model (curvature driven grain boundaries (GBs)) are presented which exhibit good agreement with one another. It is shown by this comparison that the possibility of finding by such direct methods a self similar SDF as long time asymptote can be excluded because, in contrast to the simulations based on the statistical theory, for the direct models the very large computational capacities required for the long simulation times are not available yet. The conclusion repeatedly claimed in literature that the true self similar SDF deviates from the Hillert distribution can thus be shown not to be justified.     

On the relationship between fluctuating irreversible thermodynamics and “extended” irreversible thermodynamics

The relationship between fluctuating irreversible thermodynamics and theories of irreversible processes which include the thermodynamic fluxes as independent variables is explored. It is shown that the usual fluctuating linear theory of irreversible thermodynamics is a contraction of the extended theory. This contraction contains non-Markovian effects dependent upon the relaxation times associated with the thermodynamic fluxes. In the limit that these relaxation times are small, the extended theory is shown to be equivalent to the usual fluctuating thermodynamic theory. A critique of the extended theories is given from the point of view of the mechanistic statistical theory of irreversible processes.     

Van der Waals force derived from a quantum statistical mechanical path integral method

The main purpose of this paper is to apply the quantum statistical mechanical model for polarizable fluids, used by Høye-Stell and others, to the calculation of the van der Waals force (dispersion force) between monatomics. The results are in agreement with those obtained by quantum perturbation theory. The quantum statistical method implies considerable formal simplifications.The introductory sections of the paper show, for illustrative purposes, how the dispersion forces in limiting cases also can be obtained from macroscopic quantum electromagnetic theory when two dilute dielectric media with plane-parallel surfaces are separated by a vacuum (Casimir effect).     

Thermodynamic-like transformations in information theory

This article explores the application of thermodynamic and statistical thermodynamic formalism to information theory problems. In particular, the applicability of the transformation theory of thermodynamics is investigated. After a brief tutorial discussion of thermodynamic and statistical thermodynamic methods and concepts, their information theory analogues are developed. Besides information theory entropy, the information theory counterparts of temperature, chemical potential, Helmholtz free energy, etc., are developed and related to conventional information theory concepts such as channel capacity, matching of source and channel, etc. Information theory theorems are proved via the statistical thermodynamic analogue method; and, finally, several problems are formulated and solved using thermodynamic-like transformations. This article is aimed chiefly at bridging the interface between the two disciplines, and is intended to be provocative. Therefore, no attempt has been made to have it be all inclusive.     

Interpretation of the hydrodynamical formalism of quantum mechanics

The hydrodynamical formalism for the quantum theory of a nonrelativistic particle is considered, together with a reformulation of it which makes use of the methods of kinetic theory and is based on the existence of the Wigner phase-space distribution. It is argued that this reformulation provides strong evidence in favor of the statistical interpretation of quantum mechanics, and it is suggested that this latter could be better understood as an almost classical statistical theory. Moreover, it is shown how, within this context, the Wigner and the Margenau-Hill functions are not equivalent, and that the latter is essentially unsatisfactory, as well as the associated symmetrization rule. Arguments in favor of a stochastic picture of the phenomena at the microscopic level are also presented.     

Statistical properties of the one dimensional Anderson model relevant for the nonlinear Schrödinger equation in a random potential

The statistical properties of overlap sums of groups of four eigenfunctions of the Anderson model for localization as well as combinations of four eigenenergies are computed. Some of the distributions are found to be scaling functions, as expected from the scaling theory for localization. These enable to compute the distributions in regimes that are otherwise beyond the computational resources. These distributions are of great importance for the exploration of the nonlinear Schrödinger equation (NLSE) in a random potential since in some explorations the terms we study are considered as noise and the present work describes its statistical properties.