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Fermi, Pasta, Ulam and the Birth of Experimental Mathematics

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Fermi, Pasta, Ulam and the Birth of Experimental Mathematics A reprint from American Scientist the magazine of Sigma Xi, The Scientific Research Society This reprint is provided for personal and noncommercial use. For any other use, please send a request to Permissions, American Scientist, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A., or by electronic mail to [email protected]. ©Sigma Xi, The Scientific Research Society and other rightsholders Fermi, Pasta, Ulam and the Birth of Experimental Mathematics A numerical experiment that Enrico Fermi, John Pasta, and Stanislaw Ulam reported 54 years ago continues to inspire discovery Mason A. Porter, Norman J. Zabusky, Bambi Hu and David K. Campbell our years ago, scientists around universally called, sparked a revolu- tunately, because he was at Los Ala- F the globe commemorated the cen- tion in modern science. mos in the early 1950s, he had access tennial of Albert Einstein’s 1905 annus to one of the earliest digital comput- mirabilis, in which he published stun- Time After Time ers. The Los Alamos scientists play- ning work on the photoelectric effect, In his introduction to the version of fully called it the MANIAC (MAth- Brownian motion and special relativ- LA-1940 that was reprinted in Fermi’s ematical Numerical Integrator And ity—thus reshaping the face of physics collected works in 1965, Ulam wrote Computer). It performed brute-force in one grand swoop. Intriguingly, 2005 that Fermi had long been fascinated numerical computations, allowing also marked another important anni- by a fundamental mystery of statisti- scientists to solve problems (mostly versary for physics, although it passed cal mechanics that physicists call the ones involving classified research on unnoticed by the public at large. Fifty “arrow of time.” Imagine filming the nuclear weapons) that were otherwise years earlier, in May 1955, Los Ala- collision of two billiard balls: They inaccessible to analysis. The FPU prob- mos Scientific Laboratory (as it was roll toward each other, collide, and lem was one of the first open scientific then known) released technical report shoot off in other directions. Now run investigations carried out with the LA-1940, titled “Studies of Nonlinear your film backwards. The motion of MANIAC, and it ushered in the age of Problems: I.” Authored by Enrico Fer- the balls looks perfectly natural—and what is sometimes called experimen- mi, John Pasta and Stanislaw Ulam, why not: Newton’s laws, the equa- tal mathematics. the results presented in this document tions that govern the motion of the The phrase “experimental mathe- have since rocked the scientific world. balls, work equally well for both posi- matics” might seem like an oxymoron: Indeed, it is not an exaggeration to say tive and negative times. Everyone knows that the validity of that the FPU problem, as the system Now imagine the beginning of a mathematics is independent of what Fermi, Pasta and Ulam studied is now game of billiards—actually, American goes on in the physical world. Nev- pool—with the 15 balls neatly racked ertheless, FPU’s original investigation up in a triangle and the cue ball hur- can very reasonably be described as Mason A. Porter received his doctorate from Cor- tling in to send them careening all the birth of experimental mathematics, nell University in 2002. He is currently a lecturer over the table. If we film the collision by which we mean computer-based of applied mathematics at Somerville College at the and the resulting havoc, no one who investigations designed to give insight University of Oxford. Norman J. Zabusky received has ever held a pool cue would mis- into complex mathematical and physi- his Ph.D. from California Institute of Technology take the film running forward for it cal problems that are inaccessible, at in 1959. He is now a visiting professor in the de- being run in reverse: The balls will least initially, using more traditional partment of complex physics at the Weizmann In- stitute of Science and an emeritus professor in the never regain their initial triangular ar- forms of analysis. department of mechanical engineering at Rutgers, rangement. Yet the laws governing all Today, computational studies of the State University of New Jersey. Bambi Hu of the collisions are still the same as in complex (typically nonlinear) prob- received his doctorate in 1974 from Cornell Uni- the case of two colliding billiard balls. lems are as commonplace as they are versity. He is now on the faculty in the department What then gives the arrow of time its essential, and the computer has taken of physics at the University of Houston. David K. direction? its rightful place alongside physical Campbell received his Ph.D. from the University For reasons that we will explore fur- experiment and theoretical analysis of Cambridge in 1970. He is currently provost and ther below, Fermi believed that the as a tool to study myriad phenomena a professor in the department of physics and the key was nonlinearity—the departure throughout the sciences, engineering department of electrical and computer engineering from the simple situation in which the and mathematics. Rigorous mathemat- at Boston University. The authors wish to dedicate this article to Martin Kruskal (1925–2006), one output of a physical system is linearly ical proofs, such as the one for the fa- of the pioneers in the study of solitary waves. Ad- proportional to the input. He knew mous “four-color problem,” have now dress for Porter: OCIAM, Mathematical Institute, that it would be far too complicated to been carried out with the aid of com- 24–29 St. Giles’, Oxford, U.K., OX1 3LB UK, find solutions to nonlinear equations puters. In fluid dynamics, computer- Internet: [email protected] of motion using pencil and paper. For- generated visualizations of complex, © 2009 Sigma Xi, The Scientific Research Society. Reproduction 214 American Scientist, Volume 97 with permission only. Contact [email protected]. Mark Humpage Figure 1. It’s possible to surf England’s Severn River because the Severn’s broad estuary periodically funnels exceptionally high flood tides up river, forming what’s known as a tidal bore. The waves that follow the initial onrush maintain their form for many kilometers, allowing record- breaking surfing runs. Such nondispersive waves arise in many physical systems, including a seemingly simple system of masses and springs that Enrico Fermi, John Pasta and Stanislaw Ulam studied using computational experiments at Los Alamos Scientific Laboratory in 1955. Their pioneering research ushered in the era of computer-aided discovery, deeply influencing many fields of science and mathematics. time-dependent flows have been cru- connected by springs for which motion The key question FPU wanted to cial to extracting underlying physical was allowed only along the line of the study was how long it would take the mechanisms. Modern experiments in chain. FPU’s idealized set of masses and oscillations of the string of masses and condensed-matter physics, observa- springs experienced no friction or inter- nonlinear springs to come to equilib- tions in astrophysics and data in bio- nal heating, so they could oscillate for- rium. The equilibrium they expected is informatics would all be impossible to ever without losing energy. The springs analogous to the state of thermal equi- interpret without computers. Things of this theoretical system were, howev- librium in a gas. In a monatomic gas, have come a long way since FPU’s er, not the kind studied in introductory such as helium, the thermal (kinetic) study, and in this light it becomes es- physics courses: The restoring force they energy of the molecules at equilibrium pecially important to understand how produced was not linearly proportional is equally partitioned among the three their pioneering work unfolded. to the amount of compression or exten- possible components of motion they With Pasta and Ulam, Fermi pro- sion. Instead, FPU included nonlinear can have: along the x, y or z axes. For posed to investigate what he assumed components in the mathematical rela- example, there won’t be more atoms would be a very simple nonlinear dy- tion between amount of deformation bouncing up and down than bouncing namical system—a chain of masses and the resulting restoring force. to the left and right. © 2009 Sigma Xi, The Scientific Research Society. Reproduction www.americanscientist.org 2009 May–June 215 with permission only. Contact [email protected]. Figure 2. Fermi, Pasta and Ulam modeled a series of masses connected to one another by springs. The masses move back and forth according to Newton’s law of motion f = ma (force equals mass times acceleration) along the line that connects them. Here the relevant forces are the restoring forces applied by the springs. What made the study so novel and fascinating is that the restoring forces were related nonlinearly to the amount of spring compression or extension. This notion of sharing energy even- plication of the equipartition theorem Initially, energy was shared among ly among different modes of motion is allows physicists to calculate such several different modes. After more fundamental. This precept, known as things as the heat capacity of a gas (simulated) time elapsed, their system the equipartition theorem of statistical from basic theory. returned to something that resembled mechanics, can be extended to include FPU’s premise was that they could its starting state. Indeed, 97 percent molecules that are more complicated start their system off with the masses in of the energy in the system was even- than billiard-ball-like helium, which just one simple mode of oscillation. If tually restored to the mode they had can partition energy in rotational or the system had linear springs (and no initially set up.
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