10.8. The Classical Lattice Gas
In the presence of inter-particle interactions, the relevant perturbative calculations
become rather lengthy and tedious. We shall therefore be contented to discuss only
one rather crude, but straightforward, approximation to a non-ideal classical gas
called the lattice gas model, which is of some importance during the early
development of the theory of phase transitions.
Consider then a gas whose molecules interact through a pairwise potential W(r). The
Hamiltonian for N such molecules is therefore
N
1 2 1 N
(10.94)
HN  
pi   W x i  x j
2 i  j 1
i 1 2m


Inserting this into (10.14) gives
Z can   ,V , N  
  N 1 2 1 N
 
1
3N
3N
d
x
d
p
exp


p

W
x

x





i
i
j
3N

h N!
2 i  j 1
 
  i 1 2m
1  2 m 
 3N 
h N !   

3N / 2
 1 N

3N
d
x
exp
    W xi  x j 

 2 i  j 1




(10.95)
The remaining integration can be viewed as a sum over all possible configurations
(spatial positions) of the molecules. An approximate way to specify these
configurations begins by partitioning the total volume V into a large number of cells,
each with volume v equal roughly to the ‘size’ of a molecule. A given configuration
is then specified by a list of the occupation number n of cell  for all . In the
classical regime, each molecule possesses an impenetrable hard core that defines
roughly its ‘size’. Hence, n  0,1 with
V /v
n  N .


1
So far, we have not specified the shapes of the cells.
Obviously, the approach is
meaningful only if Zcan calculated from it is insensitive to the exact sizes and shapes
of the cells. In practice, one usually choose a space- filling shape so that cells make
up a periodic lattice structure; hence the name of the model.
Since the positions xi of the molecules are approximated by cell labels , the pair

potential are likewise by approximated as W xi  x j

W  . Obviously, this is
meaningful only if W varies negligibly over distances on the order of the size of each
cell. Now, for a given set of N occupied cells, each of the N! permutations among
the N particles will contribute equally to the integral in (10.95).
1  2 m 
Z can   ,V , N   3 N 
h   
where

N
3N / 2

vN
N
n 
Hence,
 1

exp    W  n n 
 2 

n 
denotes the sum over all sets of
n 
such that
(10.96)
n  N .


By
(10.25), the grand canonical partition function is
 2 m 3/ 2  
Z gr   ,V ,      2  ve 
N 0   h  




  e  N
N 0

n 
N
N

N
n 
 1

exp    W  n n 
 2 

 1

exp    W  n n 
 2 

where
 2 m 3/ 2 

    ln  2  v 
 h  

N
Since
 
N 0 n 
N
(10.99a)
  , we have
n 


1
Z gr   ,V ,     exp     n   W  n n 
2 

n 


(10.98a)
In order to borrow the many techniques developed in the study of quantum angular
momentum, we can throw (10.96) into a form similar to that describing spin- spin
interactions. To this end, we define the spin variable s by ni 
0
 1
ni    s  
1
1
as befit the polarizations of a spin 1/2 particle.
1
1  si  so that
2
Also, for short- range interactions,
the sum
 W    W   W
(10.97)
should be independent of  for cells not too close to the boundaries. Thus, in the
thermodynamic limit, we have W  W0 for all .
Hence,
  n 

where
1
1
1
W n n    1  s   W 1  s  1  s 

2 
2 
8 

1 
1  1 
1 
1

    W0        W0  s  W  s s
2  
4  2  
2 
8 

V 
1  1 
1 
1
    W0        W0  s  W  s s
2v 
4  2  
2 
8 

V 
1  1
   W0   
2v 
4  2
V
1

v

1
s  W  s s

8 

is the total number of cells and
3/ 2
 1
1
1  2 m1 
     W0    ln  2  v   W0
2
  h  
 2
(10.99)
Thus, (10.98) becomes
Z gr   ,V ,    N
1
1

s   W  s s 
 exp  2   
8 

s 


(10.98)
where
1 
 V 
N  exp  
   W0  
4 
 2v 
(10.100’)
The interesting point is that, eq(10.98) has the same form as the grand partition
function of the Ising model for ferromagnetism [see Chapter 11]. This
experimentally well verified equivalence between 2 entirely different physical
systems will play a significant role in the theory of phase transitions.