Variational method of calculating the partition functions of the lattice-gas and Ising models

1989 ◽  
Vol 79 (3) ◽  
pp. 641-647
Author(s):  
A. A. Zaitsev
Keyword(s):  
Variational Method ◽  
Lattice Gas ◽  
Ising Models ◽  
Partition Functions

Independence polynomial and matching polynomial of the Koch network

2015 ◽  
Vol 29 (32) ◽  
pp. 1550234
Author(s):  
Yunhua Liao ◽  
Xiaoliang Xie
Keyword(s):  
Lattice Gas ◽  
Small World ◽  
Partition Functions ◽  
Complete Class ◽  
Dimer Model ◽  
Scale Free ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Classical Models ◽  
Explicit Formulae

The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “[Formula: see text]-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.

The Grassmann Representation of Ising Model

1998 ◽  
Vol 12 (20) ◽  
pp. 1995-2003 ◽  
Author(s):  
K. Nojima
Keyword(s):  
High Temperature ◽  
Ising Model ◽  
Ising Models ◽  
Integral Representations ◽  
Dimensional Case ◽  
Partition Functions ◽  
Fermion Field ◽  
Two Dimensional

The integral representations for the partition functions of Ising models are surveyed. The connection with the underlying fermion field in the two-dimensional case is discussed. The relation between the low and the high-temperature expansions is examined.

Inference for general Ising models

1982 ◽  
Vol 19 (A) ◽  
pp. 345-357 ◽  
Author(s):  
David K. Pickard
Keyword(s):  
Ising Model ◽  
Thermodynamic Parameters ◽  
Limit Theorems ◽  
Ising Models ◽  
Confidence Regions ◽  
Likelihood Inference ◽  
Partition Functions ◽  
Polynomial Equations ◽  
Interaction Energies ◽  
Asymptotic Inference

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models. In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

Partition Function and Density of States in Models of a Finite Number of Ising Spins with Direct Exchange between the Minimum and Maximum Number of Nearest Neighbors

2016 ◽  
Vol 247 ◽  
pp. 142-147
Author(s):  
Petr Dmitrievich Andriushchenko ◽  
Konstantin V. Nefedev
Keyword(s):  
Partition Function ◽  
Density Of States ◽  
Generating Functions ◽  
Energy State ◽  
Energy Levels ◽  
Nearest Neighbors ◽  
Ising Models ◽  
Partition Functions ◽  
Ising Chain ◽  
Ising Spins

The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries

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