Two-dimensional classical representations of the partition function of the spin-1/2 chain

Starting from the Trotter formula, we derive several classical representations of the partition function of a spin-1/2 chain. We investigate the rate of convergence of the different approximants by means of exact calculations for small systems. We demonstrate that it is important to use approximants that have the same symmetry properties as the original quantum model.     

New expression for the partition function of a two-dimentional classical plasma

A new expression is derived for the canonical partition function Q of a two-dimentional Coulomb system. As a by-product Q is shown to exist above the critical temperature T_c for the collapse of a pair of opposite point charges.     

A lower bound on the partition function for a classical charge symmetric system

A lower bound is obtained for the grand canonical partition function (and hence for the pressure) of a charge symmetric system with positive definite interaction. For the Coulomb interaction the lower bound on the pressure is the Debye-Hückel approximation.     

The classical partition function of a bound three-particle cluster

The task of computing the classical partition function for a bound aggregate of three particles is reduced to that of performing a 2-dimensional quadrature. A simple, analytical formula is obtained in the special case of three identical particles for which the mutual interactions are given by pair-additive square-well potentials.     

Finite-size effects and scaling for the thermal QCD deconfinementphase transition within the exact color-singlet partition function

We study the finite-size effects for the thermal quantum chromodynamics (QCD) deconfinement phase transition, and use a numerical finite-size scaling analysis to extract the scaling exponents characterizing its scaling behavior when approaching the thermodynamic limit . For this, we use a simple model of coexistence of hadronic gas and color-singlet quark gluon plasma (QGP) phases in a finite volume. The color-singlet partition function of the QGP cannot be exactly calculated and is usually derived within the saddle-point approximation. When we try to do calculations with such an approximate color-singlet partition function, a problem arises in the limit of small temperatures and/or volumes , requiring additional approximations if we want to carry out calculations. We propose in this work a method for an accurate calculation of any quantity of the finite system, without any approximation. By probing the behavior of some useful thermodynamic response functions on the whole range of temperature, it turns out that, in a finite-size system, all singularities in the thermodynamic limit are smeared out and the transition point is shifted away. A numerical finite-size scaling (FSS) analysis of the obtained data allows us to determine the scaling exponents of the QCD deconfinement phase transition. Our results expressing the equality between their values and the space dimensionality is a consequence of the singularity characterizing a first-order phase transition and agree very well with the predictions of other FSS theoretical approaches to a first-order phase transition and with the results of calculations using Monte Carlo methods in both lattice QCD and statistical physics models. Received: 11 January 2005, Revised: 7 July 2005, Published online: 30 August 2005     

An exact relation for the Gell-Mann-Low function in supersymmetric electrodynamics

We get an exact relation for the β-function in SQED.     

An improved variation-perturbation technique for the partition function of anharmonic oscillators

A new factorization of the hamiltonian is applied to statistical-variation-perturbation theory. The zero order variational hamiltonian already contains the diagonal perturbation part. The partition function (PF) of the quartic anharmonic oscillator is in very good agreement with numerical results.     

Incremental expansion of the classical partition functions

The incremental expansion of the canonical partition function of a classical many-particle system is obtained. The incremental expansion coefficients are calculated for the case of a pair interaction potential. The results are used to derive the incremental expansion of the system free energy.     

An exact solution to the classical,anisotropic Heisenberg model with long-range Kac interactions

A rigorous derivation is given for the “constant-magnetization” free energy density of the classical, anisotropic Heisenberg model with long-range Kac interactions. The derivation involves bounding arguments similar to those used for a classical fluid by Lebowitz and Penrose. The present work is carried out in a constant-magnetization ensemble. The free energy density is determined exactly under a quadruple-limiting process. The limits involved are a Lebowitz-Penrose type of triple-limiting process, followed by a final limit,x→ 0, wherex is a parameter which represents the range over which each component of the net spin density can vary. Explicit equations of state are determined for the special case of zero short-range interactions plus pure Kac-type long-range interactions.     

An exact solution for the on-axis amplitude of a circularly symmetric aperture function

The Rayleigh-Sommerfeld diffraction integral has a simple form for the case of a circularly symmetric aperture function. It is, in fact, solvable for the simple case of a constant aperture function and along the axis of symmetry. This solution can thus be utilized to obtain an exact “on-axis” solution to many circularly symmetric aperture functions.     

Symmetries,zero modes and partition function of classical systems by means of the functional integration

Summary By means of an analysis of the manifold of the equienergetic classical minima, whose dimension can change in the family of systems we are describing, we show that it is possible to give a unificated complexive termodinamical treatment in terms of functional integral. The cases of radiation in the presence of solitons of SG and DSG are then studied applying the previous results. Relative contributions to the partition function are calculated and they give the known results in the limiting values of the physical parameter on which they depend. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Functional self-similarity,scaling and a renormalization group calculation of the partition function for a non-ideal chain

The hypothesis of asymptotic self-similarity for nonideal polymer chains is used to derive the functional and differential equations of a new renormalization group. These equations are used to calculate the partition functions of randomly jointed chains with hard-sphere excluded-volume interactions. Theoretical predictions are compared with Monte Carlo calculations based on the same microscopic chain model. The excess partition function converges very slowly to its true asymptotic form δQ(N→∞)∼κ^N−1. The conventional asymptotic formula, δQ(N→∞)∼κ^N−1N^γ−1, is found to be applicable for chains of moderate length and for excluded-volume interactions appropriate to the subclass of flexible self-avoiding chains.     

Properties of exact solutions for a massive classical Yang-Mills field

Exact solutions for a massive Yang-Mills field are found and solutions of classical Wong equations and quantum Dirac equations are discussed for the field configurations obtained. A procedure for constructing constant fields is given and transition to solutions of the Yang-Mills equations in the case of a massless field is discussed.Translated from Izvestriya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 96–100, May, 1986.In conclusion, the authors express their gratitude to V. Ch. Zhukovskii and V. R. Khalilov for valuable remarks and discussions.     

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS ² classical spins asL] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has diagonal form).Supported by USNSF Grant MCS-78-01885     

Momentum autocorrelation function of a classical oscillator chain with alternating masses

A classical harmonic oscillator chain with alternating masses is studied using the recurrence relations method. The momentum autocorrelation function changes from combination of cosines to Bessel functions when the number of oscillators increases from finite to infinite bringing about irreversibility. Optic and acoustic branches of the momentum autocorrelation function are expanded in terms of even-order Bessel functions and are shown to be finite and well behaved. Irreversibility and ergodicity are discussed.     

On the rotational partition function for tetrahedral molecules

A rigorous calculation of the quantum-mechanical rotational partition function for tetrahedral XY₄ molecules yields Q_r = ( )(2I_Y + 1)⁴π ^- exp(/4), where I_Y is the spin of the Y nucleus, and ≡ Bhc/kT. This result is accurate to 1 per cent or better for all values of B and T such that < .     

A dynamical partition function for the Lorentz gas

In this paper we introduce a dynamically defined partition function for the Lorentz gas and investigate its connection with the classical ensembles and the phase-space probability measure derived from periodic orbit expansions. Numerical evidence is presented to support the equivalence of these measures and to link them to the thermodynamic quantities for the Lorentz gas. This also suggests a new dynamical basis for the assumption of equala priori probabilities in the microcanonical ensemble.     

A modular invariant partition function for the fivebrane

We compute an invariant partition function for the chiral two-form of the M theory fivebrane compactified on the six-torus T⁶. From a manifestly invariant formalism, we prove that the partition function has an additional symmetry. The combination of these two symmetries ensures invariance. Thus, whether or not a fully covariant Lagrangian is available, the fivebrane on the six-torus has a consistent quantum theory.     

An exact identity for memory

It is argued that memory functions from Time-Convolution Generalized Master Equations for probabilities of finding a system interacting with a genuine bath in arbitrary state of a full system of states have, under quite general conditions for asymmetric systems, zero time integrals provided that the latter exist and that no approximations were involved.