We further note that in Java

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We further note that
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L SiS} = N++ + N __ <ij>
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N+~ = 4N++ - 2yN+
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(14.9)
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= N + - N - = 2N + - N
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Fig. 14.1 Construction for the derivation of
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Substituting (14.9) into (14.1) and using (14.8) we obtain
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EJ(N+, N++) = -4 N++ + 2( y -H)N+ -(h -H)N (14.}(I Thus although a configuration of the system depends on N numbers the energ\ of a state depends only on two numbers. The partition function can also bl: written as
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N+~O
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L' g(N+, N++) e 4/3<N++
(14.11'
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where g( N +' N ++) is the number of configurations that has a given set of value(N +' N ++). The sum extends over all values of N ++ consistent with the fact that there are N spins of which N + are up. Since g( N +' N ++) is a complicated function, the form (14.11) is not a simplification over (14.2) for actual calcula-
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14.2 EQUIVALENCE OF THE ISING MODEL TO OTHER MODELS
By a change of names the Ising model can be made to simulate systems other than a ferromagnet. Among these are a lattice gas and a binary alloy.
Lattice Gas
A lattice gas is a collection of atoms whose positions can take on only discrete values. These discrete values form a lattice of given geometry with y neare~l neighbors to each lattice site. Each lattice site can be occupied by at most one atom. Figure 14.2 illustrates a configuration of a two-dimensional lattice gas in which the atoms are represented by solid circles and the empty lattice sites b\ open circles. We neglect the kinetic energy of an atom and assume that onh nearest neighbors interact, and the interaction energy for a pair of neareq neighbors is assumed to be a constant - O. Thus the potential energy of the system is equivalent to that of a gas in which the atoms are located only on lattice sites and interact through a two-body potential v( Iri - r) with
v(r)
(r (r
nearest-neighbor distance)
(14.121
(otherwise)
o o o o o o o
Fig. 14.2 A configuration of the lattice gas.
THE ISING MODEL
N Na N aa
= = =
total no. of lattice sites total no. of atoms total no. of nearest-neighbor pairs of atoms (14.13)
The total energy of the lattice gas is (14.14) and the partition function is Q G ( N a' T)
1 ,\,a e{J(oNaa N' L.
(14.15)
where the sum extends over all ways of distributing N distinguishable atoms over N lattice sites. If the volume of a unit cell of the lattice is chosen to be unity, then N is the volume of the system. The grand partition function is !2dz, N, T) =
Na~O
zNaQG(Na , T)
(14.16)
The equation of state is given, as usual, by
f3 PG = {
~ log !2dz, N, T)
1 1 B - = - z - 10g!2 (z NT) u N Bz G"
(14.17)
To establish a correspondence between the lattice gas and the Ising model, let occupied sites correspond to spin up and empty sites to spin down. Then Na - N +. In the Ising model a set of N numbers {Sl"'" SN} uniquely defines a configuration. In the lattice gas an enumeration of the occupied sites determines not one, but N a ! configurations. The difference arises from the fact that the atoms are supposed to be able to move from site to site. This difference, however, is obliterated by the adoption of "correct Boltzmann counting." Hence (14.18) where the function g(N +, N ++) and the sum L' are identical with those appearing in (14.11). The grand partition function is e{JNPc = !2 G(z, N, T) =
N+~O
L' g(N +' N +J e{J(oN++
(14.19)
A comparison between (14.19) and (14.11) yields the accompanying table of correspondence. Hence a solution of the Ising model can be immediately tran-
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scribed to be a solution of the lattice gas.
Ising Model
N+ N
e2{J(q- H)
( A[ N
1 + "2 Y
Lattice Gas
Na N-Na
"2 N+
1 (M[
The lattice gas does not directly correspond to any real system in nature. Ii we allow the lattice constant to approach zero, however, and then add to the resulting equation of state the pressure of an ideal gas, the model corresponds k a real gas of atoms interacting with one another through a zero-range potential Thus it may be interesting to study the phase transition of a lattice gas. The lattice gas has also been used as a model for the melting of a crystal lattice. When it is so used, however, the lattice constant must be kept finite. The kinetic energy of the atoms in the crystal lattice is appended in some ad hex: fashion. Such a model would only have a mathematical interest, because it is not clear that it describes melting.