The success of the “cluster variation method” (CVM) in reproducing quite accurately the free energies of Monte Carlo (MC) calculations on Ising models is explained in terms of identifying a cancellation of errors: We show that the CVM produces correlation functions that are too close to zero, which leads to an overestimation of the exact energy, E⁠, and at the same time, to an underestimation of −TS⁠, so the free energy F=E−TS is more accurate than either of its parts. This insight explains a problem with “hybrid methods” using MC correlation functions in the CVM entropy expression: They give exact energies E and do not give significantly improved −TS relative to CVM, so they do not benefit from the above noted cancellation of errors. Additionally, hybrid methods suffer from the difficulty of adequately accounting for both ordered and disordered phases in a consistent way. A different technique, the “entropic Monte Carlo” (EMC), is shown here to provide a means for critically evaluating the CVM entropy. Inspired by EMC results, we find a universal and simple correlation to the CVM entropy which produces individual components of the free energy with MC accuracy, but is computationally much less expensive than either MC thermodynamic integration or EMC.

1.
K. Binder, Monte Carlo Methods in Statistical Physics: An Introduction (Springer-Verlag, New York, 1992); Applications of the Monte Carlo Method in Statistical Physics, edited by K. Binder (Springer-Verlag, New York, 1987).
2.
R.
 
Kikuchi
,
Phys. Rev.
 
81
,
988
(
1951
);
Google Scholar
Crossref
Search ADS
 
For a modern review, see e.g.,
D.
de Fontaine
,
Solid State Phys.
47
,
33
(
1994
) or F. Ducastelle, Order and Phase Stability in Alloys (Elsevier, New York, 1991).
Google Scholar
Crossref
Search ADS
 
3.
Throughout this paper, all quantities are given in dimensionless units: kBT/J for temperature, energies are given normalized by J⁠, and entropies are given normalized by kB.
4.
See, for example, W. Schweika, in Structural and Phase Stability of Alloys, edited by J. L. Moran Lopez, F. Mejia-Lira, and J. M. Sanchez (Plenum, New York, 1992).
5.
These correlations are for a 163 Monte Carlo cell with 500 Monte Carlo steps (per site) discarded, and averages taken over the subsequent 30 000 steps.
6.
A. G.
Schlijper
and
B.
Smit
,
J. Stat. Phys.
56
,
247
(
1989
).
Google Scholar
Crossref
Search ADS
 
7.
J.
Lee
,
Phys. Rev. Lett.
71
,
211
(
1993
).
Google Scholar
Crossref
Search ADS
PubMed
 
8.
J. A.
Barker
,
Proc. R. Soc. London
216
,
45
(
1953
).
Google Scholar
 
9.
T.
Morita
,
J. Phys. Soc. Jpn.
12
,
753
(
1957
).
Google Scholar
Crossref
Search ADS
 
10.
L. G. Ferreira, S.-H. Wei, and A. Zunger, Int. J. Supercomputer Applications 5, 34 (1996).
11.
The results presented in Fig. 4 correspond to a grand canonical EMC for which the chemical potential, rather than composition x remained fixed. We also made canonical EMC runs by flipping pairs of opposite spins, thus maintaining constant x at the values 0.50 and 0.25. We obtain very similar results with either the canonical EMC or grand canonical EMC, provided that one normalizes both the EMC and CVM entropy with the ideal entropy, −x ln x−(1−x)ln(1−x)⁠, instead of ln 2 as in Fig. 4.
This content is only available via PDF.
© 1998 American Institute of Physics.
1998
American Institute of Physics
You do not currently have access to this content.