An upper bound on the critical temperature for a continuous system with short-range interaction

A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(, z) denotes the thermodynamic pressure at inverse temperature and activityz, then it follows from the Mayer expansion thatp(, z) is infinitely differentiable provided andz are sufficiently small. Here it is shown that there exists ₀>0 such thatp(, z) is infinitely differentiable if< ₀ andz>0. One can interpret this result as saying that ( ₀)^–1 is an upper bound on the critical temperature for the system.     

Exact distribution of cluster size and perimeter for two-dimensional percolation

Exact series expansion data of Sykes et al. are used to calculate the average numberc _n and perimeters _n of clusters of sizen20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp _n we find a sharply peaked distribution of perimeterss _n with mean s _n =((1–p _n )/p _c )n+O(n ) and width s _n ² –S _n ²n ^1.6 where1/=0.39. This perimeter s _n should not be interpreted as a cluster surface in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc _n withn is consistent with the postulated asymmetry aboutp _c : logc _n –n forn with1 forp<p _c and1/2 forp>p _c .     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Glueball spectroscopy in strongly coupled lattice gauge theories

We study the mass spectrum up to –7 (1–) log of pure three-dimensional lattice gauge theories with action (g _P) for real irreducible and small . Besides the lowest excitationm ₀–4log, we find two nearly degenerate excited statesm ₁,m ₂ withm _i–6log (i=1, 2) and (m ₁–m ₂) at leastO().Work partially supported by CNPq (Brasil)     

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by _m (L)=(Inq/E)L ^–d +O(L ^–2d ) with respect to the infinite-volume transition point _t . We also propose an alternative definition of the finite-volume transition temperature _t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections from _t .     

Four-parametric regular black hole solution

We present a regular class of exact black hole solutions of the Einstein equations coupled with a nonlinear electrodynamics source. For weak fields the nonlinear electrodynamics becomes the Maxwell theory, and asymptotically the solutions behave as the Reissner–Nordström one. The class is endowed with four parameters, which can be thought of as the mass m, charge q, and a sort of dipole and quadrupole moments and , respectively. For 3, 4, and |q|2s _c m the corresponding solutions are regular charged black holes. For = 3, they also satisfy the weak energy condition. For = = 0 we recover the Reissner–Nordström singular solution and for = 3, = 4 the family includes a previous regular black hole reported by the authors.     

Representations of an algebra of the Gell-Mann-Dashen type

LetG denote the infinitedimensional Lie algebra given by communtation relations [a _m ,a _n ] =c a _m+n (m, n=0, 1, ...), wherec are structure constants of the arbitrary centerless Lie algebra. The paper is devoted to the construction of a certain class of skewsymmetric irreducible representations (so called F-representations) of the algebraG .     

Higher quantum conserved current in a new completely integrable model

The first higher local quantum conserved current in the recently proposed new completely integrable (2 e+e^-2)₂ model is explicitly constructed thus proving absence of particle production and factorization of multiparticle scattering.     

Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

Gentle perturbations

By introducing a specific type of perturbation,A, in the Hamiltonian, we define a class of gently perturbed states, ,A, of a canonical ensemble, . The perturbations are chosen so as to preserve a relationship of the form ,A constant ×. Applications in ergodic theory and phase transitions are described.     

Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems

Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(–2mH) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn _s (x ₁, ...,x _s ; ,z) are analytic in the two variables andH if the reciprocal temperature >0 and the complex numberH is not a limit point of zeros of the grand partition function , and (ii) the Ursell functionsu _s (x ₁, ...,x _s ; ,z) tend to 0 as _s Max _{i, j} |x _i –x _j | if >0 and ReH0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value , then |u _s |<C exp [(–2 m |ReH|+) _s /] where is any small positive number andC is independent of _s . One consequence of the result (i) is that a phase transition can occur as is varied at constantH only ifH is a limit point of zeros of (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant .For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided >0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n _s andu _s must be replaced by their values integrated over small volumes surrounding each of the pointsx ₂, ...,x _s .It is shown that the pressurep is analytic in both andz, if it is analytic inz at fixed over a suitable range of values of andz, and further that, except for continuum systems without hard cores,p,n _s andu _s have convergent Maclaurin expansions in for small enoughz.Supported by the U.S. Air Force Office of Scientific Research under grant no. AF 68-1416.     

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.     

Coulomb pair density matrix I

The Coulomb pair density matrixG (r, r) for attractive and repulsive potentials is not only interesting for determining the two-particle effective potentials, but it is also essential in numerical studies of quantum systems. A high-temperature approximation is obtained for logG (r, r), in the form of simple integrals or series expansions; large-distance expansions are also given.     

On orthogonal and symplectic matrix ensembles

The focus of this paper is on the probability,E (O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (=1) and Gaussian Symplectic (=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

A Boltzmann map for quantum oscillators

We define a map on the space of quasifree states of the CCR or CAR of more than one harmonic oscillator which increases entropy except at fixed points ofx. The map is the composition of a doubly stochastic map T^* and the quasifree reductionQ. Under mixing conditions onT, iterates of take any initial state to the Gibbs state, provided that the oscillator frequencies are mutually rational. We give an example of a system with three degrees of freedom with energies ₁, ₂, and ₃ mutually irrational, but obeying a relation n₁₁+n₂₂=n₃₃,n _i . The iterated Boltzmann map converges from an initial statep to independent Gibbs states of the three oscillators at betas (inverse temperatures) ₁,₂, ₃ obeying the equation n₁₁₁+n₂₂₂=n₃₃₃. The equilibrium state can be rewritten as a grand canonical state. We show that for two, three, or four fermions we can get the usual rate equations as a special case.     

Classification of the Ricci tensor

This paper contains a classification of the Ricci tensorR . The method of derivation is analogous to the spinor version of the Petrov classification of the Weyl tensor. It is shown how the various classes are related to the number and type of eigenvectors and eigenvalues ofR . The classification is useful in the geometrization of various fields. The case of a real scalar field is treated in detail.Supported in part by the National Research Council of Canada.     

Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide

The propagation of electromagnetic surface waves along a radially inhomogeneous dielectric waveguide is investigated. The problem is formulated in terms of differential equations to be satisfied by the radially dependent parts of the electromagnetic field vectors. The dielectric waveguide is assumed to consist of a homogeneous cladding of infinite extent and a radially inhomogeneous core of higher permittivity. Numerical solutions of the differential equations in the core are obtained by two different methods, viz. by direct numerical integration and by substitution of an appropriate power series expansion. In the cladding the field is expressed in terms of modified Bessel functions. Imposing the boundary conditions at the interface of core and cladding, an equation for the unknown propagation coefficient is obtained. From this equation the propagation coefficients for the lower order modes are computed numerically. Numerical results are presented for some permittivity profiles of practical interest in single-mode transmission along optical fibres.Nomenclature a radius of the core - a _q vector coefficient in the power series expansion off() - A _i constants - A square matrix - b _q coefficient in the power series expansion of _r() - B square matrix - C square matrix - c _n unknown constant - d _n unknown constant - D() fundamental matrix - E,E _r,,z electric field vector and components - E,e _r,,z radially dependent parts ofe _r,,z - f solution vector - G _q square matrix - H,H _r,,z magnetic field vector and components - h _r,,z radially dependent parts ofE _r,,z - h reduced wavenumber - i radial mode number - j imaginary unit - k _0,m wave number - K _n modified Bessel function of the second kind and order n - n azimuthal mode number - t time - U normalized propagation constant - Z _m plane wave impedance of the cladding - r, , z cylindrical co-ordinates - p, q, s integers - propagation coefficient - increment - _0,m,r permittivity - normalized radiusr - ₀ wavelength in free space - ₀ permeability - angular frequency - d _r, differentiation with respect tor, Engineering and Professor H. J. Frankena of the Physics Department for their valuable discussions.     

The Critical Attractive Random Polymer in Dimension One

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e ^– for every self-intersection and e ^/(2d) for every self-contact to the probability of an n-step simple random walk on ^d , where , >0 are parameters. It is known that for d=1 and > the chain collapses down to finitely many sites, while for d=1 and < it spreads out ballistically. Here we study for d=1 the critical case = corresponding to the collapse transition and show that the end-to-end distance runs on the scale _n = (log n)^–1/4. We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size _n times a constant. Moreover, we derive the asymptotics of the partition function.     

On the equivalence of the relativistic theories of gravitation

Einstein's equations are rewritten in terms of a certain torsionless linear connection which differs, in general, from the Levi-Civita metric connection . The torsionless connection appears in a natural way as the canonical momentum of the gravitational field g . Einstein's equations have a simple interpretation in terms of the connection . The equivalence of the so-calledpurely metric, purely affine, andmetricaffine theories of gravitation is proved.This work has been written under the financial support of Gruppo Nazionale per la Fisica Matematica of the Italian National Research Council.