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Fermi, Pasta, Ulam and a mysterious lady Thierry DAUXOIS Université de Lyon, CNRS, Laboratoire de Physique, Ećole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon cédex 07, France [email protected] It is reported that the numerical simulations of the Fermi-Pasta-Ulam problem were performed by a young lady, Mary Tsingou. After 50 years of omission, it is time for a proper recognition of her decisive contribution to the first ever numerical experiment, central in the solitons and chaos 8 theories, but also one of the very first out-of-equilibrium statistical mechanics study. Let us quote 0 from now on theFermi-Pasta-Ulam-Tsingou problem. 0 2 The Fermi-Pasta-Ulam problem [1] was named after tionoftheLosAlamosreport. Thispapermarkedindeed n a the three scientists who proposed to study how a crys- a true change in modern science, both making the birth J tal evolves towards thermal equilibrium. The idea was of a new field, Nonlinear Science, and entering in the 0 to simulate a chain of particles, linked by a linear in- age of computational science: the problem is indeed the 1 teraction but adding also a weak nonlinear one. FPU first landmark in the development of physics computer thought that, due to this additional term, the energy in- simulations. h] troduced into a single Fourier mode should slowly drift There was howeververy few mentions of an intriguing p totheothermodes,untiltheequipartitionofenergypre- point. On the first page of the FPU Los Alamos report - dicted by statistical physics is reached. The beginning published in 1955,it is written, t s of the calculation indeed suggested that this would be ”Report written by Fermi, Pasta and Ulam. hi the case, but to their great surprise, after a longer time, Work done by Fermi, Pasta, Ulam and Tsingou”. . almost all the energy was back to the lowest frequency This remark,that Mary Tsingouwho took part in the s mode and the initial state seems to be almost perfectly c numerical study is not an author of the report, was al- i recoveredafterthis recurrenceperiod. Thus, contraryto s ways puzzling for scientists who have read this paper: y the expectations of the authors, the drift of the energy indeed,itisclearthatcodingthefirstevernumericalex- h does not occur. This highly remarkable result, known as periment on the first computer was not a direct and im- p the FPU paradox,shows that nonlinearity is not enough mediate task. Consequently, why her contribution has it [ to guarantee the equipartition of energy. been recognizedonly by two lines of acknowledgements? 1 Pursuing the solution of the FPU paradox, Zabusky Moreover,whyhasitbeenimpossibleuntiltodaytopick v andKruskalemphasizedten yearslater the link between up her track? 0 the problem in the so-called continuum limit and the PeoplemoredeeplyinvolvedintheFPUliteraturehave 9 Korteweg-de Vries equation [2], known to have spatially usuallyalsoreadthe1972paperbyTuckandMenzel[6]. 5 1 localizedsolutions. Looking to the problemin realspace A careful reading of the introduction clearly emphasizes . rather than in Fourier space, they showed how to solve that one of the author of this paper, M. T. Menzel, was 1 the paradox in terms of the dynamics of these localized codingthe originalproblem: howcanwe solvethis para- 0 excitations. It was the birth of the term solitons, for dox? 8 theselocalised(orsolitary)waveswithpropertiesofpar- 0 TheobvioussolutionisthatinthenameM.T.Menzel, : ticles (explaining the suffix on as for electron, boson,...). M is for Mary and T for Tsingou. There is no paradox, v Consequently, the numerous physical applications [3] of i thisisthesameperson,afterherwedding! However,once X solitons originates from this FPU paper. again, it has been impossible for decades to pick up her r Anotherlineofthoughtwasdevelopedinparallel. Peo- trail. We recently discovered however, that she is still a ple focused on the Fourier mode dynamics, looking for alive and present in Los Alamos, a couple of miles from non-resonance conditions that could explain the ineffi- the place where this problem, so important in the past cient energy transfer. No convincing explanation was and present of nonlinear physics [3], was devised. It is found before the discovery of the KAM theorem, which time for a proper recognition of her work. states that most orbits of slightly perturbed integrable Born in October 14th 1928 at Milwaukee, Wisconsin, Hamiltonian systems remain quasi-periodic. If the per- inaGreeknativefamily,Mary Tsingou Menzelspent turbation is so strong that nonlinear resonances ’super- herchildhoodintheUS.Asthegreatdepressionwastak- pose’, the FPU recurrence is destroyed and one obtains ingplaceintheUS,herfamilymovedtoEuropein1936, a fast convergence to thermal equilibrium. [4] whereherfatherhadapropertyinBulgaria. However,in The FPUproblemis thus of centralimportance in the June 1940 the American embassy advised them to come Solitons and Chaos theories [5]. This is the reason why, backto US for safety. They pick up the verylast Ameri- in 2005, several conferences, articles and seminars have can ship that left Italy. Almost within a week after they celebratedthe50thanniversaryoftheMay1955publica- landed in New York, Italy declared war. 2 ShegainedaBachelorofSciencein1951attheUniver- solids. Asanticipated,preliminaryresultsconfirmedthat sity of Wisconsin, and a Master in mathematics in 1955 energy initially introducedin a single Fourier mode drift at University of Michigan. In 1952, following a sugges- to other ones. However, one day, the oversight to stop tionbyhermathematicsprofessor,awoman,sheapplied thecomputerallowsonetodiscoversomeunexpectedre- for a position at Los Alamos National Laboratory. At currences which were initially hidden by the slowness of that time, womenwerenot encouragedto do mathemat- the computer. It was the start of an ongoing fruitful ics, but because ofthe Koreanwar,there was a shortage research.[5] ofAmericanyoungmenandstaffpositionswerealsopro- The algorithm used by Mary Tsingou in 1955 to sim- posedtoyoungwomen. She wasthus hiredwitha whole ulate the relaxation of energy in a model crystal on the groupofyoungpeoplerightoutofcollege,fordoinghand Maniac is reproduced in Fig. 2. Its complexity has to (cid:13)c calculations. be compared with the 15 lines Matlab code, sufficient ShewasinitiallyassignedtotheT1division(TforThe- today to reproduce the original FPU recurrences [7]. oretical) at Los Alamos National Laboratory, led during At the time, programming was a task requiring great thewarbyRudolphPeierlsandtowhichthefamousspy, insight and originality, and through the 1960s and even KlausFuchs,belonged. Butshequickly movedto T7led later, it was common to list programmers as co-authors. by N. Metropolis for workingon the first ever computer, It appears that the only reason for the mention “Work the Maniac I,that no one couldprogram. Togetherwith done by FPU+Tsingou, and report written by FPU” is Mary Hunt, she was therefore the first programmer to thatshewasnotinvolvedinthewriting. However,Fermi startexploratoryworkonit. She remembersit as pretty was not either since, as noticed in S. Ulam biographical easy because of the very limited possibilities of the com- book [8], the FPU report was never published because puter: 1000 words. Fermidiedbeforethewritingofthepaper. Consequently, Theywereworkingprimarilyonweaponsbut,inparal- Tsingou was not given credit simply because the report lel, they studied other problems like programming chess was never formally presented in a journal and its state- or studying fundamental physics’ problems. Mary Tsin- ment of credit, differentiating between the writing and gou mostly interacted with J. R. Pasta. They were the the work done, was presumably misread by later people. first ones to do actually graphicson the computer, when they considereda problem with an explosionand visual- ized it on an oscilloscope. In addition to Pasta, she interacted also with Stan Ulam, but very little with E. Fermi, at that time pro- fessor in Chicago. He was visiting Los Alamos only for short periods, mostly during the summer. However, she knewNella,Fermi’sdaughter,muchbetterbecauseNella didn’t want to stay with her parents during their visits toLosAlamos. Bothearlytwentiesgirlsweresleepingin the same dormitory, while Enrico and Laura Fermi were hosted by their good friends, Stan and Françoise Ulam. FIG.2: ReproductionofthealgorithmusedbyMaryTsingou to code the first numerical experiment. Note the date (5-20- 55) at thetop right of thefigure. In 1958, Mary Tsingou married Joseph Menzel who was also workingat Los Alamos for the ProtectiveForce oftheAtomicEnergyCommission. Shestayedherwhole life in this small city but her colleagues changed since Metropolis left Los Alamos for Chicago, Pasta went to Washington,andlaterUlamwenttoColoradoUniversity. FIG. 1: Mary Tsingou in 1955 and in 2007. She worked successively on different problems, always with computers. She became one of the early experts It was Fermi who had the genius to propose that, in- in Fortran (FORmula TRANslator) invented by IBM in steadofsimplyperformingstandardcalculus,computers 1955,andwasassignedto helpresearchersinthe labora- could be used to test a physical idea, inventing the con- tory. ceptofnumericalexperiments. Heproposedtocheckthe After her seminal programming work on the Maniac, prediction of statistical physics on the thermalization of in the beginning of the sixties she came back to the 3 FPU problem with Jim Tuck looking for recurrences [6]. husbandatLosAlamos,veryclosetotheplacewherethe ButshealsoconsiderednumericalsolutionofSchrödinger FPUproblemwasdesignedanddiscovered: itis time for equations, the mixing problem of two fluids of different a proper recognition of her contribution: let us quote densities withJ.VonNeumann, andotherproblems. Fi- from now on the Fermi-Pasta-Ulam-Tsingou problem. nally,in the eighties during RonaldReagan’spresidency, she was deeply involved in the Star Wars project calculations. Acknowledgements: I would like to thank T. J. Retiredin1991,MaryT.Menzelisstilllivingwithher Gammel and J. Barré for helps. [1] FermiE.,PastaJ.,UlamS.,“Studiesofnonlinearprob- (1966). lems. I.”, Los Alamos report LA-1940 (1955), published [5] Chaos15,Focusissue: The“Fermi-Pasta-Ulam”problem- later in Collected Papers of Enrico Fermi, E. Segré (Ed.) thefirst 50 years (2005). (UniversityofChicago Press,Chicago 1965); also inNon- [6] Tuck J. L., Menzel M. T., The superperiod of the non- linearWaveMotion,NewellA.C.Ed.,LectureinApplied linear weighted string (FPU) problem, AdvancesinMath- Mathematics 15(AMS,Providence, RhodeIsland, 1974); ematics 9, 399-407 (1972). alsoinTheMany-Body Problem,MattisC.C.Ed.(World [7] Dauxois T., Peyrard M., Ruffo S.,The Fermi-Pasta- Scientific, Singapore, 1993). Ulam “numerical experiment”: history and pedagogical [2] Zabusky N. J., Kruskal M. D., Phys. Rev. Lett. 15, perspectives, European Journal of Physics 26, S3(2005). 2403 (1965). [8] Ulam S. M., Adventures of a Mathematician, Charles [3] Dauxois T., Peyrard M., Physics of Solitons, Cam- Scribner’s Son, NewYork, (1976). bridge University Press (2006). [4] Izrailev F. M., Chirikov B. V.Sov.Phys.Dokl. 11 30