The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions byc ⁽⁾, 0 _s , the principal result is that if the initial density _{0 s} then the solution converges strongly toc ^(o), while if ₀ > _s the solution converges weak* toc ^(s). In the latter case the excess density ₀– _s corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions.     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Gentle perturbations

By introducing a specific type of perturbation,A, in the Hamiltonian, we define a class of gently perturbed states, ,A, of a canonical ensemble, . The perturbations are chosen so as to preserve a relationship of the form ,A constant ×. Applications in ergodic theory and phase transitions are described.     

Existence d'orbites quasi-periodiques dans les attracteurs de Birkhoff

For each number between the lower and the upper rotation number of the Birkhoff attractor of a dissipative monotone twist map, there is a periodic or quasi-periodic orbit with rotation number .     

Anomalous variation in electrical resistance during ordering of type Mg3Cd alloys

Using x-ray diffraction studies of the structural state of type Mg₃Cd alloys during the ordering stage accompanied by an increase in their electrical resistivity , the authors analyzed the problem of the admissibility of previously postulated interpretations of the anomalous variation in in the initial stages of ordering of certain investigated alloys. It was tentatively concluded that this anomalous variation may be associated with the nonmonotonic character of the dependence of on the degree of order.     

Humidity dependence of the atmospheric absorption coefficient in the transparency windows centered at 0.88 and 0.73 MM

A dependence is investigated of the absorption coefficient of atmospheric water vapor on absolute air humidity in the spectral intervals =11.1+12.0 and 13.2+14.3 cm^–1 by a radiospectrometer with a multipass vacuum cell. Measurements are carried out in the humidity range 3+23 g/m³ at 140 m length of the optical path.Parts of the absorption coefficient linearly and quadratically depending on have been separately defined from the experimental data. In particular, it has been obtained for the centers of transparency windows: (=0.73mm) [dB/km]=1.68 (±0.08) +0.019 (±0.004) ², (=0.88 mm) [dB/km]=0.83 (±0.06) +0.013 (±0.002) ². The given values correspond to the temperature of 306 K and the pressure of dry air of 735 Torr.A comparison is made between our results and theoretical calculations and data of other measurements.     

A study of the ϱ0 meson polarization in π− p reaction at 5 GeV/c

Experimental data on ⁰ meson polarization in ^– p reaction at 5 GeV/c are presented. Change in ⁰ meson polarization for mesons produced at smallP ² and atP ² 0·3 or produced in backward direction in CMS is demonstrated. Natural explanation of these phenomena is one pion exchange mechanism and mechanism of quark-antiquark annihilation as observed for ⁰ meson production in¯pp reactions.The authors are indebted to Ján Piút, Richard Lednický, V. S. Rumiancev and N. K. Koutsidi for discussions and critical comments.     

Magnetic susceptibility and electrical conductivity of the highest chromium germanide

Results are given of measurements over the temperature range 80–1000 ° K of the magnetic susceptibility () and the electrical resistivity () of the highest chromium germanide Cr₁₁Ge₁₉. A ferromagnetic spin ordering is indicated at T < 86 ° K. From the (T) dependence and the values of the resistivity, thermo-emf and Hall constant, it is concluded that the compound studied is a semimetal with p-type conductivity.     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

A global analysis approach to the general relativistic fluid ball problem

That a self-gravitating perfect fluid in empty space has a spherical equilibrium configuration if it is static-i.e., nonrotating-is considered physically evident, but has not yet been rigorously derived from Einstein's field equations together with suitable asymptotic conditions. In this paper the global analysis techniques developed recently mainly by Fischer, Marsden, and Cantor are used to derive the result that if a family of static perfect fluid solutions with fixed total gravitational massm and fixed equation of state(p) satisfying 0 p and 0 d/dp < depends differentiably on a parameter and contains the spherically symmetric solution then it must consist of solutions diffeomorphic to the spherically symmetric one.Partially supported by the National Sciences and Engineering Research Council, grant No. A8059.     

Dependence of characteristics of hysteresis loops on dislocation densities for low-alloy Cr-Mo steel

The microstructure of specimens of low-alloy Cr-Mo steel subjected to various heat treatments and various plastic deformations is studied by transmission electron microscopy. The observations indicate that the decisive factors which control the coercive forceH _c, the initial permeability _i and the Rayleigh constant are the density and the arrangement of dislocations. The precipitates have only a secondary effect. The observed linear dependencesH _c ^1/2, _i ^–1/2, ^–1 ( —dislocation density) are in full agreement with the theoretical predictions.Dedicated to Academician Vladimír Hajko on the occasion of his 65th birthday.     

Silver diffusion in cadmium sulfide single crystals

An experimental study was made of the penetration of silver by the thermal-diffusion method into cadmium sulfide. The silver was deposited on the samples by thermal evaporation in vacuum. Optical and electrical-probing methods were used to study the rates of surface and bulk diffusion at various temperatures. The temperature dependence of the diffusion coefficient was established for the purity used. After the penetration of the silver impurity, the low-resistivity ( 0.1 · cm), nonphotosensitive cadmium sulfide samples displayed high resistivity ( 10⁶-10⁸ · cm) and photosensitivity.Translated from Izvestiya VUZ. Fizika, No. 7, pp. 12–16, July, 1970.     

A contribution to the inequality and equality relations in structure analysis

On the basis of the non-negativity for electron density in a crystal (r), the existence of inequalities and equalities between the coefficients of the Fourier expansion of the function (r) (r+t) is shown, which limits the choice of phases of the structure factors.     

The van der Waals limit for classical systems. I. A variational principle

We consider the thermodynamic pressurep(, ) of a classical system of particles with the two-body interaction potentialq(r)+ ^v K(r), where is the number of space dimensions, is a positive parameter, and is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit 0 of the pressurep(, ) exists and is given by where the limit and the supremum can be interchanged. Here is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda ⁰() is the free energy density of a system withK=0 and density . A similar result is proved for the free energy.     

Spatial Spectrum of Artificial Ionospheric Irregularities Induced by Powerful HF Radiowaves

We present measurement results and spatial-spectrum shapes of the dependence of the relaxation time on the scale across the geomagnetic field for artificial ionospheric irregularities (AIIs) induced in the upper ionosphere by powerful HF radiation of the SURA heating facility. The irregularity diagnostics was based on observing amplitude scintillations of a 243-MHz beacon signal from a quasi-geostationary, solar-synchronous satellite and on measuring field-aligned scattering at frequencies 15 and 20 MHz. The satellite signal was received at the Kazan State University Observatory. The field-aligned scattering signals were received and analyzed by a bistatic HF radar based on the UTR-2 radio telescope located near Kharkov (Ukraine). It is shown that irregularities of the electron density, whose two-dimensional spectrum in the plane perpendicular to the geomagnetic field is the power law æ ^-p with index p2, are developed in the scale range 30–60 l 200–400 m. In this case, the relative fluctuations (N²)^1/2 of the electron density increase with decreasing scale l=2/æ. The estimate N²)^1/2 1–1.5% is obtained for a heating power of 150 MW and irregularity scales l 30–60 m at which the fluctuations are maximum. The measured dependence of the AII relaxation time, defined as the e-folding time of the amplitude-scintillation intensity, has the form _r l . If l 30–60 m, then the index is close to 2, whereas the effective diffusion coefficient D (2–3)· 10^-1 m²/s corresponds to the ambipolar cross-field diffusion coefficient in a magnetized plasma. The time _r for scales l 60–100 m is independent of l and increases with decreasing velocity of regular drift of the plasma. The Doppler-spectrum broadening (²)^1/2 0.6 Hz observed when receiving field-aligned scattered signals can be related to chaotic motions of plasma-density disturbances whose random drift velocities amount to (v²)^1/2 2–3 m/s for scales l 20 m at which the power-law index changes drastically.     

Yet another formulation of the Dirac equation

The 2-by-2 Pauli matrix algebra is used to write the 1-by-4 Dirac field in anequivalent 2-by-2 matrix . The current 4-vectors and *_µ are then compared and the latter is shown to not be easily interpretable as a probability density, and also tocontain .     

Observation of the spin alignment of the ϱ0-meson produced in ¯pp interactions at 22.4 and 5.7 GeV/c

Studying the ⁰-meson production in ¯pp interactions at 22.4 GeV/c and in 4-prong anníhilation channels of ¯pp interactions at 5.7 GeV/c, we have observed an essential ⁰-meson spin alignment. The values of the ₀₀ element of the ⁰-meson spin density matrix (thez-axis is directed along the normal to the production plane) are equal to 0.08 ± 0.07 and 0.55 ± 0.03, respectively, i.e. the ⁰-meson spin lies preferably in the production plane. The absence of such an effect in pp interactions at 12 and 24 GeV/c and also the essentially larger ⁰ production cross section in ¯pp interactions at these energies make it possible to connect the observed ⁰-meson spin alignment with the annihilation processes. The character of the observed spin alignment is unexpected from the point of view of usual models, e.g. multiperipheral models. This effect could be described by the spontaneous polarization of quarks and antiquarks during the state preceding their recombination into mesons.Dedicated to Professor Ivan Úlehla on the occasion of his sixtieth birthday.The authors would like to express their gratitude to the CERN-Prague collaboration for permission to use their data on ¯pp interactions at 5.7 GeV/c. The authors are also indebted to A. M. Baldin, S. B. Gerasimov and H. I. Miettinen for valuable discussions, to the technicians and assistants at all laboratories for their work.     

An upper bound on the free energy for classical systems with Coulomb interactions in a varying external field

The thermodynamic limit is taken using a sequence of regions all the same shape as a given region of volume ||, with a specified distribution of normal field component on . We show that with magnetostatic interactions the limiting free energy density is bounded above by jhen where (,B) is the free energy density for a system of density in a uniform external fieldB and the inf is taken over all divergence-free fieldsB with given normal component on and all densities (x) compatible with particle number constraints of the form where _i is a sub-region of . A physical argument suggests that this upper bound is the true thermodynamic limit, and that it takes account demagnetization effects. Electrostatic interactions can be treated similarly.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals

We establish an analogue of Weyl's classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractalK, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type (multifractal) measures onK. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let (x) be the number of eigenvalues less thanx. Then we show that (x) is of the order ofx ^d S/2 asx+, where the spectral exponentd _S is computed in terms of the geometric as well as analytic structures ofK. Further, we give an effective condition that guarantees the existence of the limit ofx ^–d S/2(x) asx+; this condition is, in some sense, generic. In addition, we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofK.The research of Michel L. Lapidus was partially supported by the National Science Foundation under Grant DMS-8904389 (transferred to the University of California as grant DMS-9196085) and Grant DMS-9207098, as well as by Yale University