# Monte Carlo Simulations Using the Gaussian Ensemble

## Abstract

The Gaussian ensemble [1–3] consists of a sample of size N (whose properties we are seeking) in contact with a finite bath of size N’ and having a specific functional form for its entropy. This ensemble is intermediate to the microcanonical (N’=0) and canonical (N’=∟) ensembles and can be made to attain these limiting cases by continuously varying N’. If N’ is small enough, the ensemble allows us to sample the intermediate states (with negative heat capacity/compressibility) at first-order transitions. Such states are then manifest through van der Waals’ loops in the temperature-energy or pressure-volume curves of the sample and permit a relatively easy diagnosis of the order of a phase transition. This is an advantage vis-a-vis canonical-ensemble Monte Carlo simulations using the method of Metropolis et al. [4] where the intermediate states have very low probability and make the distinction between first- and second-order transitions difficult [5]. An appealing property of this ensemble is that it can be easily implemented as a Monte Carlo process so that models such as the Ising model, which lack equations of motion, can be readily simulated in the microcanonical ensemble (The molecular dynamics method, which also uses the microcanonical ensemble, is not feasible for such stochastic models). Further, by examining the behaviour of quantities such as the average energy of the sample as a function of N’, the method allows us to investigate systematically the effects of the heat bath on static properties of the sample.

## Keywords

Ising Model Canonical Ensemble Finite Sample Microcanonical Ensemble Specific Functional Form## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J.H. Hetherington: J. Low Temp. Phys., 66, 145 (1987)ADSCrossRefGoogle Scholar
- J.H. Hetherington, D.R. Stump: Phys. Rev. D, 35, 1972 (1987)ADSCrossRefGoogle Scholar
- D.R. Stump, J.H. Hetherington: Phys. Lett. B, 188, 359 (1987).ADSCrossRefGoogle Scholar
- 2.M.S.S. Challa, J.H. Hetherington: Phys. Rev. Lett., 60, 77 (1988).ADSCrossRefGoogle Scholar
- 3.M.S.S. Challa, J.H. Hetherington: submitted to Phys. Rev. B.Google Scholar
- 4.N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys., 21, 1087 (1953). See also “Monte Carlo Methods in Statistical Physics”, ed. K. Binder (Springer-Verlag, New York, 1979) for a review of Monte Carlo methods.ADSCrossRefGoogle Scholar
- 5.M.S.S. Challa, D.P. Landau and K. Binder, Phys. Rev. B, 34, 1841 (1986).ADSCrossRefGoogle Scholar
- 6.For a review of Potts models, see F.Y. Wu: Rev. Mod. Phys., 54, 235 (1982).ADSCrossRefGoogle Scholar
- 7.R.K. Pathria, “Statistical Mechanics” (Pergamon, New York, 1972), pp. 376–377.Google Scholar
- 8.R.J. Baxter, J. Phys. C, 6, L445 (1973).ADSCrossRefGoogle Scholar
- 9.T. Kihara, Y. Midzuno and T. Shizume, J. Phys. Soc. Jpn., 9, 681 (1954).ADSCrossRefGoogle Scholar
- 10.C.N. Yang and T.D. Lee: Phys. Rev., 87, 404 (1952). ibid, p.410.MathSciNetADSMATHCrossRefGoogle Scholar