50 Years of Computer Simulation -- a Personal View
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50 Years of Computer Simulation — a Personal View
Wm. G. HooverRuby Valley Research InstituteHighway Contract 60, Boxes 598 and 601, Ruby Valley 89833, NV USA (Dated: October 21, 2018)
Abstract
In the half century since the 1950s computer simulation has transformed our understandingof physics. The rare, expensive, slow, and bulky mainframes of World War II have given wayto today’s millions of cheap, fast, desksized workstations and personal computers. As a resultof these changes, the theoretical formal view of physics has gradually shifted, so as to focuson the pragmatic and useful. General but vague approaches are being superceded by specificresults for definite models. During this evolving change of emphasis I learned, developed, anddescribed my simulation skills at Michigan, at Duke, at Livermore, and in Nevada, while formingincreasingly wide-ranging contacts around the world. Computation is now pervasive in all thescientific fields. My own focus has been on the physics of particle simulations, mainly away fromequilibrium. I outline my particle work here. It has led me to a model-based understanding ofboth equilibrium and nonequilibrium physics. There are still some gaps. There is still much todo.
PACS numbers: 02.70.Ns, 45.10.-b, 46.15.-x, 47.11.Mn, 83.10.FfKeywords: Molecular Dynamics, Computational Methods, Chaos, Fractals, Smooth Particles . INTRODUCTION Computer-induced changes in emphasis have transformed what it means to “under-stand” physics. This transformation is nowhere more striking than in the many-bodymodel-based subjects of statistical mechanics and kinetic theory. The old way was solv-ing many-body problems “in principle” (but not “in fact”), by formal expansions aroundthe ideal gas or the harmonic crystal. The new way has replaced the expansions withdirect numerical simulations of model systems. At equilibrium, ensemble-based “MonteCarlo” simulations are appropriate. “Molecular dynamics” simulations are more gener-ally useful because they apply to both equilibrium and nonequilibrium situations. Thenonequilibrium problems typically involve flow and gradients and are quite unlike theirstatic homogeneous equilibrium relatives. Writing down the governing equations for amany-body model (a sum over states, or the dynamical equations of motion, for instance)is now considered just a beginning challenge, not an end in itself.Masaharu Isobe thought my reminiscences would interest younger readers of this Jour-nal, “Ensemble”. Besides the name’s possible link to Gibbs’ ensembles, the name “Ensem-ble” also suggests cooperation, and “working together”. These are nice concepts, whichunderlie the steady progress of science, particularly in today’s electronic world, with theimmediacy of the internet and email making efficient timely international collaborationspossible.I agreed with Masaharu’s idea, and summarize some milestones of my work here.Though times and tastes change, the experiences of learning, creating, and teaching tracean enduring continuity worth considering and summarizing. Here is a bit of my ownsimulation history, with the hope it will prove useful to yours. I start out with my collegeand university days in Ohio and Michigan, and end up with retirement in Nevada. Manymore details can be found on my website [ http://williamhoover.info ].
II. OHIO, 1953-1958, AND MICHIGAN, 1958-1961
My small and isolated undergraduate college, Oberlin College in Ohio, featured andencouraged the sceptical attitude so useful in physics. Otherwise the scientific content ofmy Oberlin liberal-arts coursework was irrelevant to my research career. The classroominstruction in mathematics and physics was formal, and mired in the past. This disap-2ointing style of instruction very nearly convinced me to switch from science to economics,my Father’s field of study.Recuperation from an automobile accident kept me out of college for a semester. Iattended Harvard’s Summer School to make up the Physical Chemistry course I hadmissed. The lecturer, Stuart Rice, provided research excitement and strengthened mycommitment to science. Stuart loved kinetic theory. He reassured us students that anyshortcomings in the practical labwork portion of his course would have no influence onour grades.My graduate university, the University of Michigan in Ann Arbor, was a great im-provement over Oberlin. The University of Michigan provided useful coursework, state-of-the-art computational tools, as well as the stimulation and inspiration necessary toresearch. Shortly after my arrival there I came upon Alder and Wainwright’s “Moleculesin Motion” article in Scientific American . The so-different pictures of hard-sphere fluidand solid trajectories in that article are still worth a look today. See Figure 1 for a pairof example pictures taken from Farid Abraham’s 1980 work on the melting transition . Inthe 1950s, simulating the motion of 108 particles was a challenge. The Alder-Wainwrightpictures triggered my interests in microstructure and molecular dynamics.At Michigan, Andrew Gabriel De Rocco, a young Chemical Physics Professor andmy PhD thesis advisor, was excited by statistical mechanics. He had a rare knack forrelating formalism to the real world. What constitutes the “real world” is of course amatter of opinion. I well remember Andy’s wife Sue, who overheard one of our upstairsconversations on pair potentials, shouting up the stairs: “Andrew, all your potentials arerepulsive!” Tastes differ!As a result of Andy’s enthusiasm and support I became an expert in the Mayers’“virial” expansion, the expression of fluid pressure as a power series in the density. B n ,the n th coefficient in the series is a sum of relatively-complicated n -body integrals. Theintegrands of these integrals are products of as many as n ( n − / n particles together. For hard parallel squares and cubes these integrals can be doneanalytically , though beyond B , the topological bookkeeping requires a fast computer.By 1960 I had the patience and the training to program the FORTRAN calculationof a few million of these integrals, using the “MAD” computer [Michigan AlgorithmicDecoder] . Programming was then a bit tedious. It involved writing instructions in theform of punched cards, one card for each program line. But it had to be done. For3 igure 1: Atomistic trajectories just below (top) and just above (bottom) the melting temper-ature, taken from Reference 2. parallel cubes B required computing 468 ×
7! = 2 , ,
720 separate integrals! MADmade occasional irreproducible machine errors. These errors were useful reminders ofthe need for vigilance in numerical work. The violation of obvious requirements (suchas conservation of momentum and energy) is the usual result of logical or typographicalerrors in programming.Once the program was successfully punched out, and carefully checked, I found negative virial coefficients, both B and B , for hard repulsive parallel cubes. Negative tensilecontributions for positively repulsive particles was a big surprise! I was thoroughly hookedby the excitement of research. Of course the pace was slower then. I exchanged severalletters with “H. N. V.” Temperley about the details of the virial series. After a fewletters the “H. N. V.” changed to an informal “Neville”. A typical US ↔ UK roundtripcorrespondence took two weeks by airmail. Bob Zwanzig had published the paper whichintroduced me to hard cubes but which also wrongly contradicted my negative-coefficient4esults. It was a real thrill, and a lesson, when, in a phone conversation with Andy andme, he readily admitted his mistake. III. DUKE UNIVERSITY, DURHAM, NORTH CAROLINA, 1961-1962
In the 1960s in America a postdoctoral appointment was required before seeking out a“real job”. Andy sent me on to Duke University, where one of John Kirkwood’s students,Jacques Poirier, lived an isolated existence as a Professor of Chemistry with theoreticalinclinations. The smell of menthol from the Salem cigarette plant traveled miles in theevening to our house on a dirt road next to a turnip field. The various organic smellsduring the day at the Chemistry Department helped make this a tranquil detour frommainstream physics. Because Jacques’ ideas for our joint research turned out to be invalidI was able to pursue further work on the virial series while at Duke. In those days, whencomputer simulation was still rare, approximate integral equations for the distributionof particle pairs g ( r ) were all the rage. Because the equations were nonlinear in g ( r ),complete solutions required elaborate computation. But substituting a density expansionof g ( r ), and equating coeffients of powers of the density made it possible to compute theapproximate virial coefficients and compare them to the Mayers’ exact expressions .During the Duke year, I got in contact with George Stell, another afficianado of theMayers’ series, a skilled jazz musician, and still a great friend. I visited George in hisGreenwich Village apartment. Two memories of that visit stand out: George had his ownsauna there, and the relative calm in his apartment was broken by a sewergas explosionjust outside, strong enough to levitate a heavy manhole cover. Once back at Duke Imanaged to land job interviews at the Livermore and Los Alamos Laboratories, the tworival computing giants overseen by the University of California. Alder and Wood, at thetwo laboratories where I would soon seek a job, were rivals too, disagreeing over theirrelative priority and contributions to the understanding of the hard-disk and hard-spheremelting transitions . IV. LIVERMORE AND DAVIS, CALIFORNIA, 1962-2004
I interviewed at both Los Alamos and Livermore in 1962, flying out from Michiganto talk to a dozen or so scientists at each laboratory and giving a seminar on my virial-5eries work. Los Alamos, in the mountains of New Mexico, was physically the moreinteresting of the two bomb laboratories. Livermore was located in a once rural grapeand cattle ranching valley suffering now from pollution and urban sprawl. By the timeI arrived Alder and Wainwright were studying disk and sphere systems of about 1000particles, using both rigid and periodic boundary conditions. One of their research goalswas understanding whether or not disks and spheres could freeze and melt at high density.My hard-particle virial series expertise fitted in well with that work .At Los Alamos the relative humidity was low and the scientists wore Hawaiian shirtsand desert boots rather than the suits of the Midwest. The salaries there were nearly 20percent lower (partly a function of the longer five-week vacations due to the laboratory’sremoteness). This difference in salaries decided me on Livermore, where Berni Alderprovided me a home in the Physics Department. It was an exciting time and place.Particle, continuum, plasma, astrophysical and nuclear physicists all joined together, withweekly scientific meetings under Edward Teller’s watchful eyes. For one of the talks FrancisRee and I got Teller’s permission to pursue and present a computationally demandingsingle-occupancy explanation of the hard-disk and hard-sphere phase transitions . FrancisRee, trained by Henry Eyring, was, like me, dedicated to precise and careful statisticalanalyses. We solved Berni Alder and Bill Wood’s hard-sphere and hard-disk problemsusing both the Mayers’ virial series (for the fluid phases) and Metropolis-Rosenbluth-TellerMonte Carlo simulations (for Kirkwood’s single-occupancy solid phases). See Figure 2 .Our “Monday Morning Meeting” presentation went well.This Monday-Morning-Meeting background in a variety of disciplines was extremelyuseful to me later on, both scientifically and socially. Teller started the Department ofApplied Science at Livermore in 1963, with the idea that a wide background (nuclear,quantum, classical, mathematical, chemical, electromagnetic) was essential to trainingstudents. I taught in the Department for about thirty years, and enjoyed it thoroughly.My wife Carol was a student in one of the statistical mechanics courses I taught at DAS.The bookkeeping associated with hard-particle collisions made molecular dynamicsseem less interesting to me (or at least harder) than Monte Carlo simulation. But in theearly 1960s Rahman, Verlet, and Vineyard all followed Fermi’s 1950s work, in showinghow to carry out simulations with continuous force laws. My group leader at the time,Russ Duff, supported my interest in learning molecular dynamics. As a shockwave ex-perimentalist he liked the idea of simulating shockwave-induced melting with molecular6 igure 2: Hard-disk (above) and hard-sphere (below) equations of state. From Reference 9.The close-packed volume (area in two dimensions) is V . The lower “single-occupancy” curvescorrespond to the pressure of particles confined to space-filling cells, as shown in the insets. dynamics. I took on the project, storing thousands of long particle trajectories on mag-netic tapes. The physical stretching of some of these tapes made them unreadable oftenenough that the storage procedure was useless for long runs. The shockwave simulationsof 1967 had to be put aside temporarily, and were not resurrected until 1980 , when mag-netic tapes were obsolete, and again in 1997 , when the anisotropicity of temperaturehad caught my interest. Figure 3 illustrates that interesting result from the shockwavework: in strong shockwaves temperature becomes a tensor. The longitudinal temperature(the velocity fluctuation in the direction of the motion) greatly exceeds the transversetemperature in strong shocks.A later group leader of mine, Mark Wilkins, strongly influenced my scientific outlook.Mark was a self-taught expert in the numerical solution of the partial differential equationsof continuum mechanics, specializing in plastic flow and fracture, essential for weapons7 igure 3: Longitudinal (top), mean, and transverse (bottom) kinetic temperatures in a Lucyfluid shockwave, taken from Reference 12. simulations. Mark stressed that physics is a study of “models”, closed sets of differen-tial equations which bear some resemblance to our experiences in “real life”. Quantummechanics is such a model, and evidently quite imperfect, in that it never predicts theunique outcomes observed in the real world.At the Livermore Laboratory of the 1960s there were plenty of interesting collaboratorswith whom I could work. Al Holt was a formalist, a tensor specialist who had been trainedby chaos guru Joe Ford at Georgia Tech. Al’s interest in elastic constants combined nicelywith my own statistical and lattice dynamical skills. Dave Squire worked for the ArmyResearch Office in Durham, North Carolina. He was a practical chemist, schooled byZevi Salsburg, a colleague of Berni’s. Dave was able to visit the Livermore Laboratoryfor several months in the late 1960s, along with his wife and eight children. Al and Daveand I formulated the elastic response of crystals to strain under both isothermal andadiabatic conditions. These results, which can be evaluated by either Monte Carlo or8olecular dynamics simulations, have been rediscovered frequently.I visited the nearby IBM Research Laboratory at Almaden, which had managed to hireJohn Barker and Doug Henderson. I was invited there for a seminar and prepared a setof slides. Barker ran the slide projector while Henderson asked an occasional question.That was the whole audience at Almaden. I was fortunate, at Livermore, that dozens ofscientists were actively doing research in statistical mechanics. In the late 1960s Barkerand Henderson, along with Mansoori, Canfield, Weeks, Chandler, Andersen, Rasaiah,and Stell, developed a very successful perturbation theory for the Helmholtz free energy .Only the hard-sphere pair distribution function was needed, and a useful form for that wasavailable from the integral-equation work. The theory was actually useful for realworldequilibrium thermodynamic calculations. This equilibrium breakthrough convinced me itwas high time to switch to the study of nonequilibrium problems, where the only theoryavailable was Green and Kubo’s linear-response theory of transport .The energy crunch in the Carter administration led to mass firings at the LivermoreLaboratory and to a change of emphasis and structure: suddenly there were lots of groupleaders, lots of progress reports and proposals, and detailed budgets. When I neededdozens of hours of CRAY computer time to study the effect of Coriolis’ forces on the heatflux, Roger Minich, a favorite of one of the bomb divisions, generously gave me the timefrom his weapons-physics accounts .The cutback in basic research at Livermore made it necessary to look outside thelaboratory for collaborators. I had a lot of fun working with Brad Holian at Los Alamos.We had a common interest in statistical mechanics and simulation. Brad tried to get ajob at Livermore; I tried to get a job at Los Alamos, so that we could work together, butboth these initiatives were unsuccessful.In my attempts to satisfy the mounting laboratory pressure for “relevance” I carried outsome dynamic fracture simulations with Bill Ashurst and, in his PhD thesis work, BillMoran. Figure 4 shows a typical fracture specimen. At a meeting with the LaboratoryDirector, Mike May, Mike asked me whether or not these fracture simulations were reallyrelevant. I had to admit that atomistic models are actually quite limited in scope, andare often misleading (even today).Bill Ashurst, my first PhD student at the Department of Applied Science, was keento simulate nonequilibrium flows. We developed methods for simulating shear flows withisothermal boundaries as well as periodic homogeneous algorithms . Bill was able to9 igure 4: Fracture specimen simulation, taken from Reference 18. make movies (as was also Brad Holian at Los Alamos). These movies were a fixture ofsmall topical physics meetings in the early days of nonequilibrium molecular dynamics.An early experience with the National Science Foundation was educational and helpedsharpen my scepticism for government’s ability to select good problems to solve. In theearly 1980s I proposed a Fourier analysis of simple nonequilibrium distribution functions.The proposal was evaluated “excellent” by five reviewers and “very good” by the sixth.Despite this, the proposal was declined. This early failure reinforced my natural inclina-tion to forgo begging for research funds. V. TRAVEL, WITH SABBATICALS IN AUSTRALIA, AUSTRIA, AND JAPAN
After a first visit to France to confront Verlet, Levesque, and K¨urkijarvi’s somewhaterroneous Green-Kubo transport results with the nonequilibrium analogs I had calcu-lated with Bill Ashurst, Orsay became a regular stop for me. Carl Moser superviseda long series of workshops, seminars, and meetings, at CECAM (The European Centerfor Atomic and Molecular Simulation) which were seminal and stimulating. The locale,about an hour outside Paris in the countryside, was conducive to good work. There wasa hillside covered with wild blackberries on the way to lunch and the Parisian bistros andrestaurants attracted us in the evenings after work.10he Department of Applied Science and the Livermore Laboratory provided my fi-nancial support throughout each year. Though my salary was set by the Department,based on academic criteria, the Laboratory always tried to exert pressure toward “prac-tical applications” to justify its paying the Lion’s Share (five eighths) of my salary. Ascompensation for this pressure the “Professional Research and Teaching Leaves” avail-able at the Laboratory made it possible to get away for sabbatical research. Such leaves,augmented by support from the Fulbright Foundation, Universit¨at Wien, and the JapanSociety for the Promotion of Science, took me and my family to Australia in 1977, toAustria in 1984, and to Japan in 1989.In 1980, between my Australian and Austrian sabbaticals, I noticed that a Hamiltoniancould be constructed which reproduced exactly the geometry and energy balance for amany-body system undergoing periodic shear. This was the “Doll’s Tensor” Hamiltonian,which I described in June 1980 at Sitges (Spain) after the StatPhys organizers for theAugust 1980 Edmonton meeting turned down my proposal to speak about it in Canada.“Doll’s Tensor” was simply the tensor qp array, constructed of the Cartesian particlecoordinates { q } and momenta { p } . Adding the term ˙ ǫ P yp x to the usual many-bodyHamiltonian induces a macroscopic motion with h ˙ x i = ˙ ǫy , simple shear, or “Plane Couetteflow”. In June 1982, at Howard Hanley’s seminal meeting, “Nonlinear Fluid Behavior”,which I helped organize, I was finally able to present a talk on nonequilibrium moleculardynamics to a large mostly-American audience of interested colleagues .By 1984 my desire to present the Doll’s-tensor work as a talk at an international con-ference, the 1983 Edinburgh StatPhys meeting, had once again been frustrated. Theconsolation prize, a humorous poster detailing the history of the shear flow work, was notpublished until nine years later in the proceedings of a Sardinia meeting . In reactionto all the StatPhys frustration, I organized, with Giovanni Ciccotti, a highly-successfulEnrico Fermi Summer School meeting at Lake Como, where the new nonequilibriumalgorithms were thoroughly discussed . See particularly Denis Evans’ lecture “Nonequi-librium Molecular Dynamics”, pages 221-240 of the School’s Proceedings.Only very recently , with my wife Carol and Janka Petravic, I have quantified theerrors (nonlinear in the strainrate ˙ ǫ ) incurred by using the Doll’s and the closely relatedS’llod algorithms. The kewpie doll has a highly-interesting history (the Centennial of theDoll is 2009!) in addition to its usefulness as a mascot for statistical physics.With the shearflow problem solved, Denis Evans and Mike Gillan, working completely11ndependently, found an external field that correctly generated heat flow in 1982 – seeagain Evans’ lecture for the details. Their solution of this problem was particularlyinteresting because it provided a concrete example in which Gauss’ Principle (equivalentto Least Action for equilibrium systems) gives incorrect motion equations (inconsistentwith Green-Kubo) away from equilibrium .The Australian Sabbatical experience had been interesting, though slightly chaotic.My proposed work at the Australian National University’s Computer Centre, in Canberra,with Bob Watts, came to an abrupt end in the first week when Bob was appointed toreplace the Director of the center. Watts’ water potential, which I had intended toinvestigate in Australia, turned out to be unstable, making it possible to concentrate onwhat was for me a more interesting project, the determination of liquid and solid freevolumes. My son Nathan, having just finished high school, was in Australia with meand we worked together at the ANU Computer Centre. That work involved a “gedankenexperiment” in which a single very light particle traced out a “free volume” while itsheavier neighbors stayed put . I had used this same idea earlier to rationalize the use ofcell models . Figure 5 illustrates the difference between the fluid and solid phases fromthis perspective. It was educational to learn from these results that the free volume inthe fluid phase is actually smaller than that in the coexisting solid!My first Austrian experience also had an unexpected turn. Rather than working exclu-sively with Karl Kratky, as I had intended, I began to collaborate with Harald Posch ,whose interests in statistical mechanics and nonequilibrium simulations were similar tomine. Both Harald and I valued reproducibility and precision very highly. We oftencompared results from our two independently-written computer programs. Kratky was aformalist, and I soon lost patience waiting the months it took him to reproduce results Icould generate numerically in a matter of days. The Austrian sabbatical gave me back-ground for my first book, “Molecular Dynamics”, lectures given at Universit¨at Wien andwritten up once I was back at Livermore, in 1986. In those electronically primative timesit was necessary to edit the teX file for the manuscript in one building and to drive abouta half mile to another building to see a printout. Because car travel was limited until18:00 during the week, most of the book work had to be done at night.Prior to a 1984 CECAM workshop at Orsay I had the very good fortune to meetShuichi Nos´e on the Orly Airport train platform. (I had noticed his surname printed onhis suitcase.) This meeting eventually led to a very pleasant and creative year in Japan.12 igure 5: Fluid (left) and solid (right) hard-disk free volumes at a common density four-fifthsof close packing. The “free volumes” shown are the regions available to the center of each diskwhen the remaining disks are held fixed. Taken from Reference 27. Nos´e was isolated from the other workshop members, choosing to stay in a Japanese-style hotel. We arranged to meet at the Notre Dame cathedral. On a bench in front ofthe church we talked about his novel thermostat ideas in detail. After the workshopI visited Philippe Choquard in Lausanne to work out the consequences for a harmonicoscillator. The result was my most cited paper . Some of the oscillator orbits were quitebeautiful . Figure 6 shows a regular periodic orbit for a thermostated oscillator. Theoscillator exhibits chaotic orbits too. “Chaotic” orbits have the property of Lyapunovinstability – an infinitesimal perturbation of such an orbit grows exponentially fast withtime. Orbits for dissipative systems, in which work is converted to heat, are typically“fractal” with a fractional dimensionality less than that of the space in which they areembedded. See
Figure 7 for an example.My interest in computational thermostats was immediate and has continued to thisday. I asked Berni Alder what he thought about my energy and temperature-controlideas. He pooh-poohed the notion. I traveled to Los Alamos to ask Bill Wood for hisideas. Though a bit more diplomatic, his thoughts were the same as Berni’s: thermostatswere not a very useful idea. Fortunately, I had the freedom to spend much of the nextfew years working out the details, linking thermostats to statistical physics. Bill Moran13 igure 6: Periodic orbit for a thermostated harmonic oscillator with coordinate q , momentum p , and friction coefficient ζ , taken from Reference 33. and I generated the fractal objects which describe the collisions taking place for a verysimple problem. We studied a thermostated particle falling through a periodic “GaltonBoard” of hard-disk scatterers . The collisions formed a multifractal object, with thedimensionality of the object decreasing with increasing field strength. For a sample see Figure 7 .The finding that the dimensionality of phase-space distributions was reduced belowthat of Gibbs’ equilibrium distribution was a revealing and rewarding insight for me. Theextreme rarity of the fractal nonequilibrium states explained irreversibility. Because theequations of motion are time-reversible and the probability of choosing an initial fractalstate is of measure zero, the probability of violating the Second Law of Thermodynamics,along a time-reversed trajectory, vanishes. The fractal nature of the phase-space distribu-tions also showed that there could be no nonequilibrium entropy. This is because Gibbs’recipe for the entropy in terms of the N -body distribution function f and Boltzmann’s14 igure 7: A multifractal phase-space plot of successive collisions in the Galton Board problemwith a constant vertical gravitational field. The abscissa corresponds to the location of eachcollision on a hard disk scatterer from bottom (at the left) to top (at the right) relative tothe downward field direction. The ordinate corresponds to the tangential velocity component,which varies from − constant k , S Gibbs ≡ − k h f ln f i , diverges when f is singular.At the New York American Physical Society meeting emphasizing the hot topic of“High-Temperature Superconductors” I took long walks in Central Park, mentally es-timating the phase-space dimensionality loss in strong shockwaves and ruminating overthe paradox that time-reversible equations of motion lead to irreversible behavior. Muchlater these topics gave rise to my third book, “Time Reversibility, Computer Simulation,15nd Chaos”. Oddly enough, much of today’s research still deals with equilibrium prob-lems, though to me nonequilibrium ones are more numerous, more significant, and moreinteresting.In the years following my 1977-1978 sabbatical Down Under, nonequilibrium moleculardynamics had been developing rapidly. In this same period, my marriage was deteriorat-ing. The years from 1980 through my divorce, in 1986, were particularly difficult, but stillrelatively productive. In 1988 a very lucky chance meeting with a visitor to the Livermorelaboratory, a young researcher from Louisiana State University, a Doctor Gupta, broughtme in contact with his host, a former student of mine, Carol Griswold Tull. Carol wasworking with the Livermore supercomputers and her own marriage had ended in 1984. Wewere fortunate to share very similar interests in a “Good Life” mixture of science, nature,music, and nourishment. At last I had found a faithful woman with whom to share mylife. Our marriage was arranged for 1989, so that we would arrive for our sabbatical inJapan as an officially married couple. My son Nathan performed the ceremony in Carol’sLivermore home.The Japanese experience, at Keio University in Yokohama, was a surprise. Afterfinding that there seemed to be no plan to collaborate with Nos´e, who had invited me tovisit Keio, I prepared a list of about a dozen projects on which we might work together,and took it to his office for discussion. Still nothing. In retrospect this turned out well, atleast for me, in that it freed up my time to write another book, “Computational StatisticalMechanics”, summarizing what I had learned in my Applied Science teaching at Livermorewhile enjoying the peaceful work atmosphere of Japan. Carol and I spent many a night atthe computer laboratory near Hiyoshi station, working on the manuscript. Our HersheyHouse apartment, overlooking a busy baseball field was within walking distance so thatour working hours weren’t limited by the train schedules.At Hershey House we got weekly progress phonecalls from Tony DeGroot, back in Liv-ermore, who had built a 64-processor parallel computer capable of million-atom moleculardynamics. Working with Tony and Jeff Kallman in Livermore, with the support of IrvStowers and Fred Wooten, as well as collaborating with Taisuke Boku, Toshio Kawai, andSigeo Ihara in Japan, we worked long-distance on color movies of silicon crystal deforma-tion. A still picture from such a movie is shown here in Figure 8 . This collaboration,with nine coauthors, involved the work of more individuals than did any other of myresearch efforts . 16 igure 8: A plastic indentation pit in a 373,248-atom model of silicon. The indentor (not shown)is tetrahedral in shape and moves at one-fifth the sound velocity. Analogous two-dimensionalindentation simulations are described in Reference 35. After the Berlin Wall came down in 1989 the Livermore Laboratory was left without amission, and became a vestige of its former self, doing “work for others” and striving toappear practical. In this climate Teller’s Department of Applied Science lost its appeal tothe Laboratory, and gradually decayed. Though I continued to work at the Laboratoryfor another half dozen years, until a lucrative early-retirement package came along, theresearch excitement at Livermore had definitely disappeared. The working population atthe laboratory is now (at the end of 2008) only half its former size. My research fromabout 1994 through to the present has been carried out with my wife Carol and a largeinternational group of collaborators from outside the laboratory.17
I. LESSONS LEARNED FROM LOOKING BACKWARD
My career at Livermore, with its many pleasant overseas interludes, left me with somepowerful lessons, worth outlining here. They have to do with the value of research andits fruits and how it is best nurtured.
Publication of reproducible results is the sine quanon of science. A research project, no matter how brilliant, is quite useless unless otherscan share its results. Coworkers, administrators, and editors are the filters through whichwork must pass before it enters the literature. Their suggestions are often good ones.Some publications are not so good. George Stell suggested the journal name “Setbacksin Physics”. Setbacks in Physics was to be devoted exclusively to faulty and incorrectsolutions of significant problems already solved correctly in the literature. In my ownresearch career I came across two sets of excellent candidate articles for that journal. Bothcandidates were in research areas where I had published extensively. A series of papersby Tuckerman and coworkers , pointedly ignored the simple relationship between Nos´e-Hoover mechanics and the Second Law of Thermodynamics due to multifractal phase-space structures which I had described repeatedly in the literature. A second, remarkablylong, paper by Zhou was a specially effective setback, in that it spawned many successorsetback papers, all pushing the original claim that the (wholly-correct) microscopic formof the Virial Theorem for the pressure was incorrect, and that the kinetic part of thepressure tensor should not be included.A second category of “bad” paper, one with forged data, is more rare. The onlyexample I came across myself was one Watanabe’s free-energy calculation. His resultswere literally “too good to be true” . A phonecall to his thesis advisor revealed that thecomputer program he used to generate free energies had wild fluctuations. The programwas simply stopped when the free energy reached the “correct” value.In the past good papers were often squelched by the review process. I remember EdJaynes had George Uhlenbeck’s rejection of Ed’s seminal paper on maximum-entropytheory framed in his Washington University office. Such defects of the reviewing processare not so important now. I have never failed to publish a rejected paper elsewhere.Because there are now so many outlets for publication, including the LANL arXiv andone’s own website, arbitrary rejection is no longer a serious problem. Today an authorcan certainly publish if he wishes to do so.A half-century of simulation work has left me with some lasting lessons. Reproducibility18s paramount, and Clarity is required. Scepticism and Openness are desirable, as is alsoa sense of Perspective. Visits and discussions with others most often lead to useful ideas.Publication is in the end absolutely necessary despite the occasional frustrations of peerreview and the cronyism that discourages novelty.For me the Livermore laboratory of the 1960s provided a nearly-ideal environment forlearning about simulation – stimulating people, freedom to choose one’s own way, plentyof secretarial support and computing equipment, the possibility of travel and publication.After a few years of microscopic simulations, first equilibrium and then nonequilibrium,as described here, it was natural for me to explore continuum methods to get beyondthe limitations on system size and timescale posed by atomistic vibration lengths andtimes. The smooth-particle method, developed by Lucy and Monaghan in 1977 providesa method much like molecular dynamics for solving the continuum equations. But theparticles are not necessarily small. They can be astrophysical in size. Smooth particlesheld my interest for many years, resulting in my most recent book, which is devoted tothat technique.A Japanese colleague, Shida-san Koichiro, a Lecturer at Musashi Institute of Tech-nology in Tokyo, had been trained at Keio by Taisuke Boku and Toshio Kawai. We allmet when Carol and I visited Keio in 1989-1990. Koichiro was able to work with us atLivermore on Maxwell’s thermal-creep problem. He has very kindly translated all fourof my books into Japanese. I am very grateful to him for this and suggest that thereader interested in more details of my work look for those books, either in English orin Japanese: [1] Molecular Dynamics; [2] Computational Statistical Mechanics; [3] TimeReversibility, Computer Simulation, and Chaos; [4] Smooth Particle Applied Mechanics,the State of the Art. My website http://williamhoover.info also contains a wealth oftechnical information, including electronic forms of books [1], [2], and [4].19 II. ACKNOWLEDGMENTS
Masaharu Isobe kindly suggested this work and my wife Carol helped prepare themanuscript. B. J. Alder and T. E. Wainwright, “Molecules in Motion”, Scientific American (4), 113-130 (1959). F. F. Abraham, “Two-Dimensional Melting, Solid-State Stability, and the Kosterlitz-Thouless-Feynman Criterion”, Physical Review B , 6145-6148 (1981). R. W. Zwanzig, “Virial Coefficients of Parallel Square and Parallel Cube Gases”, Journal ofChemical Physics , 855-856 (1956). W. G. Hoover and A. G. De Rocco, “Sixth Virial Coefficients for Gases of Parallel HardLines, Hard Squares, and Hard Cubes”, Journal of Chemical Physics, , 1059-1060 (1961). W. G. Hoover and J. C. Poirier, “Determination of Virial Coefficients from the Potential ofMean Force”, Journal of Chemical Physics , 1041-1042 (1962). W. W. Wood and J. D. Jacobsen, “Preliminary Results from a Recalculation of the MonteCarlo Equation of State of Hard Spheres”, Journal of Chemical Physics , 1207-1208 (1957). B. J. Alder and T. E. Wainwright, “Phase Transition for a Hard Sphere System”, Journal ofChemical Physics , 1208-1209(1957). B. J. Alder, W. G. Hoover, and T. E. Wainwright, “Cooperative Motion of Hard DisksLeading to Melting”, Physical Review Letters , 241-243 (1963). W. G. Hoover and F. H. Ree, “Melting Transition and Communal Entropy for Hard Spheres”,Journal of Chemical Physics , 3609-3617 (1968). J. B. Gibson, A. N. Goland, M. Milgram, and G. H. Vineyard, “Dynamics of RadiationDamage”, Physical Review , 1229-1253 (1960). B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shockwave Structure via
Nonequi-librium Molecular Dynamics”, Physical Review A , 2798-2808 (1980). O. Kum, Wm. G. Hoover, and C. G. Hoover, “Temperature Maxima in Stable Two-Dimensional Shock Waves”, Physical Review E , 462-465 (1997). D. R. Squire, A. C. Holt, and W. G. Hoover, “Isothermal Elastic Constants for Argon. Theoryand Monte Carlo Calculations”, Physica , 388-397 (1969). W. G. Hoover, A. C. Holt, and D. R. Squire,“Adiabatic Elastic Constants for Argon. Theoryand Monte Carlo Calculations”, Physica , 437-443 (1969). J. A. Barker and D. Henderson, “What is Liquid? Understanding the States of Matter”,Reviews of Modern Physics , 587-671 (1976). R. W. Zwanzig, “Time Correlation Functions and Transport Coefficients in Statistical Me-chanics”, Annual Review of Physical Chemistry , 67-102 (1965). W. G. Hoover, B. Moran, R. M. More, and A. J. C. Ladd. “Heat Conduction in a RotatingDisk via
Nonequilibrium Molecular Dynamics”, Physical Review A , 2109-2114 (1981) W. T. Ashurst and W. G. Hoover, “Microscopic Fracture Studies in the Two-DimensionalTriangular Lattice”, Physical Review B , 1465-1473 (1976). W. G. Hoover and W. T. Ashurst, “Nonequilibrium Molecular Dynamics”, Advances inTheoretical Chemistry , 1-51 (1975). D. Levesque, L. Verlet, and J. K¨urkijarvi, “Computer Experiments on Classical Fluids. IV.Transport Properties and Time-Correlation Functions of the Lennard-Jones Liquid Near itsTriple Point”, Physical Review A , 1690-1700 (1973). W. G. Hoover, “Adiabatic Hamiltonian Deformation, Linear Response Theory, and Nonequi-librium Molecular Dynamics”, Lecture Notes in Physics , 373-380 (1980). W. G. Hoover, “Atomistic Nonequilibrium Computer Simulations”, Physica , 111-122(1983). W. G. Hoover, “Nonequilibrium Molecular Dynamics at Livermore and Los Alamos”, in
Microscopic Simulations of Complex Hydrodynamic Phenomena , M. Mareschal and B. L.Holian, editors (Plenum Press, New York, 1992). G. P. F. Ciccotti and W. G. Hoover, Editors, “Molecular Dynamics Simulation of Statistical-Mechanical Systems”, Proceedings of the International Enrico Fermi School of Physics,Course (1986). Wm. G. Hoover, C. G. Hoover, and J. Petravic, “Simulation of Two- and Three-DimensionalDense-Fluid Shear Viscosities via
Nonequilibrium Molecular Dynamics. Comparison of Time-and-Space-Averaged Stresses from Homogeneous Doll’s and Sllod Shear Algorithms withthose from Boundary-Driven Shear”, Physical Review E , 046701 (2008). W. G. Hoover, B. Moran, and J. M. Haile, “Homogeneous Periodic Heat Flow via
Nonequi-librium Molecular Dynamics”, Journal of Statistical Physics , 109-121 (1984). W. G. Hoover, N. E. Hoover, and K. Hanson, “Exact Hard-Disk Free Volumes”, Journal ofChemical Physics , 1837-1844 (1979). W. G. Hoover, W. T. Ashurst, and R. Grover, “Exact Dynamical Basis for a FluctuatingCell Model”, Journal of Chemical Physics , 1259-1262 (1972). Wm. G. Hoover, C. G. Hoover, and H. A. Posch, “50 Joint Explorations, 1985-2007”,Schr¨odinger Institute (Wien) Preprint Archive, S. Nos´e, “A Unified Formulation of the Constant Temperature Molecular Dynamics Meth-ods”, Journal of Chemical Physics , 511-519 (1984). S. Nos´e, “A Molecular Dynamics Method for Simulations in the Canonical Ensemble”, Molec-ular Physics , 191-198 (2002). W. G. Hoover, “Canonical Dynamics: Equilibrium Phase-Space Distributions”, PhysicalReview A , 1695-1697 (1985). H. A. Posch, W. G. Hoover, and F. J. Vesely, “Canonical Dynamics of the Nos´e Oscillator:Stability, Order, and Chaos”, Physical Review A , 4253-4265 (1986). B. Moran, W. G. Hoover, and S. Bestiale, “Diffusion in a Periodic Lorentz Gas”, Journal ofStatistical Physics , 709-726 (1987). W. G. Hoover, A. J. De Groot, C. G. Hoover, I. F. Stowers, T. Kawai, B. L. Holian, T. Boku,S. Ihara, and J. Belak, “Large-Scale Elastic-Plastic Indentation Simulations via
Nonequilib-rium Molecular Dynamics”, Physical Review A , 5844-5853 (1990). Wm. G. Hoover, D. J. Evans, H. A. Posch, B. L. Holian, and G. P. Morriss, ‘Commenton “Toward a Statistical Thermodynamics of Steady States”’, Physical Review Letters ,4103-4103 (1998). M. Zhou, “A New Look at the Atomic Level Virial Stress – On Continuum-Molecular SystemEquivalence”, Proceedings of the Royal Society of London A, , 2347-2392, (2003). B. L. Holian, H. A. Posch, and W. G. Hoover, “Free Energy via
Thermostated DynamicPotential-Energy Changes”, Physical Review E , 3852-3861 (1993)., 3852-3861 (1993).