Order-disorder statistics. I

In 1941 Kramers & Wannier discussed the statistical mechanics of a two-dimensional Ising model of a ferromagnetic. By making use of a ‘screw transformation’ they showed that the partition function was the largest eigenvalue of an infinite matrix of simple characteristic structure. In the present paper an alternative method is used for deriving the partition function, and this enables the ‘screw transformation’ to be generalized to apply to a number of problems of classical statistical mechanics, including the three-dimensional Ising model. Distant neighbour interactions can also be taken into account. The relation between the ferromagnetic and order-disorder problems is discussed, and it is shown that the partition function in both cases can be derived from a single function of two variables. Since distant neighbour interactions can be taken into account the theory can be formally applied to the statistical mechanics of a system of identical particles.

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THE SOLUTION OF THE THREE-DIMENSIONAL TWISTED GROUP LATTICES

International Journal of Modern Physics B ◽

10.1142/s0217979292000815 ◽

1992 ◽

Vol 06 (10) ◽

pp. 1631-1645 ◽

Cited By ~ 1

Author(s):

STUART SAMUEL

Keyword(s):

Partition Function ◽

Statistical Mechanics ◽

Three Dimensional ◽

Group Representations ◽

Screw Dislocations ◽

Irreducible Group ◽

Twisted Group ◽

Free Propagation

We define new lattices called d-dimensional twisted group lattices. They are similar to ordinary lattices except that abstract screw dislocations are present at the centers of all plaquettes. Some physical aspects are enumerated. We consider the statistical mechanics system of free propagation on the three-dimensional twisted group lattice. For this case, the partition function is explicitly computed by finding the irreducible group representations.

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Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

Materials Science Forum ◽

10.4028/www.scientific.net/msf.845.150 ◽

2016 ◽

Vol 845 ◽

pp. 150-153

Author(s):

Andrey N. Vakilov

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Ising Model ◽

Partition Function ◽

Critical Behavior ◽

Critical Exponents ◽

Three Dimensional ◽

Finite Size ◽

Finite Size Scaling ◽

Zeros Of Partition Function

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

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Accurate estimate ofνfor the three-dimensional Ising model from a numerical measurement of its partition function

Physical Review Letters ◽

10.1103/physrevlett.59.803 ◽

1987 ◽

Vol 59 (7) ◽

pp. 803-806 ◽

Cited By ~ 67

Author(s):

Gyan Bhanot ◽

Román Salvador ◽

Steve Black ◽

Paul Carter ◽

Raúl Toral

Keyword(s):

Ising Model ◽

Partition Function ◽

Three Dimensional ◽

Accurate Estimate

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Zeros in Partition Function and Critical Behavior of Disordered Three Dimensional Ising Model

Journal of Siberian Federal University Mathematics & Physics ◽

10.17516/1997-1397-2017-10-1-128-131 ◽

2017 ◽

Vol 10 (1) ◽

pp. 128-131

Author(s):

Andrey N. Vakilov ◽

Keyword(s):

Ising Model ◽

Partition Function ◽

Critical Behavior ◽

Three Dimensional

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Exact Solution for Three-Dimensional Ising Model

Symmetry ◽

10.3390/sym13101837 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1837

Author(s):

Degang Zhang

Keyword(s):

Boundary Condition ◽

Ising Model ◽

Partition Function ◽

Operator Algebras ◽

Three Dimensional ◽

Two Dimensions ◽

Critical Temperatures ◽

Cubic Crystal ◽

Infinite Crystal ◽

Analytical Expressions

The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.

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