An investigation of finite-size scaling for systems with long-range interaction: The spherical model

A method is suggested for the derivation of finite-size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr ^{–d –} , whered is the space dimensionality and>0. It allows for a unified treatment of short-range (=2) and long-range (<2) interaction. The asymptotic analysis is illustrated by the mean spherical model of general geometryL ^d–d× ^d subject to periodic boundary conditions. The Fisher-Privman equation of state is generalized to arbitrary real values ofd, 0d. It is shown that the-expansion may be used to study the breakdown of standard finite-size scaling at the borderline dimensionalities.     

Contribution of quantum corrections and superconducting fluctuations to the magnetic field and temperature dependence of the conductivity in amorphous CuTi and CuZr alloys

We report conductivity measurements on Cu _x Ti_100–x and Cu _x Zr_100–x glasses as a function of magnetic field (0B6T) and temperature (1.5–5. At low temperatures the measured conductivity(T,B)= ₀+(T)+(B,T) is compared quantitatively with the predictions of diagrammatic perturbation theories including quantum interference effects, spin-orbit scattering and electron-electron interaction. We obtain a set of characteristic fields for inelastic and spin orbit scattering from both the magnetic field and the temperature dependence of the measured (T, B). Below about 4K the Coulomb interaction determines (T) in consistence with the Hall effect, whereas quantum interference processes and spin-orbit scattering dominate the magnetic field dependence. In case of Cu _x Zr_100–x , (B, T) can be explained by taking into account superconducting fluctuations (Maki-Thompson and Aslamasov-Larkin parts) in addition. Superconducting fluctuations dominate the temperature dependence as well. For high magnetic fields and lower temperatures (B/T>1T/K) we find discrepancies between experiment and calculations from perturbation theory.     

On X-ray measurements of residual stresses in materials with lattice strain gradient

The non-linear distribution of lattice strain ³¹⁰vs sin² observed on the surface of a polished steel specimen can be interpreted by tri-axial stress state with surface components of stress tensor ₁₁(0)= ₂₂(0) and gradientsg ₁₁= ₁₁/T, g ₃₃= ₃₃/T. The distribution of experimental values is duscussed from the viewpoint of various ways of calculating ₁₁.The authors would like to thank Dr. J. Musil, D. Sc. of the Institute of Physics of Czechoslovak Academy of Sciences for kindly providing the samples which made possible this study.     

Kronecker products of matrices and an application to Fermi systems

An identity for the trace of an exponential function of Kronecker products of matrices is proved. This identity plays an important role for the calculation of the grand potential of interacting Fermi systems. For the HamiltonianH= _i n _i n _i wheren _i =c _i ⁺ n _i (c _i ⁺ : Fermi creation operator at the ith site with spin) we calculate the specific heat for different numbers of electrons per lattice site. Finally, we extend our calculations to find approximative solutions of the Hubbard model.     

Multicomponent composites,electrical networks and new types of continued fraction II

The outstanding problem of systematically developing rigorous bounds on the complex effective conductivity tensor * ofd-dimensional,n-component composites withn>2 is solved. The bounds incorporate information contained in successively higher order correlation functions which reflect the composite geometry. Explicit expressions are given for many of the bounds and some, but not all of them, are represented by nested sequences of circles in the complex plane that enclose, and in fact converge to, each diagonal element of *. They are derived from the fractional linear matrix transformations found in Part I that recursively link * with a hierarchy of complex effective tensors ^(j),j=0, 1, 2, ..., of increasing dimension,d(n–1) ^j . Elementary bounds on ^(j) confining the diagonal elements of ^(j) or its inverse to half-plane, wedge or open polygon regions of the complex plane, imply narrow bounds on * which converge to the exact value of * in the limit asj . When the component conductivities are real these bounds are more restrictive than the corresponding variational bounds. Besides applying to the effective conductivity *, the bounds extend to a wide class of matrix-valued multivariate functions called -functions, and thereby to conduction in polycrystalline media, viscoelasticity in composites, and conduction in multi-component, multiterminal, linear electrical networks. The analytic and invariance properties of -functions are explored and within this class of function most of the bounds are found to be optimal or at least attainable. The bounds obtained here are essentially a generalization to matrix-valued, multivariate functions of the nested sequence of lens-shaped bounds in the complex plane derived by Gragg and Baker for single variable Stieltjes functions.     

Numerical studies on the anderson localization problem

Our study of Anderson's tight binding model for strongly disordered electronic systems is extended to a numerical treatment of thed c-conductivity atT=0. For 100 × 100 square lattices, 129 × 129 triangular lattices, and for diamond lattices with 27,000 sites, the behaviour of is studied as a function of the Fermi energy and the disorder. The calculations are based on the exact eigenfunction representation of the Kubo formula, which is evaluated by the systematic application of recursion algorithms. Our results are in favour of Mott's original suggestion of a minimum metallic conductivity _min, both in two and three dimensions. In two dimensions we find the universal value of _min=(0.11 ±0.02)e ²/. Based on the thesis of J. Stein, Regensburg 1979     

Polarization switching in a ring cavity with resonantly drivenJ=1/2 toJ=1/2 atoms

We have solved within the mean field limit for the steady state behaviour of a gas ofJ=1/2 toJ=1/2 model atoms in a ring cavity excited by an incident laser field of arbitrary polarization. Results are presented for the case of zero applied magnetic field and exact resonance between the laser frequency, the atomic transition and a cavity resonance (pure absorptive case). We find that the behaviour of the polarization switching between ⁺ and ^– outputs depends on the values of the upper and lower state collisional relaxation rates, expressed via a single parameter . The case of linearly polarized input is of particular interest since optical bistability is found to occur for 2, being pre-empted by polarization switching for >2. The results are discussed in terms of an atomic feedback mechanism coupling the ⁺ and ^– modes.     

Multicomponent composites,electrical networks and new types of continued fraction I

The development of bounds on the complex effective conductivity tensor * (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for *. Here a new field equation recursion method is developed which gives an expression for * as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor ^(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of ^(j+1). These transformations combine to form the continued fraction expansion for *=⁽⁰⁾ which is exploited in the following paper, Part II, to obtain bounds: crude bounds on ^(j), forj1, give narrow bounds on *. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.     

The electrodynamics and statistical mechanics of linear plasma response functions

The purpose of this paper is to review and to extend, wherever possible, the Kramers-Kronig relations, sum rules, and symmetry properties for the electrodynamic transport tensors of a linear plasma medium. For complete generality, we consider both nonrelativistic and relativistic plasmas with and without external magnetic fields. Our study is carried out first within the framework of classical electrodynamics. We then exploit the statistical-mechanical fluctuation-dissipation theorem to further obtain the Onsager symmetry relations and Kubo sum-rule frequency moments. Of special significance is the emergence of a variety of new Kramers-Kronig formulae andf-sum rules for the inverse dispersion tensor.Nomenclature E(k,) electric field intensity - Ê(k,) electric field in absence of plasma particles, - (k,) electric field due to the plasma particles (=E-Ê) - B(k,) magnetic induction - D(k,) electric induction - H(k,) magnetic field strength - B ₀ constant external magnetic field - A ₀ vector potential corresponding toB ₀ - (k,),j(k, co) charge and current densities due to the plasma particles - (k,),J(k,) charge and current densities of the external agency - (k,,B ₀) dielectric tensor of the plasma medium in the presence of B₀ - (k,,B ₀) diamagnetic tensor - (k, co,B ₀) (k,,B ₀) – 1, electric polarizability tensor - (k,,B ₀) magnetic polarizability tensor - (k,,B ₀) ordinary conductivity tensor - (k,,B ₀) external conductivity tensor - D(k,,B ₀) n²T–(k,,B ₀), dispersion tensor, where T=1-kk is the transverse projection tensor (k being the unit vector in the direction ofk) andn = kc/ the index of refraction - n²T – 1, = vacuum wave operator (value of D in vacuum) - 1/2( + ), Hermitian part of - ^{^} 1/2( – ), Anti-Hermitian part of a - , real and imaginary parts of a - R(r,t) dissipated power per unit volume of plasma - U total energy absorbed by the plasma - R(k,) E^*(k,) · (k,,b ₀) ·E(k,) corresponding spectral energy density - W(r,t) 1/2₀E²(r, 0 + (l/2₀) B²(r,t), field energy density - W(k,) 1/2₀E^*k,) ·E(k,) + (l/2₀)B ^*(k,) · B(k,), energy content in a certain domain of (k,)-space for a single mode - x _i,p _i,v _i coordinate, momentum, and velocity of ith electron - _i [1–(_i ²/c²)]^–1/2 - X _j,P _j,V _j coordinate, momentum, and velocity of jth ion - {A _q}, {E_q} field coordinates and momenta - j_k(t),J _k(t) perturbations in the microscopic electron and ion current densities due to the presence of the small external vector potential agencyâ(r,t) = (1/L³) â_k(t) expi k ·r - Liouville distribution function = ⁰ + - ⁰ macrocanonical distribution function characterizing the equilibrium state of the system in the infinite past - small perturbation due toA - H⁰ Hamiltonian of equilibrium system which includes interaction - H Hamiltonian for the interaction between the system and the small external perturbing agencyA - ⁰ = d^R()⁰ expectation value of any quantity over the equilibrium ensemble (d^R is an element of hypervolume in -phase space) - G(12) two-particle distribution function - F(1) one-particle distribution function - g(¦x₂ – x₁ ¦) [G(12)/F(1)F(2)] – 1, pair correlation function - N total number of electron in volume L³ - n ₀ equilibrium density (of electrons) - ^–1 temperature (in energy units) - ₀ (n₀e²/m₀)^1/2, equilibrium electron plasma frequency - _c ¦e ¦–B ₀/m, electron frequency - ^–1 ( ₀/n₀e²)^1/2, Debye length - ₀ (n₀Ze²/M₀)^1/2, equilibrium ion plasma frequency - _c ZeB₀/M, ion cyclotron frequency     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

Random walks on cubic lattices with bond disorder

We consider diffusive systems with static disorder, such as Lorentz gases, lattice percolation, ants in a labyrinth, termite problems, random resistor networks, etc. In the case of diluted randomness we can apply the methods of kinetic theory to obtain systematic expansions of dc and ac transport properties in powers of the impurity concentrationc. The method is applied to a hopping model on ad-dimensional cubic lattice having two types of bonds with conductivity and ₀=1, with concentrationsc and 1–c, respectively. For the square lattice we explicitly calculate the diffusion coefficientD(c,) as a function ofc, to O(c²) terms included for different ratios of the bond conductivity. The probability of return at long times is given byP ₀(t) [4D(c,)t]^–d/2, which is determined by the diffusion coefficient of the disordered system.     

Photopyroelectric spectroscopy (P2ES) of electronic defect centers in crystalline n-CdS

The electrical and dielectric properties of mercuric iodide were studied at room temperature under various intensities and colours of light in the frequency range 1 Hz–10 kHz. In the high-frequency region (>40 Hz), the real part of the dielectric constant () is almost constant with frequency (f), colour and intensity of light. At lower frequencies, varies nearly as 1/f and monotonically increases with intensity (I) of the yellow (or green) light, whereas it is almost constant with red light intensity. This behaviour is discussed in the view of the different polarization contributions. The imaginary part of the dielectric constant () was found to vary as 1/f over the frequency range studied. This behaviour was observed whether the crystal was in dark or illuminated implying that the roomtemperature ac dark- or photo-conductivity () is independent of frequency. The observed variation of with intensity of yellow (or green) light was found to follow anI ^1/2 dependence and a weaker dependence for the red light. the red light. The conductivity behaviour is discussed in the view of the current theories.     

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.     

Mates toE=mc 2 and to the Heisenberg uncertainty relations

E=mc ² is found to be a special case ofE= ^±1cⁿ, where is any one of four susceptibilities, namely electric, magnetic, gravitational, and elastic. Letl be length,t time,t time dilation, andl a measure of Fitzgerald-Lorentz contraction. A particle is stated to be the manifestation of a collection of susceptibilities which arise when(l)/1=(t)/t. Then(E)/E=5 (t)/2t=±()/. Corresponding to susceptibility, special energy particles are postulated which exhibitSU(3) symmetry, Related to the susceptibilities are five new Heisenberg uncertainty relations. Three new conservation laws for particles are proposed.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

Cosets and ferromagnetic correlation inequalities

Consideration of subgroups of the group of all subsets ofN={1, 2, ...,n} with symmetric difference as the operation leads to new inequalities on correlations for a generalized Ising ferromagnet. An upper bound for the rate of change of ^R with respect toJ _S in terms of correlations and a new, brief proof for the monotonicity of ^R as a function ofJ _S are given.Supported by NSF GP 7946.     

The flow of electric current through the chain of nonlinear inertial resistors and capacitors

The stability conditions for an electric current flowing through a chain consisting of nonlinear inertial resistors and capacitors have been investigated. The necessary conditions for the field dependent conductivity and the inertia of the resistors and applied voltage leading to undamped oscillations in the chain have been derived. For the threshold voltage this condition has the form ₀+V ₀(/V) _v0 cos _I ^T =0, similar to that obtained in an earlier work of one of the authors from thermodynamical considerations.Computer simulation gives results for the frequency and threshold conditions in good agreement with approximate analytical solutions.     

Electrical and thermal properties of chalcogenide glass system Se75Ge25−x Sb x

The dc and thermal conductivities of five different compositions of the chalcogenide glass system Se₇₅Ge_25–x Sb _x have been studied in a temperature range below T _g. The dc conductivity results indicate that each composition has a single activation energy in the considered temperature range. The coefficient of thermal conductivity increased linearly with temperature below T _g for the compositions investigated. The increase of Sb content in the chalcogenide glass system leads to an increased coefficient of electrical conductivity , an increased coefficient of thermal conductivity , and to a decreased activation energy E and pre-exponential factor ₀. The observed compositional dependencies of and E have been correlated with the increase of weak bond density and the decrease of covalent bond density in the structure of the compositions investigated with increasing Sb content at the expense of Ge content. The decrease in ₀ and the increase in has been, respectively, correlated with the decrease in mobility and the increase in phonon velocity.     

The asymmetric contact process at its second critical value

The asymmetric contact process onZ has two distinct critical values ₁ > ₂ (at least with sufficient asymmetry). One can consider the process on {0,...,N} and analyze the time (which we call _N ) till complete vacany starting from complete occupation. Its behavior has already been resolved for all regions of except for =₂. For this value, Schinazi proved that lim _N log _N /logN=2 in probability and conjectured that _N /N ² converges in distribution. It is that result that we prove in this paper. We rely heavily on the Brownian motion behavior of the edge particle, which comes from Galves and Presutti and Kuczek.     

On the tensor product of supersingleton representations ofosp(1, 2n)

The tensor product of two supersingleton representations _n of the Lie superalgebraosp (1, 2n) is studied forn2. The main results are as follows: (a) anticommutators and commutators of the odd generators in _{n n} form a skew-symmetric representation of the Lie algebrau(n, n); (b) simple explicit form of all irreducible components of _{n n}, which are labelled by a single parameterJ=0, 1, ..., has been found. Each of them is a^*-representation ofosp (1, 2n) for which assertion (a) is valid. The dimension of its vacuum subspace equals , i.e., the nondegenerate vacuum occurs for J=0 only. Basic property of this family of irreducible^*-representations of osp(1, 2n) are analogous to those of massless representations of osp(1, 4).Dedicated to Academician Václav Votruba on the occasion of his eightieth birthday.