On the ζ-function of a one-dimensional classical system of hard-rods

The -function of a one-dimensional classical hard-rod system with exponential pair interaction is defined as the generating function for the partition function of the system with periodic boundary conditions. It is shown, here, that the -function for this system is simply related to the traces of the restrictions of the Ruelle's transfer matrix, and related operators to a suitable function space. This -function does not, in general, extend to a meromorphic function.     

Resistivity exponent of two-dimensional lattice animals

We calculate the average resistanceR(L) of lattice animals spanningL×L cells on the square lattice using exact and Monte Carlo methods. The dynamical resistivity exponent, defined asR(L) L , is found to be =1.36±0.07. This contradicts the Alexander-Orbach conjecture, which predicts 0.8. Our value for differs from earlier measurements of this quantity by other methods yielding =1.17±0.05 and 1.22±0.08 by Havlin et al.On leave from the Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China.     

Three-Mode Entangled State Representation of Continuum Variables and Optical Four-Wave Mixing

We establish a new three-mode entangled state representation , of continuum variables, which make up a complete set. Using optical four-wave mixing and a beam splitter transform we can prepare , . Based on , a new number-difference--operational-phase uncertainty relation is established and the corresponding squeezing dynamics is discussed.     

On the stability of spin waves at finite temperatures

The temperature dependent exchange stiffnessD, defined for long wavelength spin waves by=Dq ², is calculated for very weak itinerant ferromagnets. It is found that for a general single band of d-electrons,D reduces to a very simple form givingD(T)=D(0)/ ₀ in the limit ₀0. The stability condition for spin waves at finite temperatures is thenD(0) > 0 and/ ₀ > 0, where is the relative occupation ± spin sub-bands and ₀ its value at 0°K.The authors are grateful to E. P. Wohlfarth for helpful discussions.     

Partition function zeros for the one-dimensional ordered plasma in Dirichlet boundary conditions

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacity in an applied electric fieldE with Dirichlet boundary conditions. The system has a phase transition from a low-coupling phase with equally spaced particles to a high-coupling phase with particles clustered into dipolar pairs. An exact expression for the partition function is developed. In zero applied field the zeros in the plane occupy the imaginary axis from –i to –i_c and i_c to i for some _c. They also occupy the diamond shape of four straight lines from ±i_c to _c and from ±i_c to –_c. The fugacity acts like a temperature or coupling variable. The symmetry-breaking field is the applied electric fieldE. A finite-size scaling representation for the partition in scaled coupling and scaled electric field is developed. It has standard mean field form. When the scaled coupling is real, the zeros in the scaled field lie on the imaginary axis and pinch the real scaled field axis as the scaled coupling increases. The scaled partition function considered as a function of two complex variables, scaled coupling and scaled field, has zeros on a two-dimensional surface in a domain of four real variables. A numerical discussion of some of the properties of this surface is presented.     

Determinants of Laplacians on surfaces of finite volume

Determinants of the Laplace and other elliptic operators on compact manifolds have been an object of study for many years (see [MP, RS, Vor]). Up until now, however, the theory of determinants has not been extended to non-compact situations, since these typically involve a mixture of discrete and continuous spectra. Recent advances in this theory, which are partially motivated by developments in mathematical physics, have led to a connection, in the compact Riemann surface case, between determinants of Laplacians on spinors and the Selberg zeta function of the underlying surface (see [DP, Kie, Sar, Vor]).Our purpose in this paper is to introduce a notion of determinants on non-compact (finite volume) Riemann surfaces. These will be associated to the Laplacian shifted by a parameters(1–s), and will be defined in terms of a Dirichlet series (w, s) which is a sum that represents the discrete as well as the continuous spectrum. It will be seen to be regular atw=0, and our main theorem (see Sect. 1) will express exp as the Selberg zeta function of the surface times the appropriate -factor.A Sloan Fellow and partially supported by NSF grant DMS-8701865     

Self-avoiding walks in quenched random environments

The self-avoiding walk in a quenched random environment is studied using real-space and field-theoretic renormalization and Flory arguments. These methods indicate that the system is described, ford _c =4, and, for large disorder ford>d _c , by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path. This fixed point is characterized by scaling laws for the size of the walk,LN ^p withN the number of steps, and the fluctuations in the free energy,fL ^p. The bound 1/-d/2 is obtained. Exact results on hierarchical lattices yield> _pure and suggests that this inequality holds ford=2 and 3, although= _pure cannot be excluded, particularly ford=2. Ford>d _c there is a transition between strong and weak disorder phases at which= _pure. The strong-disorder fixed point for SAWs on percolation clusters is discussed. The analogy with directed walks is emphasized.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

Gravitomagnetic and Gravitoelectric Waves in General Relativity

Lower-order terms in expansions of the equations of General Relativity in powers of v/c (post-Newtonian approximations) have long been a source of analogies with em theory. A classic textbook example is the steadily spinning sphere generating a constant dipole gravitomagnetic field, with its associated vector potential B^* ₀ = × (analog of the magnetic field B of a spinning charged sphere). In the nonsteady case there are associated gravitoelectric fields E* = – _t – * also, where * is the gravitational Coulomb potential. The case of a rigid sphere spun up from rest by an external (nongravitational) torque at t = 0 is enlightening, as it demonstrates the generation of B* and E* wave fields propagating outward with the velocity of light c: for large t, B* B^* ₀. In a coordinate system for which the metric tensor is nearly equal to the Minkowski tensor, the three-vector potential obeys an equation isomorphic to the electrodynamic equation, that is, ² = –*j* with j* = –v, where is the mass density, v the three-velocity, and * = 16Gc^–2 = 3.7 × 10^–26 mksu, G being the gravitational constant. Significantly, one can construct a gauge invariant four-vector potential F* = (ic^–14*, ), obeying field equations isomorphic to Maxwell's in the Lorentz gauge F _, = 0. The traveling transient dipole field exerts torques on matter in its path, setting up shear strains that may be measurable for very large momentum transfers, for example, between massive astronomical bodies. A rough calculation suggests that such strains are in principle observable.     

Proof of a retroactive influence

Quantum theory predicts that, e.g., in a Stern-Gerlach experiment with electrons the measured spin component does not come about by an adjustment at the last moment, a forced flipping or tilting of the spin (vector), which would imply z-angular momentum exchange between particle and instrument, but will afterward appear to have had the value already before the measurement. Because an electron spin cannot have components in all directions at the same time, the measuring direction has a privileged status before the measurement, however we choose that direction, which implies a retroactive effect. A second proof of retroactivity is derived from a special case of the paradox of Einstein, Podolsky, and Rosen. It is strongly suggested by our result that, in essential respects, both Bohr and Einstein were right in their famous controversy about determinism and considering microprocesses as a whole.     

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).     

Problems with the ζ-function method

The -function method for evaluating effective actions is applied to an operator containing a potential localized on two parallel planes. This operator is characterized by a continuous spectrum and broken translation invariance. In this case, the -method leads to a divergent volume factor independent of the physical parameters. In a suitable regularization scheme, only the next to leading order term reproduces a physically interesting result.     

Low density flux fluids in high —T c superconductors: The influence of disorder

We analyze the influence of thermal and frozen-in disorder on the flux line (FL) density close to the lower critical fieldH _c1. Arguments which consider the steric repulsion of fluctuating FLs give with the roughness exponent of a single FL andd the space dimensionality. We show by a phenomenological scaling approach and a renormalization group treatment, that this is correct only fordd _c =2/–1, i.e. for . Ford>d _c the steric FL repulsion at scales more than some critical one is irrelevant and . For disordered superconductorsd _c =2 and ford=2, 3. We also found the melting line for a FL lattice in the presence of frozen-in impurities close toH _c1.     

LargeN expansion of QED: asymptotic photon propagator and contributions to the muon anomaly,for any number of loops

We calculate analytical contributions to then-loop asymptotic photon propagator from diagrams withn–1 electron loops, i.e. theO(1/N) terms in the largeN limit. The corresponding contributions to the on-shell -function, ()=6 log / logm reduced to rational combinations of _s = _p p ^–s . For the -function of the MOM scheme (i.e. the Gell-Man-Low function) we obtain theO(1/N) terms of

An Equation for the Dissipation Rate Correlation and Its Implications for the Intermittency Exponent μ in Turbulence

We derive an equation satisfied by the dissipation rate correlation function, for the homogeneous, isotropic state of fully-developed turbulence from the the Navier–Stokes equation. In the equal time limit we show that the equation leads directly to two intermittency exponents ₁=2– ₆ and ₂=z₄– ₄, where the 's are exponents of velocity structure functions and z₄ is a dynamical exponent characterizing the fourth order structure function. We discuss the contributions of the pressure terms to the equation and the consequences of hyperscaling.     

Positive similarity solutions for a discrete velocity boltzman coagulation-fragmentation model

We consider the coagulation-fragmentation model of Slemrod and coworkers which is essentially the two-dimensional Broadwell model including inelastic collisions. We construct two classes of similarity solutions (variable =x -t), positive for (- , ): the Rankine-Hugoniot solutions and the scalar Riccati similarity solutions. Previous solutions were built up with positivity along half of thex-axis. For the two classes, we determine in the parameter space, building up the solutions, domains corresponding to positive solutions.     

Modified London model for the determination of λ and ζ in a high κ superconductor

We present a modified London model suggested by Brandt [1–3] which introduces a finite vortex core size appropriate for isotropic superconductors in which the average internal field is less than approximately (1/4)H _c2. TheSR lineshape resulting from this model possesses a distinctive shape due to the magnetic penetration depth and the vortex core diameter (approximately equal to twice the coherence length ). However, for a given lineshape, there is a large range of values of and which produce nearly the same lineshape. Lineshape smearing caused by disorder in the vortex lattice increases uncertainty in values for and . If well-determined values of either (T) or (T) are not available from another technique, both of them can be determined bySR measurements alone if runs in more than one applied field at the same temperature are fit with and as shared parameters. We also present our method of estimating the degree of disorder in the vortex lattice.     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN     

ГАЛЬВАНОМАГНИТНЫЙ Э ФФЕКТ В ОБЛАСТИ ТЕМПЕРАТУРЫ КЮРИ

. - II- . .     

On Multifractality and Fractional Derivatives

It is shown phenomenologically that the fractional derivative = D u of order of a multifractal function has a power-law tail in its cumulative probability, for a suitable range of 's. The exponent is determined by the condition , where _p is the exponent of the structure function of order p. A detailed study is made for the case of random multiplicative processes (Benzi et al., Physica D 65:352 (1993)) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the _p graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g., because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data.