MacMahonesque partition functions detect sets related to primes

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacityζ 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 theζ plane occupy the imaginary axis from −i∞ to −iζc and iζc to i∞ for some ζc. They also occupy the diamond shape of four straight lines from ±iζc to ζc and from ±iζc to −ζc. The fugacityζ 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.

We consider \mathcal{N}=4𝒩=4 supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these 2/3-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large NN expression for the ABJM partition function.

A definition of a finite partition function for bound electronic states is presented for a hydrogenic ion, with the associated problems of the fall in intensity of spectral lines and the lowering of the effective ionization potential. The partition function is partly based on numerical solutions of the Schr\"odinger equation (SE) with the complete screened Coulomb potential (CSCP), where the $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$ states are considered. The CSCP is given by ${V(r),={V}_{i}(r)=\ensuremath{-}Z{e}^{2}\left(\frac{1}{r}\ensuremath{-}\frac{1}{D+A}\right), 0<~r<~A}{={V}_{0}(r)=\ensuremath{-}Z{e}^{2}\frac{D}{D+A}\frac{\mathrm{exp}[\frac{(A\ensuremath{-}r)}{D}]}{r}, r>~A}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\ensuremath{\lambda}{a}_{0}}$ and ${E}_{\ensuremath{\lambda}}=\ensuremath{-}\frac{{Z}^{2}\ensuremath{\mu}{e}^{4}}{2{\ensuremath{\hbar}}^{2}{\ensuremath{\lambda}}^{2}}$, where $\ensuremath{\lambda}$ is the CSCP quantum number, yield the standard form of the SE equation with $\ensuremath{\lambda}$ in place of $n$. The numerical solutions are obtained with a nonlinear method that is both accurate and stable. Since the accurate numerical solutions do not yield an explicit maximum-bound principal quantum number, another property of the screened solutions is used to define a decreasing probability of an electron occupying a particular Coulomb eigenstate. The quantity derived is the relative probability of the screened to the unscreened Coulomb state, given by ${\ensuremath{\Phi}}_{n,l}=\frac{{\ensuremath{\lambda}}^{2l+1}N(\ensuremath{\lambda}, l, d, a)}{{n}^{2l+1}N(n, l)},$ where the $N'\mathrm{s}$ are the normalizations in $x$ space, calculated with variations of $D$ and $A$ only. This relative occupation probability of the bound state is used to modify the Boltzmann factor in the standard expression for the electronic partition function. Correlations with observations are discussed: there is excellent agreement with the fall in intensity of hydrogen lines in the solar atmosphere and good agreement with lines from laboratory hydrogen at 21\ifmmode^\circ\else\textdegree\fi{}K. The effective ionization potential of hydrogen is calculated using the maximum detected level of hydrogen observed in the solar chromosphere. A simple analytical fit to the $\ensuremath{\Phi}$ function and a useful approximate analytical expression for the screened Coulomb partition function are given.

A general model for analysis of the infrared spectrum of atom–diatom van der Waals molecules is developed. Using energy levels and spectroscopic parameters based on the Hutson and Howard M5 potential, this model quantitatively reproduces the previously observed ArHCl spectrum in the null gap of the monomer HCl spectrum. The thermodynamics of atom–atom and atom–diatomic systems is discussed. An approximate quantum mechanical method of evaluating bound and metastable partition functions is derived. Simple expressions for the partition functions are given which are shown to accurately reproduce the well established results for atom–atom complexes. The generalization of the approach to atom–diatomics is discussed. These partition functions are used to calculate thermodynamic properties such as heat capacities and equilibrium constants.

The instanton contributions to Yang-Mills theory on the torus: localization, Wilson loops and the perturbative expansion

A study is made of a linear lattice fluid in which bonds can form between pairs of molecules on second-neighbor sites with an empty site between them. At low temperatures and pressures there is, thus, competition between local open configurations and more close-packed local configurations of higher energy. By using a matrix expression for the grand partition function, the density and effective coordination number are found as functions of the absolute temperature T and one-dimensional pressure p. Equivalent results are shown to follow from a constant-pressure partition function based on an explicit expression for the configuration number. It is found that a pressure p0 exists such that for any p < p0 the open configuration is stable at T = 0, while the density passes through a maximum as T increases. For any p > p0, on the other hand, the close-packed configuration is stable at T = 0 and the density decreases monotonically with T. Using a constant-pressure partition function, we also consider continuous models with the ``bonding'' represented by a potential well separated from the hard core. With a well of parabolic shape, similar results to those of the lattice model are obtained, while with a square well the curve of density against T displays a minimum as well as a maximum for any p < p0.

Generalized direct-summation procedure for the thermal conductivity in moderately dense gases

Parafermion Hall states from coset projections of abelian conformal theories

The polymer threading a membrane transition (PTM) which is a first‐order thermodynamic phase transition for an isolated linear polymer in the limit of infinite molecular weight is coupled to the helix to random coil (HRC) transition and the polymerization (P) transition. In each case an exact expression for the partition function of the coupled pair is given as a 1‐dimensional summation over products of the individual partition functions. For HRC‐HRC there are 8 possible translocation modes. For example, as we raise the temperature we can have H1→H2→R1→R2 in obvious notation. The PTM‐P mode is most interesting because polymer can be created in the PTM side even though the P side is in the depolymerization regime! These exact model solutions provide a base for the study of the kinetics of significant technological problems such as the translocation of DNA through pores in membranes. They also throw light on the nature of polymer‐membrane‐pore interactions in living cells and viruses.

The polymer threading a membrane transition (PTM), which is a first-order thermodynamic phase transition for an isolated linear polymer in the limit of infinite molecular weight, is coupled to the other four phase transitions of the isolated polymer molecule. They are (1) the helix–random coil (HR) phase transition which can be diffuse (polypeptides), second-order (DNA) or first-order (collagen) depending on the number of strands, (2) the collapse (C) transition which is usually second-order but can be first-order for polymeric solvents, (3) adsorption onto a surface (SA) which is second-order, (4) a model of equilibrium polymerization (P) which is first-order. In each case an exact expression for the partition function of the coupled pair is given as a one-dimensional summation over products of the individual partition functions corresponding to sides 1 and 2. Using a procedure analogous to evaluation of the grand canonical ensemble the summation can be performed and the character of the transition elucidated in the limit of infinite molecular weight. Given that the solutions on either side are sufficiently diverse there are 15 possible translocation pair couplings. They are PTM–PTM, HR–HR, C–C, SA–SA, P–P, PTM–HR, PTM–C, PTM–SA, PTM–P, HR–C, HR–SA, HR–P, C–SA, C–P, SA–P. The PTM–P coupling is most interesting because one can create polymer in the PTM side even though the P side is in the depolymerization regime. For HR–HR there are eight possible translocation modes. For example, as we raise the temperature we can have H1→H2→R1→R2 in obvious notation. These exact model solutions provide a thermodynamic base for the study of the kinetics of significant technological problems such as the translocation of DNA through pores imbedded in membranes. They also throw light on the nature of polymer–membrane–pore interactions in living cells and viruses.

This paper systematically studies and improves the statistical theory of the kink-phonon interactive mixture in a one-dimensional model for the displacive phase transition. By taking applicable forms of the path integral for both the partition functions of kink and phonon and altering the method in calculating the path integral of phonon, a more reasonable and general expression of the partition function of the Grand ensemble is obtained. Under the lowest eigenstate approximation, the Grand.partition function may be reduced to a result similar to that of reference[6]. Under the classical approximation, the average density of the kink calculated from the Grand partition function is in agreement with the result of the computer simulation better than those obtained in earlier publications.

An approximate expression for the partition function in a generalization of Bethe's theory of superlattices including long range interaction is obtained as an integral of the approximate energy expression in the generalized Bethe's theory. An alternative form of the energy expression is also considered, and higher approximations are treated. The paper ends with an application to the problem of adsorption.

We characterize the breaking of analyticity with respect to the replica numberwhich occurs in random energy models via the complex zeros of the momentof the partition function. We perturbatively evaluate the zeros in the vicinityof the transition point on the basis of an exact expression for the moment ofthe partition function utilizing the steepest descent method, and examine anasymptotic form of the locus of the zeros as the system size tends to infinity. Theincident angle of this locus indicates that analyticity breaking is analogous toa phase transition of the second order. We also evaluate the number of zerosutilizing the argument principle of complex analysis. The actual number of zeroscalculated numerically for systems of finite size agrees fairly well with the analyticalresults.

The author constructs a three-dimensional generalisation of the brick lattice and shows that the partition function for a suitably modified SO(n) or SU(n) lattice gauge theory can be written exactly as a sum over closed surfaces on this lattice. For Ising lattice gauge theory, the author derives a formal expression for the partition function.

A-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle S⊥1, probing a transverse ℝ4/ℤM background. Below the scale of the radius of S⊥1, these theories resemble a circular quiver gauge theory with M nodes of gauge group U(N) and matter in the bifundamental representation (or adjoint in the case of M = 1). In this paper, we study these LSTs in the presence of a surface defect, which is introduced through the action of a ℤN orbifold that breaks the gauge groups into [U(1)]N. We provide a combinatoric expression for the non-perturbative BPS partition function for this system. This form allows us to argue that a number of non-perturbative symmetries, that have previously been established for the LSTs, are preserved in the presence of the defect. Furthermore, we discuss the Nekrasov-Shatashvili (NS) limit of the defect partition function: focusing in detail on the case (M, N) = (1, 2), we analyse two distinct proposals made in the literature. We unravel an algebraic structure that is responsible for the cancellation of singular terms in the NS limit, which we generalise to generic (M, N). In view of the dualities of higher dimensional gauge theories to quantum many-body systems, we provide indications that our combinatoric expression for the defect partition are useful in constructing and analysing quantum integrable systems in the future.