Study ofA+A→0 with probability of reaction and diffusion in one dimension and in fractal substrata

We present Monte Carlo simulations of annihilation reactionA+A0 in one dimensional lattice and in three different fractal substrata. In the model, the particles diffuse independently and when two of them attempt to occupy the same substratum site, they react with a probabilityp. For different kinds of initial distributions and in the short an intermediate time regimes, the results for 0<p1 show that the density ofA particles approximately behaves as (t)=(t=0)f(t/t ₀), with the scaling functionf(x)1 forx1,f(x)x ^–y forx1. The crossover timet ₀, behaves ast _{0 0eff} ^–1y where theeffective initial density _0eff depends on (t=0) and on the kind of initial distribution. For a given substratum of spreading dimensiond _s, the exponenty(d _s/2<y<1) depends only onp and its value increases asp decreases (y1 whenp0). In the very long time regime it is expected thatp(t)t ^–ds/2 independently ofp.     

F-Information, a Unitless Variant of Fisher Information

A new information matrix [F] with elements F _mn = (y _m - a _m )(y _n - a n) ( ln p(y | a)/a _m ) ( ln p(y | a)/a _n ) is analyzed. The PDF p(y | a) is the usual likelihood law. [F] differs from the Fisher information matrix by the presence of the first two factors in the given expectation. These factors make F _mn unitless, in contrast with the Fisher information. This lack of units allows F _mn values from entirely different phenomena to be compared as, for example, Shannon information values can be compared. Each element F _mn defines an error inequality analogous to the Cramer-Rao inequality. In the scalar case F _mn F, for a normal p(y|a) law F = 3, while for an exponential law F = 9. A variational principle F = min (called FMIN) allows an unknown PDF p(x) to be estimated in the presence of weak information. Under certain conditions F obeys a Boltzmann F-theorem F/t 0, indicating that F is a physical entropy. Finally, the trace of [F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.     

Effect of hydrogen doping on the magnetic properties of Gd2CuO4

We report the results of ac-susceptibility and dc-magnetization measurements for H_yGd₂CuO₄ (0y0.54). It is shown thatH doping lowers the weak ferromagnetic component in the material. The distinct hysteresis loops observed atT=77 K for both non- and hydrogenated samples change its shape withy. The magnetic ordering temperatures T _N ^Cu and T _N ^Gd , as determined from the temperature dependencies of ac-susceptibility, remain unchanged with sample's hydrogenation. This result seems to indicate that extra electrons are not doped onto the Cu-O planes of Gd₂CuO₄. The frequency dependencies ofx(, T) andx(, T) for bothy=0 andy=0.15 samples are analysed., The maximums ofx andx found at about 200K are considered in terms of susceptibility dependence on the spin-lattice relaxation time (). The anomalies in ac-susceptibility found recently in Gd₂CuO₄ atT _a=8 K andT _b=9.5 K decrease significantly withy. Results are discussed in the context of available data on 214T-type compounds.     

Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions

Let ₁(x) and ₂(y) be two local fields in a conformal quantum field theory (CQFT) in two dimensional spacetime. It is then shown that the vector-valued distribution ₁(x)₂(y)|0 is a boundary value of a vectorvalued holomorphic function which is defined on a large conformally invariant domain. By group theoretical arguments alone it is proved that ₁(x)₂(y)|0 can be expanded into conformal partial waves. These have all the properties of a global version of Wilson's operator product expansions when applied to the vacuum state |0. Finally, the corresponding calculations are carried out more explicitly in the Thirring model. Here, a complete set of local conformally covariant fields is found, which is closed under vacuum expansion of any two it its elements (a vacuum expansion is an operator product expansion applied to the vacuum).Work supported by Deutsche Forschungsgemeinschaft     

Self-diffusion in simple models: Systems with long-range jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self-diffusion coefficientD _N() in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set ofN neighboring sites. We obtain positive upper and lower bounds onF _N()=N{(1–)–[D_N()/D_N(0)]}/[(1–)]x for [0, 1]. Computer simulations for the square, triangular, and one-dimensional lattices suggest thatF _N becomes effectively independent ofN forN20.     

Statistical mechanics of the isothermal lane-emden equation

For classical point particles in a box with potential energy H^(N)=N ^–1(1/2) _ij=1 ^N V(x _i,x _j) we investigate the canonical ensemble for largeN. We prove that asN the correlation functions are determined by the global minima of a certain free energy functional. Locally the distribution of particles is given by a superposition of Poisson fields. We study the particular case =[–L, L] andV(x, y)=}- cos(x–y),L}>0, }>0.References     

Measurement of the refractive index and optical absorption spectra of epitaxial bismuth substituted yttrium iron garnet films at uv to near-ir wavelengths

Refractive-index and optical-absorption spectra of Bi-substituted yttrium iron garnet films, epitaxially grown by liquid-phase epitaxy, have been measured in the spectral regime 0.26 m1.9 m by thin-film interference for 0.52 m and by ellipsometry for0.52 m. The Y_3–x–y Bi _x Pb _y Fe_5–z Pt _z O₁₂ films contain bismuth in the range Ox 1.42, lead in the range 0.01 y0.08 and platinum in the range 0.005<=z0.03. There is satisfactory coincidence between the results from ellipsometry and thin-film interference in the overlapping wavelength region. The materials investigated are the same as reported earlier from this laboratory in ter mof their magnetic and magnetooptic properties.     

On Strong Superadditivity of the Entanglement of Formation

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation E_F and the conjugate function E^* of the entanglement function E()=S(Tr_A). We then consider the conjectured strong superadditivity of the entanglement of formation E_F()E_F(_I)+E_F(_II), where _I and _II are the reductions of to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E^*. Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1.     

New class of Finsler metrics

A class of Finsler spaces is introduced which is determined by the metric functionF(x, y)=[ +kB B )y y ]^1/2, whereB =B x, y andk is a constant. Various properties of these spaces are developed. A particular choice ofB is shown to produce a geodesic equation which is equivalent to the Lorentz equation of motion for a charged particle. Some general arguments for the physical applicability of Finsler spaces are also given.     

Determination of random variations in ionosphere height from phase fluctuations of VLF signals

Conclusions A feature of our problem of restoration of fluctuations in ionosphere height is that the experimental data(t, _k) obtained at a fixed reception point are functions of time, whereas the function ₁(u, y₂) to be restored is a function of the coordinates. If we use the assumption that the irregularities migrate transversely, coordinate y₂ can be exchanged for time t. Restoration with respect to the second coordinate u=x_i–x₀/2 is in effect obtained by using data(t, _k) on some set of carrier frequencies.The resultant solution of the restoration problem is in the form of an expansion of ₁(u, y₂) in known functions determined from the observed data(t, _k). We have evaluated the solution accuracy, which depends on the overall power signal-to-noise ratio at all frequencies used. We have demonstrated that the restoration algorithm contains an optimum number (with respect to accuracy) of coefficients to be evaluated.Khar'kov Aviation Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 20, No. 8, pp. 1138–1145, August, 1977.     

Mixing properties,differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature

For the Ising model with nearest neighbour interaction it is shown that the spin correlations _{A B} - _{A B} decrease exponentially asd(A, B) in a pure phase when the temperature is well belowT _c. This is used to prove that the free energyF(,h) is infinitely differentiable in and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(,h).     

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ _x,y const/¦x–y¦². For translation-invariant systems with well-defined lim _x x ² J _x =J ⁺ (possibly 0 or ) we establish: (1) There is no long-range order at inverse temperatures withJ ⁺1. (2) IfJ ⁺>q, then by sufficiently increasingJ ₁ the spontaneous magnetizationM is made positive. (3) In models with 0<J ⁺< the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM( _c )1/( _c J ⁺)^1/2. (4) For Ising (q=2) models withJ ⁺<, it is noted that the correlation function decays as xy()c()/|x–y|² whenever< _c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.     

Optical Pumping of Samarium Atoms in a Bichromatic Laser Field in the Presence of Velocity-Changing Collisions

Dark resonances in the ¹⁵⁴Sm -system 4f ⁶6s ²(⁷ F ₀) 4f ⁶6s6p(⁹ F ₁ ⁰) 4f ⁶ s ²(⁷ F ₁) are observed alongside the velocity selective optical pumping. The shape of the resulting spectra strongly depended on the buffer gas (He, Ar) pressure due to velocity-changing collisions (VCC): the sign of the effect could be reversed from the dark to the bright resonance. The observed spectra are interpreted within the framework of the hard-sphere collision model. The role of VCC in the formation of the dark state in the -system is discussed.     

Random walk with persistence

The exact analytic result is obtained for the Fourier transform of the generating functionF(R,s)= _n=0 s ⁿ P(R,n), whereP(R,n) is the probability density for the end-to-end distanceR inn steps of a random walk with persistence. The moments R ²(n), R ⁴(n), and R ⁶(n) are calculated and approximate results forP(R,n) and R ^–1(n) are given.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

The contrast gain in a guest-host nematic display due to a periodic boundary condition

The mean square tilt angle of a nematic slab with finite anchoring energy and periodic boundary conditions has been theoretically investigated, as a function of the slab geometry and of the reduced extrapolation length. If the anchoring strength is free-surfacelike, the contrast is affected by a loss 10% at room temperature if the ratio between the anchoring pitch and the cell thickness is 0.5.Glossary anchoring pitch - h cell thickness - /h - ( = x/, = y/h) reduced coordinates - (, ) local tilt angle - elastic constant - w_a anchoring energy anisotropy - b=/w _a de Gennes-Kleman extrapolation length - B=b/h reduced extrapolation length - T _NI nematic-isotropic transition temperature - :=(T/T _NI ) – 1 reduced temperature - easy axis direction - _MAX - _± ² mean square tilt angle along the boundary - () absorbance coefficients of the p-dye - r /: dichroic ratio - c contrast - G contrast gain - S order parameter     

Theory of fiber bundle with local metric of internal space and gravitation with torsion

In this paper we propose a new theory of a fiber bundle provided with a local metric of internal space. The fibers differ from usual fibers, having an enlarged factor. The enlargement may be procured by a differential mapping(x) from structure groupG to the fiberF _x atx M, and(x)R. The torsion presented stems from the local metric of internal space and the local metric stems from a induced mapping _*(x) of(x). From the theory we can get the Brans-Dicke theory with torsion. If we assume the spin density of the gauge field determines the enlarged factor of the fiberF _x, our theory is an extended Cartan theory.     

Functional equation for dynamical zeta functions of Milnor-Thurston type

A Milnor-Thurston type dynamical zeta function _L (Z) is associated with a family of maps of the interval (–1, 1). Changing the direction of time produces a new zeta function _L (Z). These zeta functions satisfy a functional equation _L (Z) _L (Z)= ₀(Z) (where amounts to sign changes and, generically,₀1). The functional equation has non-trivial implications for the analytic properties of _L (Z).     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.