Some remarks on the location of zeroes of the partition function for lattice systems

We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.     

Complex zeros of the partition function for two-dimensionalU(N) lattice gauge theories

The complex zeros of the partition function for the two-dimensionalU(N) lattice gauge model in the variable β or ξ=N/β are calculated analytically for large β and numerically. The zeros formN trajectories which forN→∞ fill a domain of the ξ plane densely. This domain touches the real ξ axis at ξ=1 with a kink that causes the Gross-Witten phase transition.     

Distribution of zeros of the partition function of the lattice gases of fisher and of temperley

The distributions of zeros of the partition function of lattice gas models of Fisher (at some temperature) and of Temperley are obtained. The former is a closed loop crossing the real axis and the latter is a part of the negative real axis.     

Asymptotics of the partition function of a general Markov random field on an infinite rectangular lattice

We investigate theoretically and numerically the asymptotics of the partition function of a general Markov random field (MRF) on an infinite rectangular lattice. We first propose the general local energy function (LEF)-parameterized MRF. Then we prove that the thermodynamic limit of the free energy of the MRF can be exactly characterized by the Perron root of the fundamental transfer matrix of a particular Markov additive process (MAP). This matrix possesses a special structure and many interesting properties that enable parallel computation of the Perron root and may be beneficial for deriving an analytical form of the free energy. We also develop another transfer matrix for numerical computation of the desired Perron root. Specifically, the former is a site-to-site transfer matrix on a twisted cylindrical lattice, while the latter is the one associated with a row-to-row transition on a vertical strip. Numerical results show that our methods exhibit consistent finite-size scaling behavior even for small values of the lattice width. This study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general MRFs within the scope of matrix algebra.     

Strutinsky smoothing and the partition function approach

The Strutinsky shell correction is analytically shown to be equivalent to the shell correction obtained from the partition function approach, provided certain restrictions on the Strutinsky smoothing parameter are met.     

Coherent states and partition function

The concept of coherent states for arbitrary Lie group is suggested as a tool for explicitly obtaining an integral representation of the partition function, whenever the Hamiltonian has a dynamical group. Two examples are thoroughly discussed: the case of the nilpotent group of Weyl related to a generic many-body problem with two-body interactions, and the case of SU(1, 1)() relevant for a superfluid system.     

Rotational partition function of methane

An analytical formula for the quantum-mechanical rotational partition function of tetrahedral XY₄ molecules is shown to be accurate to better than 1% for methane at temperatures T ≥ 40°K.     

Quantum vibrational partition function in the non-extensive Tsallis framework

The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures.     

Group structure analysis for classical lattice systems with constraints: II. Zeroes of the partition function and unicity of states

The Asano-Ruelle method is used to discuss the zeroes of the partition function of arbitrary lattice systems with constraints. The group structure associated with these systems yields necessary and sufficient conditions to build up the partition function by Asano contractions. For a large class of systems with constraints, uniqueness of the symmetric equilibrium state, as well as analyticity properties of the free energy and the correlation functions, is established at sufficiently low temperature. Explicit analyticity domains are obtained for several models with constraints. Some properties of power sets are derived.     

Rapidly calculating the partition function of macroscopic systems

It has remained an open problem to accurately compute the partition function of macroscopic systems since the establishment of statistical physics. A rapid method approaching this problem was presented and was strictly tested by molecular dynamic(MD) simulations on Ar atoms in both dense gaseous and liquid states. The outcomes from the method on the internal energy and the work of isothermal expansion(and therefore the free energy) are in good agreement with the MD simulations, suggesting the method would be immediately applied in vast areas.     

The partition function of the three-dimensional anharmonic oscillator

The thermodynamic perturbation theory is applied to the three-dimensional anharmonic oscillator. Numerical results are obtained for the partition function of the quartic anharmonic oscillator.     

Modular invariant partition function of the Hubbard model

Summary By making use of the Abelian bosonization procedure, we obtain a Coulomb-gas picture of the continuum limit of the one-dimensional Hubbard model. It is shown clearly that the semi-direct product of two Virasoro algebras (c=1) denotes symmetry of excitations of the Hubbard model. A systematic study of modular invariant partition function for the Hubbard model is presented. Correlation functions are calculated explicitly and the result is in good agreement with those of numerical simulations and Tomonaga-Luttinger model.     

Exact quantum partition function of the BCS model

The exact calculation of the reduced BCS model quantum partition function (QPF) in the thermodynamic limit is carried out by the path integration method. The expression for the QPF and the phase transition temperatureT _c in the regular phase coincide with the results of Bogolyubov. In the nonregular phase a temperature singularity appears in the expression for the QPF: the QPF diverges in the region of temperaturesT _c which are smaller than some critical temperatureT _c ^* , and it turns out that in all casesT _c ^* > T _c and the differenceT _c ^* –T _c is not small. The interpretation of the temperatureT _c ^* is given.     

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.     

Densities of states and the complex partition function

A Fourier transform technique based on the method of Dubner and Abate is here applied to the energy-level density problem for quantum partition functions. It is shown that, for typical molecular systems at realistic energies, the integrated level count, W(E), is obtainable to near ‘spectroscopic’ accuracy and in reasonable computing time. A smooth Fourier resolution of the simple density function, g(E), is likewise possible. Though somewhat less economical, this procedure entirely removes the slight discrepancies found in the steepest-descent method and can reasonably be claimed to dispose once and for all of the problem of quantum level-density computations in molecular rate theory.