A Brief History of the Use of the Metropolis Method at LANL in the 1950s

Transcription

1 A Brief History of the Use of the Metropolis Method at LANL in the 1950s William W. Wood Los Alamos National Laboratory Los Alamos, New Mexico Abstract. A personal view of the use of the Metropolis Algorithm in statistical mechanics calculations at Los Alamos during the 1950s will be presented, based on [1] and [2]. INTRODUCTION I came to Los Alamos in the fall of 1950 to work in the explosives division under Duncan P. Macdougall and Eugene H. Eyster. I had nothing to do with the invention of the Metropolis Algorithm, and indeed did nothing with it until several years later. I had done my doctoral work with Professor John G. Kirkwood at Cal Tech, who was then a consultant to the explosives division. He and I worked closely together until his death in 1959, and I came to admire him greatly. The first time I heard about what later would become known as the Metropolis Algorithm was when Marshall Rosenbluth gave me a preprint of [3] in early 1953, which led me to mention it briefly to Kirkwood in a letter dated March 3, But he may well have heard about it the previous summer during his consulting visit. Kirkwood quickly realized the potential of the method and urged that we should use it to calculate the equation of state of Lennard Jones molecules, that is to say, the equation of state of particles interacting with the Lennard Jones 6 12 potential V r ε r r 12 r r We had already been using that potential for some calculations based on the Lennard Jones cell theory, for the equation of state of detonation products. We began our work with the Metropolis Algorithm in late 1953 or early 1954, I do not recall which. (Many of my research notes of that period later fell victim to bureaucratic enthusiasm for reducing the amount of material in the files.) It was, of course, quite natural for us to begin our work in the belief that the Rosenbluths had done as well as could be done on the equation of state of hard spheres [4] with the available calculators of that time (MANIAC, in their case). It was a simple matter to modify a Lennard Jones code to calculate for hard spheres, and we did indeed make such a run as a check on the correctness of our program. Unfortunately, we happened to choose a density somewhat above what would later become known as the transition region for our IBM 704 run, and found reasonable agreement with the Rosenbluths results. 6 (1)

2 In the meantime, Kirkwood s enthusiasm for the Metropolis Algorithm had temporarily waned. This was due to the fact that the Rosenbluths results for the equation of state of hard spheres [4] disagreed badly with the results from the work that he and his students, including Alder, had done using the superposition approximation in solving the Born Green Kirkwood integral equation, which was a difficult calculation in its own right on the then-available calculators. To make matters worse, the simple cell theory gave somewhat better agreement with the Rosenbluths results than did the integral equation results. To see the nature of his objections, we will need to go slightly into the details of the Metropolis Algorithm. THE METROPOLIS ALGORITHM The essence of the Metropolis Algorithm is the generation of a certain kind of random walk in the configuration space of the system. The simplest kind of system to consider is one in which there is a finite number of possible configuration states, say M. Then the method consists of the generation of the type of random walk known as a Markov chain. Suppose we want to estimate the canonical average of some function f i of the state i of the system, f M 1 f i exp Q i 1 M βe i Q exp βe i (3) i 1 with β 1 k B T,andE i being the energy of state i. The realization average over S steps of the Markov chain is S S 1 f S f i n (4) n 1 i n being the state of the system on step n. Then S f (2) f (5) in probability as S tends to, provided p i j exp βe i p j i exp βe j (6) (microscopic reversibility), along with other important conditions regarding the ergodicity of the chain; p i j is the one-step probability that, if the system is in state i on step n, then it will be in state j on step n 1. One way in which the microscopic reversibility condition can be satisfied is to let p i j A i j π i j j i (7) with A i j being a symmetric stochastic matrix (i.e., M j 1 A i j 1, for all i),

3 and 1 if E π i j exp β E 0 if E 0; here E E j E i. Then the necessary normalization condition can be satisfied by letting for each i. (8) M p i i 1 j p 1! i j (9) i KIRKWOOD S OBJECTIONS For a system of hard spheres Eqs.(8 9) become and π i j 1 if E 0 0 if E p i i 1 f i (11) where f i is the fraction of states which can be reached from state i, but which result in an overlap between two (or more) hard spheres. It follows that one must count state i n again in Eq.(4) in order to satisfy the essential normalization condition, but it was this prescription to which Kirkwood first objected, in a letter to me on November 16, 1953, in which he attributed the opposite procedure (keep trying until a successful move is found, without counting state i n again) to Frankel and Lewinson. There then ensued a lively correspondence between Kirkwood and me, and between Kirkwood and Marshall Rosenbulth. Marshall wrote up a more detailed proof of the method, and the two Rosenbluths did a calculation for a one dimensional system of hard rods, finding good agreement with the (exact) Tonks equation of state. I did a calculation, both analytically and numerically, for a system of one hard rod confined between hard walls. These efforts temporarily convinced Kirkwood of the validity of the procedure, according to a letter to me on December 21,1953. But when I visited Kirkwood in New Haven in early January, 1954, he raised a second objection, this time to the bias in Eq.(8) in favor of a state of lower energy. He proposed that one should use instead the more symmetric π i j exp βe j exp βe i exp βe j (12) Both Marshall and I agreed that Eq.(12) was a valid procedure, but we intuitively thought Eq.(8) was a better choice. Finally, on April 24, 1954, Kirkwood wrote to me I finally was able to find the time to devote a weekend to thinking about the Monte Carlo problem. He now agreed that Eq.(8) was one of the correct ones, but not to let us (10)

4 off too lightly, he continued with Marshall s arguments were at no time helpful, and your modes of presentation were opaque. From that time on he was an active proponent of the method and was urgently prompting me to publish our results as soon as possible. By October, 1954, we had results for LJ systems on the k B T ε 2 74 isotherm over the fluid range V V $ 0 95, and by April, 1955, we had seen the first indication of the fluid solid phase transition, in the form of the now familiar two level structure of the pressure versus Markov chain time. Those results were not published until 1957 [5], because we got distracted by the disagreement of the Alder and Wainwright molecular dynamics results and the Rosenbluths results for hard spheres, as will be described in the next section. THE HARD SPHERE TRANSITION In August, 1956, there was a symposium on the theory of transport processes in Brussels, which I did not attend, at which Alder and Wainwright [6] presented a variety of results from their molecular dynamics calculations. As far as I am aware, this was the first public mention of the molecular dynamics method. There they called attention to the significant differences between their results and those of the Rosenbluths [4]. I was unaware of those differences until I received a letter from Alder, written on October 8, 1956, which mentioned it. As soon as I became aware of the discrepancy, I suspected that the disagreement was due to the shortness of the runs made by the Rosenbluths, typically only a few hundred attempted moves per particle, as well as to their choice of system size. We had already noticed that in order to obtain reliable results for Lennard Jones particles, something on the order of several thousand moves per particle were required. In November, 1956, Alder and Wainwright visited Los Alamos, and our conversations resulted in my agreeing that we would repeat the Rosenbluths calculations in the region around V V 0 1 6(V 0 being the close packed volume) where the disagreement was largest. There then ensued a period of intense collaboration between us at Los Alamos and Alder and Wainwright at Livermore. It was of course an easy matter to convert our Lennard Jones code to calculate for hard spheres, although the code was not very efficient for systems of more than N 32 particles, since it examined all N 1 interactions of the displaced particle. So we concentrated our efforts on systems of 32 hard spheres, until a more efficient code could be developed [7]. By mid-december, 1956, we had made Monte Carlo runs at V V 0 = 1.5, 1.6, and 1.7 for systems of N=32 and 108 particles and had observed the transient melting of the initial lattice configuration, interpreting the results as evidence of the existence of a first order phase transition in the hard sphere system. Runs started from the initial lattice configuration tended to remain in the initial ordered arrangement, with all particles remaining relatively near their initial positions, suggestive of the solid state, for some number of attempted moves per particle; but then, over a small number of moves per particle, the initial lattice would suddenly disorder into a state of relatively rapid diffusion, suggestive of the fluid state, with an accompanying increase in the computed pressure. Those results were communicated to Alder and Wainwright in letters written on

5 December 4, 1956, and January 4, 1957, and also, of course, to Kirkwood. Kirkwood was delighted, as he had previously predicted [8] such a transition. Alder and Wainwright were skeptical of this interpretation at first, because they were not able to observe the disordering of the initial lattice configuration for the larger systems they were using, and because they were using a somewhat slower UNIVAC calculator than the IBM 704. Both groups of course realized that such results were very far from definitively proving the existence of such a phase transition. In January, 1957, a symposium on the many body problem was held at the Stevens Institute of Technology. I was not present, but Kirkwood presented our results, including the phase transition interpretation. Alder argued against it, at the time. The proceedings were not published until 1963 [9], after Kirkwood s death in 1959, at which time Alder and Wainwright remarked in their published paper... but it is clear that some transition is occurring in that region, presumably a first order one from a fluid to a solid phase. In March, 1957, the molecular dynamics results and the Monte Carlo results at V V 0 = 1.6 and N = 32 were in serious disagreement, the latter giving higher values of the pressure, after discarding the initial, solid-like portion of the run. It turned out that the molecular dynamics runs were too short to disorder the initial lattice, as became clear after the two groups interchanged their final configurations, which were then used as the initial configurations for additional runs by the other group. In April, 1957, I arranged an invitation for Alder to come to Los Alamos that summer. He accepted, and brought with him a molecular dynamics code for the IBM 704. Between June and August, 1957, he used about 180 hours of time on those machines, during which time we prepared our presentations for the IUPAP meeting in Varenna, in September, which were published in Nuovo Cimento [10],[11] and as adjacent Letters to the Editor in the Journal of Chemical Physics [12],[13]. EPILOGUE Until recently, I had always recalled this period of friendly collaboration with Alder and Wainwright as a very pleasant one. Thus I was quite surprised to learn, after a colleague pointed it out to me, that Alder, at least, recalls it very differently: In the proceedings of a meeting in Alghero (Sardinia) in July 15 17, 1991, he writes [14]: So then we went back to our first love which was the hard sphere transition. I had to know whether Bill Wood had done his job right by Monte-Carlo and it turned out that he had not. By molecular dynamics we found a phase transition and subsequently Wood also found it and that s a long story. REFERENCES 1. Wood, W. W., Early History of Computer Simulations in Statistical Mechanics in Molecular Dynamics Simulation of Statistical Mechanics Systems, edited by G. Ciccotti and W. G. Hoover, North Holland, New York, 1986, pp Wood, W. W., Chapter 36, pp , in Monte Carlo and Molecular Dynamics of Condensed Matter Systems, edited by K. Binder and G. Ciccotti, Italian Physical Society, 1996.

6 3. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., J. Chem. Phys. 21, (1953). 4. Rosenbluth, M. N., and Rosenbluth, A. W., J. Chem. Phys. 22, (1954). 5. Wood, W. W., and Parker, F. R., J. Chem. Phys (1957). 6. Alder, B. J., and Wainwright, T. E., Molecular Dynamics by Electronic Computers, in International Symposium on the Statistical Mechanical Theory of Transport Processes, Brussels, 1956, edited by I. Prigogine, Interscience, New York, 1958, pp Wood, W. W., and Jacobson, J. D., Monte Carlo Calculations in Statistical Mechanics, in Proceedings of the Western Joint Computer Conference, San Francisco, 1959, pp Kirkwood, J. G., Crystallization as a Cooperative Phenomenon, in Phase Transformations in Solids, edited by R. Smoluchowski, J. E. Mayer and W. A. Weyl (Wiley, New York, 1951), pp Alder, B. J., and Wainwright, T. E., Investigation of the Many Body Problem by Electronic Computers, Chapter 29 in The Many Body Problem, edited by J. K. Percus, Interscience, New York, 1963, pp Wood, W. W., Parker, F. R., and Jacobson, J. D., Supplement to Nuovo Cimento 9, (Series 10), (1958). 11. Alder, B. J., and Wainwright, T. E., Supplement to Nuovo Cimento 9, (Series 10), (1958). 12. Wood, W. W., and Jacobson, J. D., J. Chem. Phys. 27, (1957). 13. Alder, B. J., and Wainwright, T. E., J. Chem. Phys. 27, (1957). 14. Alder, B. J., in Microscopic Simulations of Complex Hydrodynamic Phenomena, edited by M. Mareschal and B. L. Holian, Plenum Press, New York, 1992, pp