A general, feasible approach is presented for the evaluation of the statistical thermodynamics of interacting lattice gases. Exact solutions are obtained for lattice systems of infinite length and increasing finite width, using the matrix method which treats all densities on an equivalent basis. Through the application of symmetry reduction and the use of an electronic computer to perform logical as well as arithmetical operations, widths of up to 24 sites for two‐dimensional lattices can be handled. For examples studied, rapid convergence is obtained away from transition regions and in the vicinity of phase transitions the behavior appears to be a sufficiently regular function of the width to allow meaningful extrapolation to systems of infinite width. Two problems of two‐dimensional lattice gases are solved as illustrations of the technique: the square and triangular lattice gases with infinite repulsive interactions preventing the simultaneous occupancy of adjacent lattice sites (excluded‐volume effect). Both systems exhibit phase transitions which are most likely second order at densities of 74.2% (square) and 83.7% (triangular) of the close‐packed density. For both lattices the compressibility is infinite at the transition point, becoming infinite linearly with the logarithm of the width of the lattice for the square lattice and perhaps slightly more strongly for the triangular lattice.

1.
R. H.
Fowler
 and
G. S.
Rushbrooke
,
Trans. Faraday Soc.
33
,
1272
(
1937
).
Google Scholar
Crossref
Search ADS
 
2.
H. A.
Kramers
 and
G. H.
Wannier
,
Phys. Rev.
60
,
252
,
263
(
1941
).
Google Scholar
Crossref
Search ADS
 
3.
E. A.
DiMarzio
 and
F. H.
Stillinger
, Jr.,
J. Chem. Phys.
40
,
1577
(
1964
).
Google Scholar
Crossref
Search ADS
 
4.
J. F. Nagle, thesis, Yale University, 1965.
5.
D. A. Chesnut and F. H. Ree (private communication).
6.
T. L. Hill, Statistical Mechanics (McGraw‐Hill Book Co., Inc., New York, 1956), Chap. 7.
7.
G. S.
Rushbrooke
 and
H. I.
Scoins
,
Proc. Roy. Soc. (London)
A230
,
74
(
1955
).
Google Scholar
 
8.
E. Bodewig, Matrix Calculus (Interscience Publishers, Inc., New York, 1959), p. 269.
9.
Only Stage one has been executed for M = 24. This required 1 h of 7040 time. To obtain numerical results would have required an additional 24 h or more, an investment not deemed necessary for the solution of problems solved so far.
10.
C. Jordan, Calculus of Finite Differences (Röttig and Romwalter, Sopron, Hungary, 1939), p. 164.
11.
For the square lattice and equivalences induced by DM it is readily seen for any element h of DM that if h(ψi) = ψk and h(ψj) = ψl, then Pij = Pkl; this is in fact stronger than the requirement for an admissible group.
12.
L. K.
Runnels
,
Phys. Rev. Letters
15
,
581
(
1965
).
Google Scholar
Crossref
Search ADS
 
13.
D. S.
Gaunt
 and
M. E.
Fisher
,
J. Chem. Phys.
43
,
2480
(
1965
); several earlier references are contained in this work.
Google Scholar
Crossref
Search ADS
 
14.
L.
Onsager
,
Phys. Rev.
65
,
117
(
1944
).
Google Scholar
Crossref
Search ADS
 
15.
To achieve the greatest accuracy possible here (and also for the heat‐capacity maxima) a parabola was fit to the three points nearest to each extremum.
16.
D. M.
Burley
,
Proc. Phys. Soc. (London)
75
,
262
(
1960
).
Google Scholar
Crossref
Search ADS
 
17.
D. M.
Burley
,
Proc. Phys. Soc. (London)
85
,
1173
(
1965
).
Google Scholar
Crossref
Search ADS
 
18.
D. A. Chesnut (private communication).
19.
D. S. Gaunt and M. E. Fisher (private communication).
20.
The strong symmetry shown in Footnote 11 for the square lattice now holds only if h is a rotation about the M‐fold axis. If h is a twofold rotation perpendicular to the main axis, then it may well happen that Pij≠Pkl. There is, however, a unique state ψl′ associated with each triple (i, j, k) for which Pij = Pkl′, and this is sufficient to satisfy the admissibility requirement. The prescription for l′ is ψl′ = Cm[h(ψj)], where h is a twofold rotation and CM is a rotation of 2π/M about the M‐fold axis.
21.
B. J.
Alder
 and
T. E.
Wainwright
,
Phys. Rev.
127
,
359
(
1962
).
Google Scholar
Crossref
Search ADS
 
22.
C.
Domb
,
Advan. Phys.
9
,
149
(
1960
).
Google Scholar
Crossref
Search ADS
 
23.
C. Berg, The Theory of Graphs (John Wiley & Sons, Inc., New York, 1962), p. 35.
24.
J. Riordan, An Introduction to Combinationatorial Analysis (John Wiley & Sons, Inc., New York, 1958), Chap. 3.
25.
Reference 10, p. 548.
26.
M. Hamermesh, Group Theory (Addison‐Wesley Publ. Co., Inc., Reading, Mass., 1962), p. 105.
27.
E. W.
Montroll
,
J. Chem. Phys.
9
,
706
(
1941
).
Google Scholar
Crossref
Search ADS
 
28.
O.
Perron
,
Math. Ann.
64
,
248
(
1907
).
Google Scholar
Crossref
Search ADS
 
29.
The matrix P is an irreducible nonnegative matrix; see F. R. Gantmacher, Applications of the Theory of Matrices (Interscience Publishers, Inc., New York, 1959), Chap. 3.
This content is only available via PDF.
© 1966 American Institute of Physics.
1966
American Institute of Physics
You do not currently have access to this content.