The partition function of degenerate quadratic functional and Ray-Singer invariants

The partition function of degenerate quadratic functional is defined and studied. It is shown that analytic torsion and similar invariants can be interpreted as partition functions of quadratic functionals.     

Semiflexible polymers in straining flows

We introduce a model for semiflexible polymer chains based on the integral of an appropriate Gaussian process. The stiffness is characterized physically by adding a bending energy. The degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity, the limiting case reduces to the Brownian model of completely flexible chains studied in earlier work. The calculation of the partition function for the configuration statistical mechanics (i.e., the distribution of shapes) of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated Gaussian process. An exact formula for the partition function is presented; however, in practice, this formula is too complicated for most computations. We therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first-order term which gives the first-order deviation from the completely flexible case. In addition to the partition function, the method presented here can also deal with other quadratic functionals such as the “stochastic area” associated with two polymer chains.     

On polymer conformations in elongational flows

We consider various models of polymer conformations using paths of Gaussian processes such as Brownian motion. In each case, the calculation of the law of the moment of inertia of a random polymer structure (which is equivalent to the calculation of the partition function) is reduced to the problem of finding the law of a certain quadratic functional of a Gaussian process. We present a new method for computing the Laplace transforms of these quadratic functionals which exploit their special form via the Ray-Knight Theorem and which does not involve the classical method of eigenvalue expansions. We apply the method to several simple examples, then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory.     

Shallow acceptors in ionic semiconductors with a degenerate band

The energy spectrum of an impurity-phonon system in semiconductors with a degenerate valence band ₈ edge is calculated. The matrix Hamiltonian of the problem is represented as a sum of two matrices containing operators of the s and d type respectively. Neglecting the latter matrix, the partition function of the system is represented as a path integral, which is calculated with the help of the variational principle based on the introduction of a trial quadratic Lagrangian. The matrix with operators of the d-type is taken into account as a perturbation. The results are compared with experiment.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 51–53, January, 1991.     

量子变换理论及其应用概述

概括地介绍了量子变换理论的基本思想,并就这一基本理论在指数上二次型的正规与反正规乘积表示、二次型哈密顿量的配分函数和波函数,以及量子信息中连续变量系统的纠缠度和保真度的计算做了简要的说明。     

多维相空间中任意指数二次型算符的矩阵元

在多维相空间中,利用指数二次型算符的正规乘积和反正规乘积表示式,给出了任意指数二次型算符矩阵元的严格表达式．在能谱和能量本征函数未知的条件下,由此得到了哈密顿量为二次型系统的配分函数和波函数 关键词：     

On a classical two-component plasma with a quadratic interaction

We calculate the canonical partition function for a two-component classical plasma with a quadratic interaction in d dimensions. The equation of state is that of an ideal gas.     

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.     

Analytic Calculation of Arbitrary Matrix Elements for Boson Exponential Quadratic Operators

By means of the transformation relation between the ordinary form of boson exponential quadratic operators (BEQO) and its anti-normal product form, we present an effective method to conveniently calculate arbitrary matrix elements of BEQO. By this method, many important matrix elements can be calculated analytically. As a direct application, we obtain the exact solutions of the density matrix and partition function for general boson quadratic Hamiltonian without any information about the energy level.     

Asymptotic expansion of the one-loop Ω-potential of quantum fields with a quadratic dispersion law

Nonextensive contributions of free quantum fields with quadratic dispersion law to the partition function are investigated. Nonextensive corrections are caused by finite sizes of the system. The general procedure of finding such corrections is developed. As an example, a model of the ideal gas of nonrelativistic fermions in a cube is considered. The equations of state are obtained for this system.     

Influence of rotational isomerism on two-photon absorption properties of FTC chromophores

The influence of rotational isomerism on the two-photon absorption(TPA) of FTC chromophores has been investigated using the quadratic response theory with the B3LYP functional.Eight rotamers induced by three rotatable single bonds in the molecule are fully optimized,and it is found that their conformational energies are nearly degenerate.Our calculations demonstrate that rotational isomerism has an important effect on the TPA cross sections.For a certain rotamer,the maximum TPA cross section is enhanced significantly.In addition,in the longer wavelength region,the rotational isomerism could lead to a large shift of the TPA position.     

Oscillatory critical amplitudes in hierarchical models

We study the oscillatory critical amplitudes of theq-states Potts model on a diamond hierarchical lattice. We consider an example of the generic case (finite critical index), as well as the degenerate case (essential singularity). In both cases, we compare the magnitude of the oscillations with geometrical characteristics of the Julia set of zeroes of the partition function.     

Quantum Statistics for General Quadratic System

By the generalized linear quantum transformation theory, we concisely derive the analytic expressions of partition function for general quadratic systems of multi-mode boson and fermion Fock space without any information for the energy spectrum. Under a general condition, we firstly acquire the exact expressions of energy spectra for these systems.     

ON THE APPROXIMATE PARTITION FUNCTION IN THE STATISTICAL THEORY OF ADSORPTION

Wang's theory for determining the approximate configurational partition function of the adsorbed layer is modified in two different ways. One is to assume that the configurational energy should be corrected: the other to advocate that the deficiency due to a wrong expression for the a priori probability of the. central site is more significant.The configurational partition function is evaluated is both methods and the adsorptipn isotherin and the beat of adsorption computed for the case of quadratic lattice With dipole interaction. values for the last two quantities when a uniform continuous distribution of the distant adsorbed particles is assumed are further given for comparison. The second method, which surpasses the first, is compared with Kirkwood's method. in the case of hexagonal lattice with neighbour interaction. Numerical work is also carried out in this case.     

Electron capture in collisions between negative and positive ions

Collisions between negative and positive atomic ions are investigated. The ionic wave function is expressed in terms of the Coulomb Green’s function. Normalizing this function allows the system of two ions to be described completely. The exchange matrix elements turn out to be the sums of products of the Coulomb wave functions over degenerate states. These sums are expressed in terms of the quadratic form of the wave function for a state with zero angular quantum numbers, l=m=0. The nonadiabatic coupling of quasi-crossing terms with other terms of the system is analyzed; this effect significantly increases the cross section for single-electron capture.     

Newton-Leibniz integration for ket-bra operators (II)—application in deriving density operator and generalized partition function formula

Via the route of applying Newton-Leibniz integration rule to Dirac’s symbolic operators, we show that the density operator e^−βH, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring convenience in deriving partition functions.     

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.     

Steepest descent path for the microcanonical ensemble-resolution of an ambiguity

The microcanonical entropy plays an essential role in the equilibrium statistical mechanics of gravitating systems. A peculiar feature of many of these systems is the existence of stable thermodynamic equilibrium configurations with negative heat capacities. Different methods have been developed for calculating the microcanonical entropy involving multivariate integrals of constraints and functional integrations. An apparent ambiguity between an approach due to Hawking and Gibbons, based on an entropy definition involving an inverse Laplace transform of the partition function, which they developed to treat quantum systems with gravity, and a different approach developed by Horwitz and Katz defining the entropy as an equal weight sum over a constant energy surface developed originally to treat Newtonian and classical GR systems is shown here to be spurious, at least at the level of quadratic fluctuations of all variables about the extremal solutions. The two approaches involve distinct contours for different orders of integration, each of which is shown to be the appropriate steepest descent path corresponding to the given order of investigation. Up to quadratic fluctuations both methods yield identical results. However, they represent different perturbation expansions for the gravitational modes of freedom with different radii of convergence. The discussion is made in terms of a particular convenient model, a system of point particles interacting via Newtonian forces, confined to a sphere, but results are quite general.     

The partition function of the SO(8192) bosonic string

We discuss the partition function of the SO(8192) bosonic string at arbitrary genus. As in the genus-one case, diagrams defined in terms of involutions acting in the same way on the homology basis are weighted by identical mulitplicity factors. This property emerges as the basis feature of the SO(8192) theory. It corresponds to the picture of this model as a parameter-space orbifold of the closed bosonic string. By factorization we recover amplitudes in the degenerate limit.     

Coulomb-gas representation on higher-genus surfaces

In this paper we use the Coulomb-gas approach to construct the minimal-model conformal blocks of higher-genus Riemann surfaces. We define the higher-genus blocks by sewing, and write them in terms of the rational blocks of a compactified scalar field. We show that spurious states decouple, which implies that the blocks degenerate correctly. As an example, we compute the genus-two partition function, and verify modular invariance for the subset of minimal models which only require one type of screening charge.