A Note on the Faddeev–Popov Determinant and Chern–Simmons Perturbation Theory

A refined expression for the Faddeev–Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern–Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, pointing out that previous results on the finiteness and formal metric-independence of perturbative expansions of the partition function continue to hold.     

Probing a subcritical instability with an amplitude expansion: An exploration of how far one can get

We explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite-wavelength instability. We do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers. This is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic (laminar) state. We apply the method to two model equations, a subcritical generalization of the Swift–Hohenberg equation and a novel extension of the Kuramoto–Sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate “from infinity” and that can nevertheless be probed perturbatively. We explore the reliability and robustness of such an expansion, with a particular focus on the use of these methods for determining the existence and approximate properties of finite-amplitude stationary solutions. Such methods obviously are to be used with caution: the expansions are often only asymptotic approximations, and if they converge their radius of convergence may be small. Nevertheless, expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results.     

Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

One-loop omega-potential of quantum fields with ellipsoid constant-energy surface dispersion law

Rapidly convergent expansions of a one-loop contribution to the partition function of quantum fields with ellipsoid constant-energy surface dispersion law are derived. The omega-potential is naturally decomposed into three parts: the quasiclassical contribution, the contribution from the branch cut of the dispersion law, and the oscillating part. The low- and high-temperature expansions of the quasiclassical part are obtained. An explicit expression and a relation of the contribution from the cut with the Casimir term and vacuum energy are established. The oscillating part is represented in the form of the Chowla–Selberg expansion of the Epstein zeta function. Various resummations of this expansion are considered. The general procedure developed is then applied to two models: massless particles in a box both at zero and nonzero chemical potential, and electrons in a thin metal film. Rapidly convergent expansions of the partition function and average particle number are obtained for these models. In particular, the oscillations of the chemical potential of conduction electrons in graphene and a thin metal film due to a variation of size of the crystal are described.     

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovász local lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer⁽⁹⁸⁾ and explicitly by Dobrushin.^(37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for soft dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.     

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.     

The Modified Group Expansions for Construction of Solutions to the BBGKY Hierarchy

A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as well as local conservation laws. As a result, modified group expansions are proposed. On the basis of these expansions, a generalized kinetic equation for hard spheres and a generalized Bogolubov–Lenard–Balescu kinetic equation for a dense electron gas are derived within the polarization approximation.     

Polymer Gas Approach to N-Body Lattice Systems

We give a simple proof, based only on combinatorial arguments, of the Kotecký–Preiss condition for the convergence of the cluster expansion. Then we consider spin systems with long-range N-body interactions. We prove directly, using the polymer gas representation, that the pressure may be written in terms of an absolutely convergent series uniformly in the volume when the interaction is summable in a suitable sense. We also give an estimate of this radius of convergence. In order to get the proof we use a method introduced by Cassandro and Olivieri in the early 1980s. We apply this method to various concrete examples.     

Surface Tension and the Ornstein–Zernike Behaviour for the 2D Blume–Capel Model

We prove existence of the surface tension in the low temperature 2D Blume–Capel model and verify the Ornstein–Zernike asymptotics of the corresponding finite-volume interface partition function.     

Distributional Borel Sum for the PT-Symmetric Hénon–Heiles Model

The proof is outlined of the distributional Borel summability of the Rayleigh–Schrödinger perturbation expansions of the quantum Hénon–Heiles model.     

Path Integrals and Statistics of Identical Particles

We summarize the essential ingredients, which enabled us to derive the path-integral for a system of harmonically interacting spin-polarized identical particles in a parabolic confining potential, including both the statistics (Bose–Einstein or Fermi–Dirac) and the harmonic interaction between the particles. This quadratic model, giving rise to repetitive Gaussian integrals, allows to derive an analytical expression for the generating function of the partition function. The calculation of this generating function circumvents the constraints on the summation over the cycles of the permutation group. Moreover, it allows one to calculate the canonical partition function recursively for the system with harmonic two-body interactions. Also, static one-point and two-point correlation functions can be obtained using the same technique, which make the model a powerful trial system for further variational treatments of realistic interactions.     

Letter: Spherically Symmetric Metrics with Shear Controlling Expansion

An isotropic spatially inhomogeneous spacetime with the stress tensor satisfying the limiting case of the strong energy condition [T₀₀ + 1/2)T] = 0 in the locally inertial coordinates where the observer's four-velocity is u ^a = ₀ ^a satisfying the constraint u ^a u_a = –1 is studied. Special metrics with accelerating expansions of the inhomogeneous spacetime merely controlled by the shear are presented as an alternative model.     

Time Development of Exponentially Small Non-Adiabatic Transitions

Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two–level analytic Hamiltonian functions. Our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function. Our results confirm the physics predictions of Sir Michael Berry in the sense that the switching function for this family of Hamiltonians has the form that he argues is universal.Partially supported by National Science Foundation Grants DMS–0071692 and DMS–0303586.     

Polymer Chain Size from Geodesic Path and Geometrical Bolyai–Lobachevskij Partition Function. Application to Swelling of Macromolecules in Solution and Micellar Growth

The Bolyai–Lobachevskij (BL) formula, relating parallelism angle and distance in a Non-Euclidean space, is used to introduce a geometrical partition function. Employing a correspondence between Boltzmann factor and BL characteristic length, allows us to get a simple relation for average size and space curvature, which is the analogy to the equation for the mean energy derived from the ordinary partition function. Due to the equivalence, recently proposed, between a chain molecule in a liquid and a geodesic path in a relativistic space, the equation obtained is expected to be suitable for describing geometrical phenomena in polymer-like networks. Simple applications to swelling of polymer solutions and micellar growth are presented and discussed.     

Universal Solutions of Quantum Dynamical Yang–Baxter Equations

We construct a universal trigonometric solution of the Gervais–Neveu–Felder equation in the case of finite-dimensional simple Lie algebras and finite-dimensional contragredient simple Lie superalgebras.     

Surface free energy of a finite system without interfaces at low temperatures

We derive the specific surface free energy for a rather general system at low temperatures that can be rewritten as a gas of non-interacting contours (polymers). To this end, we use a standard cluster expansion series for the system?s partition function. A specific regime of ‘weak’ boundary conditions is assumed to ensure that no interfaces or large droplets occur in the system. We illustrate the general results, using a simple lattice–gas model.     

Statistical Mechanics Approach to Coding Theory

We propose a method based on cluster expansion to study the optimal code with a given distance between codewords. Using this approach we find the Gilbert–Varshamov lower bound for the rate of largest code.     

The O(n) Loop Model on the 3–12 Lattice

The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3–12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n = 0 this gives the recently found exact value = 1.711041... for the connective constant of self-avoiding walks on the 3–12 lattice. The exact critical points are recovered for the Ising model on the 3–12 lattice and the dual asanoha lattice at n = 1.     

Wick-Ordered Entire Functions of the Indefinite Metric Free Field

We present a simple and general method for constructing Wick-ordered entire functions of free fields with an indefinite metric, based on using an appropriate generalization of the Paley–Wiener–Schwartz theorem.