Scattering contributions to the internal partition function of a diatomic molecular system

An analytical expression for the phase shift contribution to the internal partition function for the Morse potential is derived by using an approximate Jost function. This function is shown to be a convergent sum. The numerical results obtained for H₂ and HCl show the partition function to be a monotonically increasing function of temperature. This observation agrees with the results of Rogers and co-workers. Work supported in part by the Department of Atomic Energy, Government of India.     

Time partition function analysis of a neural network

The time characteristics of a linear network in the brain are obtained by the method of the time partition function, which is analogous to a grand partition function or a distribution function in statistical mechanics. The analogy between the average density in a many-particle system and the reciprocal of the frequency in a network is shown. By this method, the frequency distribution functions are obtained with respect to a network composed of two layers, the network used in information retrieval and the network generating a brain wave.     

The partition function of a degenerate functional

The partition function of a degenerate quadratic functional is defined and studied. It is shown that Ray-Singer invariants can be interpreted as partition functions of quadratic functionals. In the case of a degenerate non-quadratic functional the semiclassical approximation to the partition function is considered.     

Some Exact Results for a Two-Dimensional System of Particles

The Laughlin variational wave function of a quantum liquid used in studying the fractional quantum Hall effect may be interpreted as a two-dimensional system of particles interacting via a logarithmic interaction, with a centripetal force proportional to the square of the distance from the origin. We present exact calculations of the partition function, radial distribution function, and particle density for systems of up to six particles. Received April 25, 1994; revised August 5, 1994; accepted for publication August 30, 1994     

Exact Solution of a Generalized ANNNI Model on a Cayley Tree

We consider the Ising model on a rooted Cayley tree of order two with nearest neighbor interactions and competing next nearest neighbor interactions restricted to spins belonging to the same branch of the tree. This model was studied by Vannimenus who found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus’s results are based on an analysis of the recurrence equations (relating the partition function of an n ? generation tree to the partition function of its subsystems containing (n ?1) generations) and most results are obtained numerically. In this paper we analytically study the recurrence equations and obtain some exact results: critical temperatures and curves, number of phases, partition function.     

Modelling free energy of a rigid protein in solid water: Comparison between rigid-body motions and harmonic oscillators

We test two methods to estimate a partition function of a system consisting of multi-atom molecules with intermolecular interaction. Our test case is a protein in water. The first method is based on rigid-body motions. The space in which the protein (bovine pancreatic trypsin inhibitor) would be moving is limited, so by making a correction to a partition function of a rigid body, we can obtain the function. The function depends on temperature and space. The second method is based on harmonic oscillators. Under the potential field produced by surrounding water molecules, a protein behaves like a set of harmonic oscillators. We obtain the partition function for the oscillators within a harmonic approximation. The function also depends on temperature and the strength of potential energy between the protein and waters. Comparison of the two methods indicates that the second method is better for estimating a partition function for a protein in water.     

Partition function for an electron in a random potential

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.Work supported by the National Science Foundation.     

Multiplicity distributions in canonical and microcanonical statistical ensembles

The aim of this paper is to introduce a new technique for the calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand canonical partition function. A Taylor expansion of the generating function is used to separate contributions to the partition function in their power in volume. We employ Laplace’s asymptotic expansion to show that any equilibrium distribution of multiplicity, charge, energy, etc. tends to a multivariate normal distribution in the thermodynamic limit. A Gram–Charlier expansion additionally allows for the calculation of finite volume corrections. Analytical formulas are presented for the inclusion of resonance decay and finite acceptance effects directly into the partition function of the system. This paper consolidates and extends previously published results of the current investigation into the properties of statistical ensembles.     

Topological aspects of gauge and spin potts models

For the d-dimensional lattice gauge Potts model the representation of the partition function is derived, where the field configuration summation is reduced to the summation over submanifolds (oriented and nonoriented), constructed from the plaquettes of the lattice. The topological invariants (Betti numbers) of these two-dimensional submanifolds are used in an essential way. Some possible applications of this representation for the partition function are discussed.     

Partition function of a “Supersymmetric” three-dimensional gauge ising model on a regular lattice

The partition function is calculated for a Z ₂ gauge model coupled to Majorana fermions on a simple cubic lattice.     

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.     

Monte Carlo calculation of the thermodynamic properties of a one-dimensional fermion lattice model

The generalized Trotter formula is used to derive two different classical representations of the partition function of a one-dimensional fermion model. Shortchain calculations are used to study the corresponding approximants for the energy and specific heat. A Monte Carlo technique has been used to calculate the temperature-dependent properties of a chain of 64 sites.     

Quantum decay rate of a nonlinear dissipative system with fission-like potential nearT c

We use a thermodynamic scheme (imaginary free energy method) in terms of the path integral technique to study the quantum decay rates of a metastable state system coupled to a heat bath in the crossover temperature (T _c) region. In this region the transition between thermally activated decay and tunneling occurs. A nonlinear coupling form factor is used to overcome the divergent integral in the partition function nearT _c. The decay rate formula based on the steepest descent approximation has been improved. A method is developed to calculate the real and imaginary parts of the partition function which combines a random walk method with fast-Fourier transform Monte-Carlo evaluation. For a nonlinear dissipative system with a damping correlation kernel of exponential form, the accurate numerical calculations are presented. The effects of nonlinear and frequency-dependent damping on the rate are shown.     

A Simple Inductive Approach to the Problem of Convergence of Cluster Expansions of Polymer Models

We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of polymer models. We give a very simple proof of the Dobrushin–Kotecký–Preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used.     

A statistical view of exclusive and inclusive cross sections

The interrelations between exclusive and inclusive cross sections are investigated by means of efficiency matrices. A partition function Z is used as a generating functional for these cross sections which yields similarly also topological and other mixed cross sections. Regarding the partition function itself as a cross section, the general properties of Z and of statistical quantities derived from ln Z are investigated as functions of the chemical potentials. These statistical methods are used to analyze three simple examples of data for multiplicity distributions, in particular for the third distribution from the Echo Lake experiment a behaviour of Z is found which has similarities with a phase transition in statistical mechanics.     

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.     

Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex SOS model, we derive the recursion relations of the partition function for the eight-vertex model with a generic non-diagonal reflecting end and domain wall boundary condition. Solving the recursion relations, we obtain the explicit determinant expression of the partition function. Our result shows that, contrary to the eight-vertex model without a reflecting end, the partition function can be expressed as a single determinant.     

Level density of a Fermi gas with pairing interactions

The paper presents the level density for a Fermi gas with pairing interactions. First an approximate partition function is derived in terms of a universal function. Then a modified saddle-point method is used for the inverse Laplace transformation to obtain an algebraic expression which does not diverge for the ground state. This method was applied to the normal Fermi gas as well as to the paired Fermi gas. The results are compared to the standard and to the backshifted Fermi-gas level density.