Random surfaces with two-sided constraints: An application of the theory of dominant ground states

We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints _x m/2. The main result is that for ₀, where ₀ does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.     

Atomic versus ionized states in many-particle systems and the spectra of reduced density matrices: A model study

We study the spectrum of appropriate reduced density matrices for a model consisting of one quantum particle (electron) in a classical fluid (of protons) at thermal equilibrium. The quantum and classical particles interact by a shortrange, attractive potential such that the quantum particle can form atomic bound states with a single classical particle. We consider two models for the classical component: an ideal gas and the cell model of a fluid. We find that when the system is at low density the spectrum of the electron-proton pair density matrix has, in addition to a continuous part, a discrete part that is associated with atomic bound states. In the high-density limit the discrete eigenvalues disappear in the case of the cell model, indicating the existence of pressure ionization or a Mott effect according to a general criterion for characterizing bound and ionized electron-proton pairs in a plasma proposed recently by M. Girardeau. For the ideal gas model, on the other hand, eigenvalues remain even at high density.     

ИЗЛУЧЕНИЕ ЭЛЕКТРИЧЕ СКОГО ДИПОЛЯ НАД ГРАН ИЦЕЙ ДВУХ СРЕД

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Single crystal diffraction patterns of germanium. Part II

The rocking curves of Ge (111), (220), (333) for CuK ₁ radiation were measured by means of the triple-crystal diffractometer. Perfect silicon single crystals, cut parallel to the (111) plane were used in the monochromator part of the triple-crystal diffractometer. The results prove the suitability of such a monochromator for studying diffraction patterns.

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. II

(rocking curves) (111), (220), (333) CuK ₁ . , (111). .

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In conclusion the authors thank A. Haruý for preparing the germanium single crystals and they are indebted to V. Smutná and A. Irra for the care with which they carried out various tasks.     

On the theory of ferromagnetic thin films

As is known, the second approximation in the calculation of the partition function by the traces method of ferromagnetic thin films gives wrong results for the coordination number equal to eight. In order to obtain correct results even for this case, the third order approximation of the partition function is developed and thus the magnetic properties of body-centred cubic iron thin films are studied. The dependence of the Curie temperature on the thickness, for different values of the ratio between the anisotropy constant and the exchange energy between two neighbours, is discussed. A value can be chosen for this ratio such that the thin film becomes ferromagnetic only for a thickness greater than a definite value.

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, . , , (Fe). . , - .

     

МОДЕЛИРОВАНИЕ ЭЛЕКТ РОННО-ОПТИЧЕСКИХ СИС ТЕМ С ОСЕВОЙ СИММЕТРИЕЙ П РИ ПОМОЩИ СЕТИ СОПРОТИВЛЕНИЙ

The paper explains the theory of modelling electrostatic fields by a resistance network. The conditions, which the resistance network must satisfy, are derived and the question of modelling electrodes of different shapes is solved. The finished network and the results obtained on it when modelling a jet for a linear h-f accelerator of electrons are described. Particular attention is paid to the influence of a space charge, the modelling of which is an advantage of this method.

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Photo-luminescence and growth defects in ZnS: Cu monocrystals

The paper describes microscopic investigation into the spectral distribution of photo-luminescence and its polarization on hexagonal ZnS monocrystals in relation to the structural defects of such crystals. These observations, as well as birefringent observations, show that the values of the birefringence and the degree of polarization in different stripes of the same crystal are the same within the limits of observational errors.

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ZnS: Cu

ZnS . , , , .

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In conclusion the author would like to thank K. Pátek, L. Souková and M. Mokonová for providing the monocrystals and for fruitful discussions.     

On the upper critical dimension of lattice trees and lattice animals

We give a rigorous proof of mean-field critical behavior for the susceptibility (=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of spread-out or long-range models in which trees and animals are constructed from bonds of various lengths, above eight dimensions. This provides further evidence that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk.     

A contribution to the semi-focusing method with a flat polycrystalline sample

Condition are derived for the approximate focusing of diffraction lines in the chosen, relatively wide angular region, using the semi-focusing method with a flat sample, if a bent crystal monochromator is employed. The form of the absorption factor for a flat polycrystalline sample is derived for the case of uniform oscillation of the sample in the angular region ± around the axis of the camera.

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On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.     

ИЗОТОПЫ ФТОРА F19 И F21 В МОДЕЛИ НИЛЬССОНА

Energy levels and electromagnetic properties of fluorin isotopes in the Nilsson's model with Coriolis mixing are investigated. Position of the F¹⁹ levels is satisfactorily explained up to the energy of 4·5 MeV. Concerning the F²¹ it seems that the shell model will better correspond to the F²¹ nucleus.

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The low-temperature phase of Kac-Ising models

We analyze the low-temperature phase of ferromagnetic Kax-Ising models in dimensionsd2. We show that if the range of interactions is ^–1, then two disjoint translation-invariant Gibbs states exist if the inverse temperature satisfies –1^N, where =d(1–)/(2d+2)(d+1), for any >0. The proof involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous-spin system which is suitable for the use of a variant of the Peierls argument.     

Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.     

Methodological imperfection and formalizations of scientific activity

Any mathematical formalization of scientific activity allows for imperfections in the methodology that is formalized. These can be of three types, dirty, rotten, and dammed. Restricting mathematical attention to those methods that cannot be construed to be imperfect drastically reduces the class of objects that must be analyzed, and relates all other objects to these more regular ones. Examples are drawn from empirical logic.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.     

ЧАстИЧНО пОгРУжЕННы И БлОк В кВАДРАтНОИ РЕ шЕткЕ

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Partially inserted block in square lattice

The paper gives a theory of the partially inserted block (control rod) in a reactor of finite height, consisting of a square lattice of fuel blocks and a moderator. The partially inserted block replaces the block of the original lattice which was located in the axis of cylindrical symmetry of the reactor. The results obtained agree with the relations for a homogeneous medium in paper [1] and are thereby justified for homogeneized reactors.

     

Correlation inequalities for multicomponent rotators

A recent approach to G.H.S. and Lebowitz inequalities is used to prove Griffiths' second inequality for 3 and 4 component models (e.g. Classical Heisenberg model, ||⁴ Euclidean fields). Applications include monotonicity of the mass gap in the external field, and two-sided inequalities between parallel and transverse correlations.     

Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem

In a nongeometrical interpretation of gravity,the metric g(x) = + (x)is interpreted as an effective metric, whereas(x) is interpreted as afundamental gravitational field, propagated in spacetime which isactually flat. Some advantages and disadvantages of suchan interpretation are discussed. The main advantage isa natural resolution of the flatness problem.     

Finite-Time Lyapunov Exponents for Products of Random Transformations

I show how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decomposition into angular, diagonal, and shear degrees of freedom. In the case of a Gaussian process a set of variables, adapted to the Iwasawa decomposition and still having a Gaussian distribution, is introduced and used to compute the statistics of the finite-time Lyapunov spectrum of the process. The variables also allow to show the exponential freezing of the shear degrees of freedom, which contain information about the Lyapunov eigenvectors.     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.