Asymptotic behaviour of the positional and mixed Ursell correlation functions in classical polar fluids

Rigorous asymptotic expressions have been obtained for the positional and mixed Ursell correlation functions in classical equilibrium systems of particles with dipole-dipole interaction at large distance.     

On the exponential decay of correlation functions

We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.     

Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ⁴ models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ⁴ field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.     

Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.     

Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>T _c). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.     

Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Bose-Einstein transition in a dilute interacting gas

We study the effects of repulsive interactions on the critical density for the Bose-Einstein transition in a homogeneous dilute gas of bosons. First, we point out that the simple mean field approximation produces no change in the critical density, or critical temperature, and discuss the inadequacies of various contradictory results in the literature. Then, both within the frameworks of Ursell operators and of Green's functions, we derive self-consistent equations that include correlations in the system and predict the change of the critical density. We argue that the dominant contribution to this change can be obtained within classical field theory and show that the lowest order correction introduced by interactions is linear in the scattering length, a, with a positive coefficient. Finally, we calculate this coefficient within various approximations, and compare with various recent numerical estimates. Received 15 July 2001     

One-dimensional Ising model in arbitrary external field

The external potential needed to produce an arbitrary equilibrium density profile for a one-dimensional lattice gas with nearest neighbor interactions is solved exactly. The resulting sequence of direct correlation functions is shown to be of short range, and in the ferromagnetic case the even members alternate in sign at zero spin. The even Ursell distributions in this case likewise alternate in sign.Supported in part by U.S. ERDA under contract E(11-1)-3077.     

Equilibrium properties of a multi-component ionic mixture: I. Sum rules for correlation functions

Equilibrium statistical methods are used to derive sum rules for two- and three-particle correlation functions of a multi-component ionic mixture. Some of these rules are general consequences of the electrostatic character of the interaction, whereas others depend on specific thermodynamic properties of the system. The first group of rules follows from the BBGKY hierarchy and a clustering hypothesis for Ursell functions. The sum rules of the second group are obtained by describing the system with the help of a restricted grand-canonical ensemble in which the particle numbers of the various components in the mixture fluctuate under the condition that the total charge in the system remains constant.     

A spin-coefficient approach to Weyssenhoff fluids in Einstein-Cartan theory

The generalized Newman-Penrose formalism is used to analyze semiclassical aligned spin fluids satisfying the Weyssenhoff restriction in the framework of Einstein-Cartan theory. Some general properties are derived and the formalism is then used to obtain two classes of exact solution. One has a flat metric, but the fluid has in general nonzero acceleration, expansion, and shear. It is characterized by two arbitrary constants and two functions of two variables satisfying one partial differential equation. In the other class the fluid has nonzero acceleration and vorticity, and the free gravitational field is of typeD. It is characterized by three arbitrary constants and an arbitrary function of two spacelike coordinates.     

Stochastic aspects in the theory of spectral-line broadening. II. Noncommutative cluster expansions

Generalizing ideas of von Waldenfels we develop a systematic procedure to define truncatedn-point operators which are reminiscent of Ursell functions of statistical mechanics. The truncation procedure is adapted to factorization relations obeyed by the operators in question. The results are applied to spectral-line broadening in plasmas. We derive cluster expansions for the line-shape function in terms of these truncated operators, where the ions are treated quasistatistically. The first order approximation for the line-shape function is discussed. The results are carried over to several moving perturber species, in particular to nonquasistatic ions.     

The Feynman-Schwinger Representation in QCD

The proper time path integral representation is derived explicitly for Green's functions in QCD. After an introductory analysis of perturbative properties, the total gluonic field is separated in a rigorous way into a nonperturbative background and valence gluon part. For nonperturbative contributions the background perturbation theory is used systematically, yielding two types of expansions, illustrated by direct physical applications. As an application, we discuss the collinear singularities in the Feynman-Schwinger representation formalism. Moreover, the generalization to nonzero temperature is made and expressions for partition functions in perturbation theory and nonperturbative background are explicitly written down.     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Large numbers hypothesis. IV. The cosmological constant and quantum physics

In standard physics quantum field theory is based on a flat vacuum space-time. This quantum field theory predicts a nonzero cosmological constant. Hence the gravitational field equations do not admit a flat vacuum space-time. This dilemma is resolved using the units covariant gravitational field equations. This paper shows that the field equations admit a flat vacuum space-time with nonzero cosmological constant if and only if the canonical LNH is valid. This allows an interpretation of the LNH phenomena in terms of a time-dependent vacuum state. If this is correct then the cosmological constant must be positive.     

Ground-state correlations and finite temperature properties of the transverse Ising model

We present a semi-analytic study of Ising spins on a simple square or cubic lattice coupled to a transverse magnetic field of variable strength. The formal analysis employs correlated basis functions (CBF) theory to investigate the properties of the corresponding N-body ground and excited states. For these states we discuss two different ansaetze of correlated trial wave functions and associated longitudinal and transverse excitation modes. The formalism is then generalized to describe the spin system at nonzero temperatures with the help of a suitable functional approximating the Helmholtz free energy. To test the quality of the functional in a first step we perform numerical calculations within the extended formalism but ignore spatial correlations. Numerical results are reported on the energies of the longitudinal and the transverse excitation modes at zero temperature, on critical data at finite temperatures, and on the optimized spontaneous magnetization as a function of temperature and external field strength.     

Decay of correlations in Fermi systems at nonzero temperature

The locality of correlation functions is considered for Fermi systems at nonzero temperature. We show that for all short-range, lattice Hamiltonians, the correlation function of any two fermionic operators decays exponentially with a correlation length which is of order the inverse temperature for small temperature. We discuss applications to numerical simulation of quantum systems at nonzero temperature.     

Asymmetry of the scattering amplitude of atoms in the field of short counterpropagating light pulses

Rectification in bipotential scattering of a beam of atoms in the field of short pulses of traveling and standing waves is studied: As a result of the coherence induced by the traveling-wave pulse, the momentum transferred to the atomic beam during scattering by the standing wave is nonzero. The magnitude and sign of the asymmetry in the scattering amplitude are oscillatory functions of the duration of the traveling-wave pulse and the detuning of the frequency of the field from atomic resonance. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 920–923 (25 June 1996)     

Violation of chiral symmetry in curved spacetime with a magnetic field

The phase structure of a four-fermion (4F) model in curved spacetime with a magnetic field is investigated (in the framework of a 1/N expansion and an approximation linear in the curvature). The effective potential for the combined fields is calculated for the following situations: a) nonzero curvature and b) nonzero curvature and nonzero magnetic field. It is shown that the gravitational field may offset the magnetic-field effect, with restoration of chiral symmetry as the result. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 43–47, July, 1997.