On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.     

Attractor States and Quantum Instabilities in de Sitter Space

The asymptotic behavior of the energy–momentum tensor for a free quantized scalar field with mass m and curvature coupling in de Sitter space is investigated. It is shown that for an arbitrary, homogeneous, and isotropic, fourth-order adiabatic state for which the two-point function is infrared finite, T _ab approaches the Bunch–Davies de Sitter invariant value at late times if m ² + R > 0. In the case m = = 0, the energy–momentum tensor approaches the de Sitter invariant Allen–Folacci value for such a state. For m ² + R = 0 but m and not separately zero, it is shown that at late times T _ab grows linearly in terms of cosmic time leading to an instability of de Sitter space. The asymptotic behavior is again independent of the state of the field. For m ² + R < 0, it is shown that, for most values of m and , T _ab grows exponentially in terms of cosmic time at late times in a state dependent manner.     

Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x =lim _L (L)/L and the first four magnetization moment ratios V _2n = ²ⁿ / ^{2 n} . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*_2n . We confirm these predictions by a high-precision Monte Carlo simulation.     

On the dynamics of excitations in disordered systems

A new, time-local (TL) reduced equation of motion for the probability distribution of excitations in a disordered system is developed. ToO(k²) the TL equation results in a Gaussian spatial probability distribution, i.e, P(r, t) = [(2)^1/2]^–dexp(-r²/2²), where = (t) is a correlation length, andr = ¦r¦. The corresponding distribution derived from the Hahn-Zwanzig (HZ) equation is more complicated and assumes the asymptotic (r ) form: P(r, s)(s ^d )^–1exp(–r/) · (r/)^(1-d)/2 where = (s),d is the space dimensionality, ands is the Laplace transform variable conjugate tot. The HZ distribution generalizes the scaling form suggested by Alexanderet al. ford= 1. In the Markov limit (t)t, (s)1/s, and the two distributions are identical (ordinary diffusion).     

Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ; this result is translated into the GHP formalism using a new generalised Lie derivative operator with respect to a vector field . We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator in the presence of a Killing vector field . This new operator also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.     

Lorentzian Warped Products and Singularity

We show that to any convex function f: ⁿ there correspondinfinitely many geodesically complete metricsds² such that Ric() 0 for anynonspacelike vector . These metrics are constructedas the warped products of the natural metric in and the inner metric of a convexhyperface (the graph of f) in ^{n + 1}.     

Acoustomagnetoelectric effects at meso-ultrasonic frequencies in anisotropic semiconductors in arbitrary (classical) magnetic fields

Anisotropic acoustomagnetoelectric (AME) effects at meso-ultrasonic frequencies are calculated analytically in semiconductors with an anisotropic mobility () in arbitrary classical magnetic fields. For Bq(q is the ultrasonic wave vector) and an arbitrary direction of q two transverse components of the AME field (E ^B q E _y ^B ) occur in the crystal, and the longitudinal acoustoelectric field changes under the action of a longitudinal magnetic field (E _q ^B =E _q ^B -E _q ⁰ ),E ^B is even, and E ^B is odd in B; for B 1 the component E _y ^B E ^B /B, andE ^B and E _q ^B are independent of B and can be commensurate with the zero-field acoustoelectric field E _q ⁰ if the anisotropy of is large (hexagonal ZnS and ZnO or n-Ge highly compressed along [111]). The transverse AME field E _st ^B is calculated in the configuration E _st ^B qBE _st ^B (standard AMEeffect). For B >> 1 the field B 1E _st ^B B ^–3, so thatE ^B , E _y ^B , and _q ^B can be greater than E _st ^B here. The acoustoelectric analog of the Grabner effect (E _G ^B ), i.e., the component of the AME field along a transverse magnetic field (E _G ^B Bq) is also calculated. For pB > 1 the componentE _G ^B B ^–3.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 93–97, June, 1989.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

Crossover finite-size scaling at first-order transitions

In a recent paper we developed a method which allows one to control rigorously the finite-size behavior in long cylinders near first-order phase transitions at low temperature. Here we apply this method to asymmetric transitions with two competing phases, and to theq-state Potts model as a typical model of a temperature-driven transition, whereq low-temperature phases compete with one high-temperature phase. We obtain the finite-size scaling of the firstN eigenvalues (whereN is the number of competing phases) of the transfer matrix in a periodic box of volumeL × ... ×L ×t, and, as a corollary, the finite-size scaling of the shape of the order parameter in a hypercubic box (t=L), the infinite cylinder (t=), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover length _L is given by O(L^w)exp(L^v), where is the inverse temperature, is the surface tension, and w=1/2 if v+1=2 whilew=0 if v+1 >2. For the standard Ising model we also consider free boundary conditions, showing that _L=exp[L^v+O(L^v– 1)] for any dimension v+12. For v+1=2 we finally discuss a class of boundary conditions which interpolate between free (corresponding to the interpolating parameter g=0) and periodic boundary conditions (corresponding to g=1), finding that _L=O(L^w)exp(L ^v) withw=0 forg=0 andw=1/2 for 0<g1.     

On a ferromagnetic spin chain

The quotient (s-1)/(s) of Riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures.     

Yang-Lee theory and the conductor-insulator transition in asymmetric log-potential lattice gases

A feature of a conducting phase at low density is that there is a singularity in the fugacity expansion of the pressure, whereas the same expansion in the insulating phase gives an analytic series. The Yang-Lee characterization of a phase transition thus implies that in the conducting phase the zeros of the grand partition function must pinch the real axis in the complex scaled fugacity () plane at =0, whereas in the insulating phase a neighborhood of =0 must be zero free. Exact and numerical calculations are presented which suggest that for two-component log-potential lattice gases in one dimension with dimensionless coupling, the zeros pinch the point =0 for<2, while for2 a neighborhood of =0 is zero free. The conductor-insulator transition therefore takes place at=2 independent of the density and other parameters in the model.     

Radiative decays of the Z boson

We calculate the Stokes parameters of the photons produced in the decays of neutral vector bosons Z, Z 1+¯1+ and Z q+¯q+, wherel=e, , or , and q is a quark. In the decays of unpolarized Z bosons (with the production of unpolarized leptons or quarks) the nonzero Stokes parameters for ₂ (circularly polarized photons) and ₃ (linearly polarized photons). The magnitude of ₃ does not depend on the parameters of the netural weak current of the leptons and the quarks (if their mass is neglected). The anomalous magnetic moment of the Z boson is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–43, February, 1986.     

On the Born-Oppenheimer approximation for excited states of polyatomic Schrödinger operators

In this Letter, excited states of polyatomic molecular Schrödinger operators are investigated with the help of the Born-Oppenheimer approximation. The ratio of electronic and nuclear mass plays the role of a semi-classical parameter h ². Asymptotic series of eigenvalues at the bottom of the spectrum are constructed up to any order in h. Mathematically, this leads to the discussion of the semi-classical limit of pseudo-differential operators with the principal symbol po(x,) = ² + , where has a degenerate minimum (a whole manifold).     

Isometric motions of a perfect charged fluid

Space-times with perfect charged fluids as sources, that admit groups G_r of isometric motions, are investigated. It is assumed that the velocity vector of the fluid is collinear to the timelike Killing vector ⁱ of group G_r. It is shown that the macroscopic motion of a perfect charged fluid can occur only in the direction of such a Killing vector ⁱ that defines an operator in an invariant subgroup or, in particular, an operator of the center of the group. Parametric representations of the generalized equations of state for the pressure p, the energy density of the fluid, , and the electric charge density are established. All these quantities are functions of the norm of the Killing vector ⁱ and the projection of the 4-potential of the electromagnetic field onto this vector ⁱ. In the approximation of the weak field in the coordinate system where ⁱ= ₄ ⁱ , these functional dependences imply that p, , and are functions of Newtonian and electrostatic potentials.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 25–29, October, 1987.     

On nonlinear stationary half-space problems in discrete kinetic theory

We show that every steady discrete velocity model of the Boltzmann equation on the real line, _i·(d/dx)f ⁱ=C ⁱ(f), which satisfies anH-theorem and for which all _i0, has solutions on the half-line (0, ) which take prescribed non-negativef ⁱ(O) if _i>0 and approach a certain manifold of Maxwellians asx. Such solutions give the density distribution in a Knudsen boundary layer in the discrete velocity case.     

High frequency gravitational radiation in Kerr-Schild space-times

Vaidya has obtained general solutions of the Einstein equationsR _ab= _{a b} by means of the Kerr-Schild metricsg _ab= _ab +H _{a b} . The vector field _a generates a shear free null geodetic congruence both in Minkowski space and in the Kerr-Schild space-time. If in addition it is hypersurface orthogonal, the Kerr-Schild metric may be interpreted as the background metric in a space-time perturbed by a high frequency gravitational wave. It is shown that Vaidya's solutions satisfying this additional condition are of only two types: (1) Kinnersley's accelerating point mass solution and (2) a similar solution where a space-like curve plays the role of the time-like curve describing the world line of the accelerating mass. The solution named by Vaidya as the radiating Kerr metric does not satisfy the hypersurface orthogonal condition.Supported in part by National Science Foundation Grant MPS 741029.     

Fluctuation-enhanced conductivity and magnetoconductivity of high-quality YBa2Cu3O7−δ crystals

We present measurements of the in-plane resistivity _ab of YBa₂Cu₃O_7– single crystals withT _c 92 K and _ab (100 K)50 cm. The temperature dependence of the fluctuation conductivity and of the magnetoconductivity aboveT _c is analyzed in terms of direct and indirect fluctuation contributions for layered superconductors. The combination of fluctuation conductivity and magnetoconductivity allows to determine both coherence lengths _ab (0) and _c (0) as well as the phase-relaxation time of the pairs in an unequivocal manner. Evidence for clean limit type-II superconductivity in our crystals is given by large values of the mean free pathl _{ab ab} (0).Dedicated to Prof. Dr. F. Hund on the occasion of his 95th birthday     

Multiatom interactions,order and stability in binary transition metal alloys

Interface delocalization or depinning transitions such as wetting or surface induced disorder are considered. At these transitions, the correlation length for transverse correlations parallel to the surface diverges. These correlations are studied in the framework of Landau theory. It is shown the t^–1/2 at all types of transitions for systems with short-range forces wheret measures the distance from bulk coexistence.     

The Countable Number of Models in the Cosmology with Nonminimal Coupling and Torsion

Cosmological models of flat space with a nonminimally coupled scalar field and ultrarelativistic gas are studied within the Einstein–Kartan theory. Exact general solutions are derived for two-component models and those containing only scalar field for an arbitrary coupling constant . It is shown that both singular and countable number of nonsingular models is possible depending on the type of scalar field and the sign of . The special values of and restrictions on are found for the above solutions. The role of relativistic gas in the evolution of models is revealed.