THE BEGINNING of the MONTE CARLO METHOD by N. Metropolis he year was 1945. Two earth- Some Background first electronic computer—the ENIAC— shaking events took place: the at the University of Pennsylvania in Phil- successful test at Alamogordo Most of us have grown so blase about adelphia. Their mentors were Physicist and the building of the first elec- computer developments and capabilities First Class John Mauchly and Brilliant tronicT computer. Their combined impact -even some that are spectacular—that Engineer Presper Eckert. Mauchly, fa- was to modify qualitatively the nature of it is difficult to believe or imagine there miliar with Geiger counters in physics global interactions between Russia and was a time when we suffered the noisy, laboratories, had realized that if electronic the West. No less perturbative were the painstakingly slow, electromechanical de- circuits could count, then they could do changes wrought in all of academic re- vices that chomped away on punched arithmetic and hence solve, inter alia, dif- search and in applied science. On a less cards. Their saving grace was that they ference equations—at almost incredible grand scale these events brought about a continued working around the clock, ex- speeds! When he’d seen a seemingly renascence of a mathematical technique cept for maintenance and occasional re- limitless array of women cranking out known to the old guard as statistical sam- pair (such as removing a dust particle firing tables with desk calculators, he’d pling; in its new surroundings and owing from a relay gap). But these machines been inspired to propose to the Ballistics to its nature, there was no denying its new helped enormously with the routine, rela- Research Laboratory at Aberdeen that an name of the Monte Carlo method. tively simple calculations that led to Hi- electronic computer be built to deal with This essay attempts to describe the de- roshima. these calculations. tails that led to this renascence and the John von Neumann, Professor of Math- roles played by the various actors. It is The ENIAC. During this wartime pe- ematics at the Institute for Advanced appropriate that it appears in an issue ded- riod, a team of scientists, engineers, and Study, was a consultant to Aberdeen and icated to Stan Ulam. technicians was working furiously on the to Los Alamos. For a whole host of Los Alamos Science Special Issue 1987 125 Monte Carlo reasons, he had become seriously inter- that was realistically calculable. (There ested in the thermonuclear problem being was a small interlude at Alamogordo!) spawned at that time in Los Alamos by The war ended before we completed our a friendly fellow-Hungarian scientist, Ed- set of problems, but it was agreed that we ward Teller, and his group. Johnny (as he continue working. Anthony Turkevich was affectionately called) let it be known joined the team and contributed substan- that construction of the ENIAC was near- tially to all aspects of the work. More- ing completion, and he wondered whether over, the uncertainty of the first phase of Stan Frankel and I would be interested the postwar Los Alamos period prompted in preparing a preliminary computational Edward Teller to urge us not only to com- model of a thermonuclear reaction for the plete the thermonuclear computations but ENIAC. He felt he could convince the to document and provide a critical review authorities at Aberdeen that our problem of the results. could provide a more exhaustive test of the computer than mere firing-table com- The Spark. The review of the ENIAC putations. (The designers of the ENIAC results was held in the spring of 1946 had wisely provided for the capability of at Los Alamos. In addition to Edward much more ambitious versions of firing Teller, the principals included Enrico Fer- tables than were being arduously com- mi, John von Neumann, and the Direc- puted by hand, not to mention other quite tor, Norris Bradbury. Stanley Frankel, different applications.) Our response to Anthony Turkevich, and I described the Stanislaw Ulam von Neumann’s suggestion was enthusi- ENIAC, the calculations, and the con- astic, and his heuristic arguments were clusions. Although the model was rel- dition, however, Stan’s extensive mathe- accepted by the authorities at Aberdeen. atively simple, the simplifications were matical background made him aware that In March, 1945, Johnny, Frankel, and I taken into account and the extrapolated statistical sampling techniques had fallen visited the Moore School of Electrical En- results were cause for guarded optimism into desuetude because of the length and gineering at the University of Pennsylva- about the feasibility of a thermonuclear tediousness of the calculations. But with nia for an advance glimpse of the ENIAC. weapon. this miraculous development of the We were impressed. Its physical size Among the attendees was Stan Ulam, ENIAC—along with the applications Stan was overwhelming—some 18,000 double who had rejoined the Laboratory after must have been pondering—it occurred to triode vacuum tubes in a system with a brief time on the mathematics faculty him that statistical techniques should be 500,000 solder joints. No one ever had at the University of Southern California. resuscitated, and he discussed this idea such a wonderful toy! Ulam’s personality would stand out in with von Neumann. Thus was triggered The staff was dedicated and enthusi- any community, even where “characters” the spark that led to the Monte Carlo astic; the friendly cooperation is still re- abounded. His was an informal nature; he method. membered. The prevailing spirit was akin would drop in casually, without the usual to that in Los Alamos. What a pity that a amenities. He preferred to chat, more or The Method war seems necessary to launch such revo- less at leisure, rather than to dissertate. lutionary scientific endeavors. The com- Topics would range over mathematics, The spirit of this method was consis- ponents used in the ENIAC were joint- physics, world events, local news, games tent with Stan’s interest in random pro- army-navy (JAN) rejects. This fact not of chance, quotes from the classics—all cesses—from the simple to the sublime. only emphasizes the genius of Eckert and treated somewhat episodically but always He relaxed playing solitaire; he was stim- Mauchly and their staff, but also suggests with a meaningful point. His was a mind ulated by playing poker; he would cite that the ENIAC was technically realizable ready to provide a critical link. the times he drove into a filled parking even before we entered the war in Decem- During his wartime stint at the Labora- lot at the same moment someone was ac- ber, 1941. tory, Stan had become aware of the elec- commodatingly leaving. More seriously, After becoming saturated with indoc- tromechanical computers used for implo- he created the concept of “lucky num- trination about the general and detailed sion studies, so he was duly impressed, bers,” whose distribution was much like structure of the ENIAC, Frankel and I re- along with many other scientists, by the that of prime numbers; he was intrigued turned to Los Alamos to work on a model speed and versatility of the ENIAC. In ad- by the theory of branching processes and 126 Monte Carlo contributed much to its development, in- tain spatial position. The next decisions cluding its application during the war to are the position of the first collision and neutron multiplication in fission devices. the nature of that collision. If it is deter- For a long time his collection of research mined that a fission occurs, the number of interests included pattern development in emerging neutrons must be decided upon, two-dimensional games played according and each of these neutrons is eventually to very simple rules. Such work has lately followed in the same fashion as the first. emerged as a cottage industry known as If the collision is decreed to be a scatter- cellular automata. ing, appropriate statistics are invoked to John von Neumann saw the relevance determine the new momentum of the neu- of Ulam’s suggestion and, on March 11, 1947, sent a handwritten letter to Robert Richtmyer, the Theoretical Division lead- er (see “Stan Ulam, John von Neumann, and the Monte Carlo Method”). His let- ter included a detailed outline of a pos- sible statistical approach to solving the problem of neutron diffusion in fission- able material. Johnny’s interest in the method was contagious and inspiring. His seemingly relaxed attitude belied an intense interest and a well-disguised impatient drive. His talents were so obvious and his coopera- tive spirit so stimulating that he garnered the interest of many of us. It was at that time that I suggested an obvious name for the statistical method—a suggestion not unrelated to the fact that Stan had an John von Neumann uncle who would borrow money from rel- atives because he “just had to go to Monte tron. When the neutron crosses a material Carlo.” The name seems to have endured. boundary, the parameters and characteris- The spirit of Monte Carlo is best con- tics of the new medium are taken into ac- veyed by the example discussed in von count. Thus, a genealogical history of an Neumann’s letter to Richtmyer. Consider individual neutron is developed. The pro- a spherical core of fissionable material cess is repeated for other neutrons until a surrounded by a shell of tamper material. statistically valid picture is generated. Assume some initial distribution of neu- trons in space and in velocity but ignore Random Numbers. How are the vari- radiative and hydrodynamic effects. The ous decisions made? To start with, the idea is to now follow the development computer must have a source of uni- of a large number of individual neutron formly distributed psuedo-random num- chains as a consequence of scattering, ab- bers.