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Equation of State Calculations by Fast Computing Machines Nicholas Metropolis, Arianna W Equation of State Calculations by Fast Computing Machines Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller Citation: J. Chem. Phys. 21, 1087 (1953); doi: 10.1063/1.1699114 View online: http://dx.doi.org/10.1063/1.1699114 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v21/i6 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 23 Aug 2013 to 128.252.11.181. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions THE 0 R Y 0 F T RAe KEF FEe T SIN R A D I 0 L Y SIS 0 F W ATE R 1087 instead, only water molecules with different amounts of paths (a) and (b) can be designated H 20* and those excitation energy. These may follow any of three paths: following path (c) can be designated H 20t. It seems reasonable to assume for the purpose of these calcula­ (a) The excitation energy is lost without dissociation tions that the ionized H 0 molecules will become the into radicals (by collision, or possibly radiation, as in 2 H 20t molecules, but this is not likely to be a complete aromatic hydrocarbons). correspondence. (b) The molecules dissociate, but the resulting radi­ In conclusion we would like to emphasize that the cals recombine without escaping from the liquid cage. qualitative result of this section is not critically de­ (c) The molecules dissociate and escape from the pendent on the exact values of the physical parameters cage. In this case we would not expect them to move used. However, this treatment is classical, and a correct more than a few molecular diameters through the dense treatment must be wave mechanical; therefore the medium before being thermalized. result of this section cannot be taken as an a priori In accordance with the notation introduced by theoretical prediction. The success of the radical diffu­ Burton, Magee, and Samuel,22 the molecules following sion model given above lends some plausibility to the occurrence of electron capture as described by this 22 Burton, Magee, and Samuel, J. Chern. Phys. 20, 760 (1952). crude calculation. Further work is clearly needed. THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion. I. INTRODUCTION II. THE GENERAL METHOD FOR AN ARBITRARY POTENTIAL BETWEEN THE PARTICLES HE purpose of this paper is to describe a general T method, suitable for fast electronic computing In order to reduce the problem to a feasible size for machines, of calculating the properties of any substance numerical work, we can, of course, consider only a finite which may be considered as composed of interacting number of particles. This number N may be as high as individual molecules. Classical statistics is assumed, several hundred. Our system consists of a squaret con­ only two-body forces are considered, and the potential taining N particles. In order to minimize the surface field of a molecule is assumed spherically symmetric. effects we suppose the complete substance to be periodic, consisting of many such squares, each square contain­ These are the usual assumptions made in theories of ing N particles in the same configuration. Thus we liquids. Subject to the above assumptions, the method define dAB, the minimum distance between particles A is not restricted to any range of temperature or density. and B, as the shortest distance between A and any of This paper will also present results of a preliminary two­ the particles B, of which there is one in each of the dimensional calculation for the rigid-sphere system. squares which comprise the complete substance. If we Work on the two-dimensional case with a Lennard­ have a potential which falls off rapidly with distance, Jones potential is in progress and will be reported in a there will be at most one of the distances AB which later paper. Also, the problem in three dimensions is can make a substantial contribution; hence we need being investigated. consider only the minimum distance dAB. * Now at the Radiation Laboratory of the University of Cali­ t We will use thl~ two-dimensional nomenclature here since it fornia, Livermore, California. is easier to visualize. The extension to three dimensions is obvious. Downloaded 23 Aug 2013 to 128.252.11.181. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 1088 MET R 0 POL IS, R 0 SEN B L U T H, R 0 SEN B L U T H, TEL L E R, AND TEL L E R Our method in this respect is similar to the cell configurations with a probability exp ( - E/ kT) and method except that our cells contain several hundred weight them evenly. particles instead of one. One would think that such a This we do as follows: We place the N particles in any sample would be quite adequate for describing any one­ configuration, for example, in a regular lattice. Then phase system. We do find, however, that in two-phase we move each of the particles in succession according systems the surface between the phases makes quite a to the following prescription: perturbation. Also, statistical fluctuations may be X~X+a~l sizable. (3) If we know the positions of the N particles in the square, we can easily calculate, for example, the poten­ where a is the maximum allowed displacement, which tial energy of the system, for the sake of this argument is arbitrary, and h and ~2 N N are random numbers§ between (-1) and 1. Then, after E=! L L V(dii)· (1) we move a particle, it is equally likely to be anywhere ;=lj=1 ;~j within a square of side 2a centered about its original position. (In accord with the periodicity assumption, (Here V is the potential between molecules, and dij is if the indicated move would put the particle outside the the minimum distance between particles i and j as square, this only means that it re-enters the square from defined above.) the opposite side.) In order to calculate the properties of our system we We then calculate the change in energy of the system use the canonical ensemble. So, to calculate the equi­ b.E, which is caused by the move. If b.E<O, i.e., if librium value of any quantity of interest F, the move would bring the system to a state of lower energy, we allow the move and put the particle in its FexP(-E/kT)d2NPd2Nq]/ new position. If b.E>O, we allow the move with F=[f probability exp( - b.E/kT); i.e., we take a random number ~3 between 0 and 1, and if ~3 < exp ( - b.E/ kT), we move the particle to its new position. If ~3 [f eXP(-E/kT)d2NPd2Nq} (2) > exp( - b.E/kT), we return it to its old position. Then, whether the move has been allowed or not, i.e., where (d2 npd2 nq) is a volume element in the 4N-dimen­ whether we are in a different configuration or in the sional phase space. Moreover, since forces between original configuration, we consider that we are in a new particles are velocity-independent, the momentum in­ configuration for the purpose of taking our averages. So tegrals may be separated off, and we need perform only M the integration over the 2N-dimensional configuration F= (l/M) L Fh (4) space. It is evidently impractical to carry out a several i=1 hundred-dimensional integral by the usual numerical methods, so we resort to the Monte Carlo method.t where F i is the value of the property F of the system The Monte Carlo method for many-dimensional in­ after the jth move is carried out according to the com­ tegrals consists simply of integrating over a random plete prescription above. Having attempted to move a sampling of points instead of over a regular array of particle we proceed similarly with the next one. points. We now prove that the method outlined above does Thus the most naive method of carrying out the choose configurations with a probability exp(-E/kT). integration would be to put each of the N particles at a Since a particle is allowed to move to any point within random position in the square (this defines a random a square of side 2a with a finite probability, it is clear point in the 2N-dimensional configuration space), then that a large enough number of moves will enable it to calculate the energy of the system according to Eq.
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