Discrete systems in statistical mechanics

Systems whose microscopic states are defined by specifying the states at each point (site) of a given spatial lattice. One of their applications is their use as models of a solid body in which a microscopic motion, resulting in changes of state at the lattice points, is studied, and each change is considered to be independent of the others. One of the simplest systems — the Ising model (1925) — is characterized by the Hamiltonian [1]

$$ H = - h \sum _ {1 \leq i \leq N } \sigma _ {i} - \sum _ {1 \leq i \leq j \leq N } J ( i , j ) \sigma _ {i} \sigma _ {j} , $$

where $ i = \mathbf r _ {i} $ are the coordinates of the lattice points and $ \sigma _ {i} = \pm 1 $.

This model is utilized in studying substitution alloys, magnetic substances, rarified gases, etc. [2]. Discrete systems of this type characteristically display long-range order at temperatures below a transition point — that is, a general regularity in the direction of the spins $ \sigma _ {i} $ of the magnetic substances, or a regular sequence of different atoms in binary alloys, which is lost as the temperature is increased at a point $ \theta $( the transition point), with a characteristic singularity of the heat capacity $ c _ {v} $. In the case of a short-range order — a correlation between a given point and the neighbouring points — on the contrary such a sudden change does not take place. A qualitative description of ordering phenomena is obtained by theories of the type of molecular field theory. Despite the mathematical simplicity of the model, an exact general solution was obtained only for a one-dimensional model and for a planar ferromagnetic lattice $ ( J ( i , j ) > 0 ) $ with an interaction of nearest neighbours only in the case $ h = 0 $. A one-dimensional model does not involve phase transitions, while the two-dimensional model has a singularity of logarithmic type of the heat capacity (only if $ N \rightarrow \infty $). In the general case approximation methods for low and high temperature ranges have been developed.

Other extensively used models include the Heisenberg model of magnetic substances, with a Hamiltonian which differs from the Hamiltonian of Ising by the fact that the numbers $ \sigma _ {i} $ are replaced by $ \sigma _ {i} ^ {z} $ and that the product $ \sigma _ {i} \sigma _ {j} $ is replaced by $ ( \sigma _ {i} , \sigma _ {j} ) $, where $ \sigma _ {i} $ are the Pauli matrices.

An asymptotically exact (as $ N \rightarrow \infty $) study by the method of approximation Hamiltonians [3] is valid in the case of discrete systems of a certain class with a certain type of interaction between the lattice points.

The investigation of discrete systems has stimulated the development of some basic ideas in scaling theory and the Wilson procedure (renormalization group) in the recent theory of phase transitions and critical phenomena [4].

References

[1]	K. Huang, "Statistical mechanics" , Wiley (1963)
[2]	J.M. Ziman, "Principles of the theory of solids" , Cambridge Univ. Press (1972)
[3]	N.N. Bogolyubov jr., "Method for studying model Hamiltonian" , Pergamon (1972) (Translated from Russian)
[4]	K.G. Wilson, J. Kogut, "The renormalization group and the $\epsilon$-expansion" Phys. Rep. , 12c (1974) pp. 75–199