Abstract
Monte Carlo simulation techniques are discussed, with special emphasis on those technical aspects that are important for the simulation of dense liquids and solids. In these notes, the Metropolis sampling scheme is introduced as a special case of importance sampling. The choice of the optimum trial move is discussed, as are the problems encountered when sampling orientational and internal degrees of freedom. After introducing the MC method as a technique to measure averages in the canonical ensemble, we briefly look at simulations in other ensembles, in particular: microcanonical, isobaric-isothermal and grand-canonical. The latter ensemble forms the starting point for a discussion of particle insertion techniques to measure the chemical potential and more general acceptance-ratio and overlapping distribution schemes. We thereupon discuss the application of computer simulation techniques to the study of phase transitions. Although the emphasis is on first-order transitions, a few remarks are made about the technical problems posed by continuous phase transitions. We end with a discussion of the relative merits of Monte Carlo and Molecular Dynamics simulations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. P. Feynman, Statistical Mechanics, Benjamin, Reading (Mass.), 1972.
L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd edition, Pergamon Press, London, 1980.
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J.Chem. Phys. 21:1087 (1953).
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987.
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes, Cambridge U.P., Cambridge, 1986.
Although almost all published MC simulations on atomic and molecular systems generate trial displacements in a cube centered around the original center-of-mass position, this is by no means the only possibility. Sometimes it is more convenient to generate trial moves in a spherical volume, and it is not even necessary that the distribution of trial moves in such a volume is uniform, as long as it has inversion symmetry. For an example of a case where another sampling scheme is preferable, see: W. G. T. Kranendonk and D. Frenkel, Mol. Phys. 64:403 (1988).
The situation is of course different for trial moves that affect not all particles, but only a non-interacting subset. In that case the acceptance or rejection can be decided for each particle individually. Such a procedure is in fact very suited for Monte Carlo simulations on both vector and parallel computers.
F. J. Vesely, J. Comp. Phys. 47:291 (1982).
M. Fixman, Proc. Nat. Acad. Sci. 71:3050 (1974).
G. Ciccotti and J. P. Ryckaert, Comp. Phys. Reports 4:345 (1986).
Unfortunately, the referees of this paper thought otherwise: A. Cassandro, G. Ciccotti, V. Rosato and J. P. Ryckaert, unpublished.
M. Creutz, Phys. Rev. Lett. 50:1411 (1983).
W. W. Wood, J. Chem. Phys. 48:415 (1968).
I. R. McDonald, Mol. Phys. 23:41 (1972).
Actually, there is no need to assume a real piston. The systems with volume V and V 0 — V may both be isolated systems subject to their individual (periodic) boundary conditions. The only constraint that we impose is that the sum of the volumes of the two systems equals V 0.
R. Eppenga and D. Frenkel, Mol. Phys. 52:1303 (1984).
M. Parrinello and A. Rahman, Phys. Rev. Lett. 45:1196 (1980).
M. Parrinello and A. Rahman, J. Appl. Phys. 52:7182 (1981).
H. C. Andersen, J. Chem. Phys. 72:2384 (1980)
R. Najafabadi and S. Yip, Scripta Metall. 17:1199 (1983).
Except that one should never use the constant stress method for uniform fluids, because the latter offer no resistance to the deformation of the unit box, and very strange (flat, elongated etc.) box-shapes may result. This may have serious consequences because simulations on systems that have shrunk considerably in any one dimension tend to exhibit appreciable finite-size effects.
D. C. Wallace, in: Solid State Physics, H. Ehrenreich, F. Seitz and D. Turnbull, eds., Academic Press, New York, (1970), Volume 25, p 301.
J. R. Ray and A. Rahman, J. Chem. Phys. 80:4423 (1984).
G. E. Norman and V. S. Filinov, High Temp. Res. USSR 7:216 (1969).
D. J. Adams, Mol. Phys. 28:1241 (1974).
D. J. Adams, Mol. Phys. 29:307 (1975).
D. J. Adams, Mol. Phys. 32:647 (1976).
D. J. Adams, Mol. Phys. 37:211 (1979).
L. A. Rowley, D. Nicholson and N. G. Parsonage, J. Comp. Phys. 17:401 (1975).
J. Yao, R. A. Greenkorn and K. C. Chao, Mol. Phys. 46:587 (1982).
M. Mezei, Mol. Phys. 40:901 (1980).
J. P. Valleau and K. L. Cohen, J. Chem. Phys. 72:3935 (1980).
W. van Meegen and I. Snook, J. Chem. Phys. 73:4656 (1980).
D. Frenkel in: Molecular Dynamics Simulations of Statistical Mechanical Systems, Proceedings of the 97th International School of Physics ‘Enrico Fermi’, G. Ciccotti and W. G. Hoover, editors. North-Holland, Amsterdam, 1985, p.151.
A. Z. Panagiotopoulos, Mol. Phys. 61:813 (1987).
A. Z. Panagiotopoulos, N. Quirke, M. Stapleton and D. J. Tildesley, Mol. Phys. 63:527 (1988).
A. Z. Panagiotopoulos, Mol. Phys. 62:701 (1987).
B. Widom, J. Chem. Phys. 39:2808 (1963).
K.S. Shing and S. T. Chung, J. Phys. Chem. 91:1674 (1987).
P. Sindzingre, G. Ciccotti, C. Massobrio and D. Frenkel, Chem. Phys. Lett. 136:35 (1987).
J. L. Lebowitz, J. K. Percus and L. Verlet, Phys. Rev. 153:250 (1967).
P. Sindzingre, C. Massobrio, G. Ciccotti and D. Frenkel, Chemical Physics (to appear).
K. S. Shing and K. E. Gubbins, Mol. Phys. 46, 1109 (1982).
K. S. Shing and K. E. Gubbins, Mol. Phys. 49, 1121 (1983).
C. H. Bennett, J. Comput. Phys. 22, 245 (1976).
O. G. Mouritsen, Computer Studies of Phase Transitions and Critical Phenomena, Springer, Berlin, 1984.
K. Binder, Applications of the Monte Carlo Method in Statistical Physics, Springer, Berlin, 1984.
W. G. Hoover and F. H. Ree, J. Chem. Phys. 47:4873 (1967).
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd edition, Academic Press, London, 1986.
D. Frenkel and A. J. C. Ladd, J. Chem. Phys. 81:3188 (1984).
D. Frenkel and B. M. Mulder, Mol. Phys. 55:1171 (1985).
A. Stroobants, H. N. W. Lekkerkerker and D. Frenkel, Phys. Rev. A36:2929 (1987).
D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants, Nature 332:822 (1988).
R. Zwanzig and N. K. Ailawadi, Phys. Rev. 182:280 (1969).
G. Jacucci and A. Rahman, Nuovo Cimento D4:341 (1984).
D. Frenkel and R. Eppenga, Phys. Rev. A31:1776 (1985).
P. J. Rossky, J. D. Doll and H. L. Friedman, J. Chem. Phys. 69:4628 (1978).
S. E. Koonin, Computational Physics, Benjamin/Cummings, Menlo Park, 1986.
J. P. Valleau and S. G. Whittington, in Statistical Mechanics, part A, B. J. Berne (editor), Plenum, New York (1977), p. 137
J. P. Valleau and G. M. Torrie, ibidem in Statistical Mechanics, part A, B. J. Berne (editor), Plenum, New York (1977), p 169.
D. W. Heermann, Computer Simulation Methods in Theoretical Physics, Springer, Berlin, 1986.
G. Ciccotti, D. Frenkel and I. R. McDonald, Simulation of Liquids and Solids, North-Holland, Amsterdam, 1987. This reprint collection contains, among others, references [3,14,18,19,32,41,45,48,50,54]
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Frenkel, D. (1990). Monte Carlo Simulations. In: Catlow, C.R.A., Parker, S.C., Allen, M.P. (eds) Computer Modelling of Fluids Polymers and Solids. NATO ASI Series, vol 293. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2484-0_4
Download citation
DOI: https://doi.org/10.1007/978-94-009-2484-0_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7621-0
Online ISBN: 978-94-009-2484-0
eBook Packages: Springer Book Archive