The basic equations of the dynamics of the continuous distribution of dislocations I. General theory

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell's equations are derived in a series of papers I, II, III. The analogy of the elastic and electromagnetic fields is analyzed. In part I the basic equations of the continuous distribution of dislocations are derived for the dynamic case (especially for small deformations and for the Poisson ratio equal to zero in a continuum, infinite with respect to space and time, from the variational problem by means of the formal apparatus of the special theory of relativity.

---

I.

, . . , , .

     

The basic equations of the dynamics of the continuous distribution of dislocations II. Interpretation of general theory

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell equations are derived in a series of papers [I, II, III]. The analogy of the elastic and electromagnetic fields is analyzed. In part [II] a physical interpretation is given of the tensors of stress and distortion and expressions are considered for the energy density of the elastic field of the dislocations. The basic equations are generalized for the case of a non-zero Poisson ratio and a finite body. A detailed interpretation is given of the tensor components of dislocation current density.

---

II.

, . . , . . .

     

Linear Einstein equations. III

The Cauchy problem for equations of the theory of gravitation, linearized around the non-Galilean metric , can have a nonunique solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 95–98, July, 1982.     

Differential equations of spin dynamics and asymptotic expansion of quantum mean values near the classical limit. III

A study is made of the asymptotic behavior of the functional analogs of the coefficients of vector addition (Clebsch-Gordan coefficients) introduced by means of atomic coherent states for large values of the angular momenta. The cases J₁ J₂, J₃ 1 and J₁, J₃ J₂, where J₁ and J₂ are the original angular momenta and J₃ is the resultant, are investigated. It is shown that the investigated functions are transformed for large J into narrow distributions, which makes it possible to expand integrals containing such functions in asymptotic series in powers of J^–1. The quantum rule for adding angular momenta, formulated in functional language, goes over into the classical one in the limit J The possibility of using these relations to describe molecules with rotational degrees of freedom is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 142–148, April, 1977.     

Cauchy problem for the Einstein equations. III

The equations of small perturbations of the metric in synchronous coordinates are equivalent to a first-order system for which the Cauchy problem is correct only upon the satisfaction of several conditions imposed on the lowest symbols of the operator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 105–109, November, 1982.     

Separation of variables in the Klein-gordon equations. III

This paper supplements [1] and [2]. All kinds of external electromagnetic fields are found which contain arbitrary functions admitting complete separation of variables in the Klein-Gordon equations by using one first and two second order differential symmetry operators. The curvilinear coordinates in which the variables separate are presented, and equations in the separated variables are written down.     

The distribution of dislocations in silicon iron single-crystals grown from the melt

The distribution of dislocations in a crystal of an Fe-4.2% Si alloy prepared from the melt is studied microscopically and, by X-ray diffraction after rendering them visible by anodic dissolving. The dislocations are also studied inside the crystal by successively grinding the surface. The density of the dislocations inside the blocks is determined by calculation from microphotos, the density of the dislocations forming the boundaries between the blocks by measuring the angles between the lattices of neighbouring blocks. It is found that the dislocations are distributed very unevenly in the crystal and most of them form complicated boundaries of blocks, similarly as with ionic crystals. The distribution of dislocations is also discussed from the point of view of their formation and conclusions are reached as to the preparation of single crystals having a smaller number of dislocations.

---

Fe-Si,

- Fe-4,2%Si, , . . , , — . , , . .

---

---

The author is indebted to Candidates of Mathematics and Physics, F. Kroupa and J. Kaczér, for careful reading and advice whereby they helped to give greater depth and accuracy to this paper. He thanks Z. Tahal and M. Honegrová for initiative shown in helping to prepare the single crystals and during exacting experimental work, and S. Kadeková for aid in adjusting the spectrometer.     

A kinematic model for continuous distributions of dislocations

In continuum theory of defects the notion of a flat connection is employed. This paper gives a characterisation of these connections via injective ³-valued differential forms. For material structures with continuous distributions of dislocations, a configuration space in the sense of global analysis is introduced and analysed. A kinematics for these dislocations is formulated which generalises from elasticity.     

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

Solutions of the einstein equations with zero coupling. III

A transformation of the Kerr-Schild zero-coupling type is used to obtain new exact solutions of the Einstein-Maxwell equations from the known steady-state solutions in vacuum, including the Kerr solution describing the gravitational field of a uniformly rotating ring.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 76–81, November, 1971.In conclusion the author thanks V. I. Rodichev for valuable discussions of the topics discussed in this paper.     

The method of interpolating integral equations for quantum Fluids.—III

Summary A method for interpolating between the (F)HNC and (F)PY approaches in order to take into account elementary contributions has been presented in two preceding papers concerned with the properties of zero-temperature quantum fluids, described by short-range correlated wave functions. In the present paper, both for Bose and for Fermi systems, the technique is extended to the case in which the two-body radial distribution function contains a long-range tail, going asr ⁻⁴. The results obtained for the energy per particle and momentum distribution of liquid⁴He, polarized hydrogen and³He are presented in correspondence to variational wave functions containing only two-particle correlations.

---

Riassunto In due precedenti lavori è stata sviluppata una tecnica di interpolazione tra le approssimazioni (F)HNC e (F)PY per tenere conto dei cosiddetti contributi elementari nel calcolo delle proprietà dei fluidi quantistici a temperatura zero e descritti da funzioni d’onda con fattori di correlazione a corto raggio. Nel presente lavoro il metodo è esteso, per sistemi sia bosonici che fermionici, al caso in cui la funzione radiale di distribuzione contenga una coda con andamentor ⁻⁴. Sono presentati i risultati ottenuti per l’energia per particella e la distribuzione di quantità di moto dell’⁴He liquido, dell’idrogeno polarizzato e dell’³He liquido, in corrispondenza a funzioni d’onda variazionali contenenti solo correlazioni di coppia.

---

Резюме В двух предыдущих работах был предложен метод интерполяции между (F)HNC и (F)PY подходами. В этих работах рассматривались свойства квантовых жидкостей при нулевой температуре, которые описываются волновыми функциями с короткодействующими корреляциями. В данной работе, для Бозеи Ферми-систем, предложенная тенхика обобщается на случай, когда двух-частичная радиальная функция распределения содержит длиннодействующий хвост, который спадает какr ⁻⁴. Приводятся результаты для энергии на одну частицу и импульсное распределение для жидкого⁴He, поляризованного водорода и³H, причем соответствующие вариационные волновые функции содержат только двух-частичные корреляции.

     

The construction of special solutions to partial differential equations

Almost all the methods devised to date for constructing particular solutions to a partial differential equation can be viewed as manifestations of a single unifying method characterized by the appending of suitable “side conditions” to the equation, and solving the resulting overdetermined system of partial differential equations. These side conditions can also be regarded as specifying the invariance of the particular solutions under some generalized group of transformations.     

Dynamics of solidification in binary melts during rapid cooling I. The basic equations

In this paper we deal with the analysis of the general complex model which describes the solidification of the binary melt. Within the framework of this model we consider the mass and energy transport in the system and the kinetics of the phase transformation. The paper is divided into two parts. In the first part the set of equations is derived, which describe the evolution of the system within the framework of the theory of stochastic processes. The emphasis is laid on the rapid changes of temperature in the system. In the second part the analysis of the mentioned set of equations is presented. It results in the delimitation of the conditions under which the adiabatic approximation and the kinetic phase diagrams at the phase interface can be applied.     

The Maxwell-Vlasov equations as a continuous hamiltonian system

The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.     

The scale equations in the critical dynamics of fluctuating systems

The scale equation method is applied to the investigation of the critical dynamics of systems described by Ginzburg-Landau functionals of the most general form. The method does not require renormalizability of the Ginzburg-Landau functional and does not make use of the scaling invariance hypothesis.     

The dynamics of coupled nonlinear model Boltzmann equations

A multispecies gas described by coupled nonlinear Boltzmann equations is studied as a dynamical system. Properties are determined of theN coupled nonlinear ODEs for the number densities obtained from the Boltzmann equations for the spatially uniform system ofN species undergoing binary scattering, removal, and regeneration in the presence of an external force field and a reservoir of background gas. The physically realizable setQ, the nonnegative cone in theN-dimensional phase space of species number densities, is established as invariant under the flow. The fixed-point equations for the ODEs are shown to be equivalent to 2 ^N linear systems, and conditions for the stability and instability of the fixed points are then established. Stable fixed points are demonstrated to exist inQ by showing that they enter via a sequence of transcritical bifurcations as physical parameters are varied. For the two-species case the typical global structure of the solutions is established. Various particular cases are described including one which possesses an infinite family of periodic solutions and one that depends delicately upon initial conditions due to a separatrix that separatesQ into two invariant sets.     

Hypercomplex equations of dynamics

On the basis of the hypercomplex formalism, a system of dynamics equations for material points is introduced, which is a generalization of Newton's equations, and which treats spatial translations and proper rotations in a similar fashion. The notion of generalized forces is formulated in such a way that these have a welldefined physical meaning. Some simple consequences of these equations are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 95–98, January, 1991.     

Quantitative depth distribution of dislocations by planar channeling

Planar channeling analysis with MeV He backscattering is shown to give depth profile of the dislocation density with advantages over axial channeling analysis due to the larger dechanneling cross section of dislocations for planar channeling.     

Effects of size on the dynamics of dislocations in ice single crystals

Single crystals of ice subjected to primary creep in torsion exhibit a softening behavior: the plastic strain rate increases with time. In a cylindrical sample, the size of the radius affects this response. The smaller the radius of the sample becomes while keeping constant the average shear stress across a section, the softer the response. The size-dependent behavior is interpreted by using a field dislocation theory, in terms of the coupled dynamics of excess screw dislocations gliding in basal planes and statistical dislocations developed through cross slip occurring in prismatic planes. The differences in the results caused by sample height effects and variations in the initial dislocation microstructure are discussed.