## Fermi-Pasta-UlamApril 20, 2015 Of all the scientific disciplines, physics is the most mathematical. It's apparent, as Eugene Wigner wrote in 1960, that mathematics is unreasonably effective in describing the world around us.[1] The use of mathematics in the natural sciences can be traced back at least as far as the third century BC, when Greek mathematician, Eratosthenes, used trigonometry to determine the radius of the Earth to within two percent of its established value. Physicists were quick to use the first computers, including mechanical calculators made by Marchant and Friden. During the Manhattan Project, several wives of the Los Alamos scientists, including Edward Teller's wife, Mici, acted as human computers by doing calculations on such machines.[2] The versatile Richard Feynman acted as ade facto calculator repairman. Feynman teamed for calculator repair with Nicholas Metropolis. Metropolis was instrumental in early physics computing, as I wrote in a previous article (Nicholas Metropolis, June 11, 2010).
As Feynman wrote in his book, Surely You're Joking, Mr. Feynman!, Los Alamos eventually obtained an IBM punched-card computer.[3] This type of computer had proven its worth in calculating planetary orbits, so it was a safe bet. The first task of the Los Alamos IBM computer was a calculation to simulate a shaped-charge implosion. As a test of the efficacy of this new machine, the calculations were done simultaneously by both the ladies on the calculators and the IBM computer. The ladies were able to calculate at the same pace, but the IBM could function continually, without rest, so it won the competition.[3-4]
John von Neumann, who originated the shaped-charge idea, had an early interest in computers, and he arranged to have an unclassified Los Alamos calculation computed on the Harvard University Mark I electromechanical calculator. The Mark I took longer to calculate, but it did the calculation with more precision than the Los Alamos IBM computer.[2] Von Neumann was subsequently involved with ENIAC, the first electronic general-purpose computer, and its successor, the 6,000 vacuum tube EDVAC, one of the first stored-program computers.
1/ε, where ^{8}ε is a very small system parameter.[8] The mode mixing does not have a threshold, and the equipartition of energy will happen even for small nonlinearity.[8] Says Yuri Lvov, an author of the paper and a professor of mathematical sciences at Rensselaer Polytechnic Institute,
"My collaborators and I have shown that interactions of triads, quartets, and quintets are reversible; in other words, they do not bring the FPU system closer to thermal equilibrium. However, the interaction of waves in sixtets does lead to irreversible transfer of energy. It takes the cooperation of six different waves to produce an interaction that is irreversible and, because of that, the process is extremely weak and very slow. That is why it takes so long to approach thermal equilibrium for the FPU system."[9]
## References:- Eugene P. Wigner, "The unreasonable effectiveness of mathematics in the natural sciences," Communications on Pure and Applied Mathematics, vol. 13, no. 1 (February 1960), doi:10.1002/cpa.3160130102, pp. 1-14. An online transcription is available, here, and a PDF file is available here.
- Manhattan Project History - Evolving from Calculators to Computers, The Manhattan Project Heritage Preservation Association, Inc..
- Richard Feynman, "Surely You're Joking, Mr. Feynman! (Adventures of a Curious Character)," W. W. Norton & Company, April 17, 1997, 352 pp. (via Amazon)
- Computing and the Manhattan Project, Atomic Heritage Foundation.
- N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, "Equation of State Calculations by Fast Computing Machines," Journal of Chemical Physics, vol. 21, no. 6 (1953), pp.1087-1092. For a review of this paper, see its Wikipedia page. John R. Pasta (1918 – June 5, 1981), Chess Programming at Wikispaces.
- Thierry Dauxois, "Fermi, Pasta, Ulam, and a mysterious lady," Physics Today, January, 2008, pp. 55-57 (PDF File).
- J.A.N. Lee, "Computer Pioneers - John R. Pasta," IEEE Computer Society.
- Miguel Onorato, Lara Vozella, Davide Proment, and Yuri V. Lvov, "Route to thermalization in the α-Fermi–Pasta–Ulam system," Proc. Natl. Acad. Sci., Published online before print, March 24, 2015, doi: 10.1073/pnas.1404397112
- Mary L. Martialay, "A Mathematical Explanation for the Fermi-Pasta-Ulam System Problem First Proposed in 1953," Rensselaer Polytechnic Institute Press Release, March 23, 2015.
- UEA mathematician solves 60-year-old problem, University of East Anglia Press Release, March 23, 2015.
Linked Keywords: Science; scientific; discipline; physics; mathematics; mathematical; Eugene Wigner; The Unreasonable Effectiveness of Mathematics in the Natural Sciences; mathematics is unreasonably effective; natural science; third century BC; Greek; mathematician; Eratosthenes; trigonometry; radius of the Earth; physicist; computer; mechanical calculator; Marchant calculator; Friden, Inc.; Manhattan Project; wife; Los Alamos National Laboratory; Edward Teller; human computer; calculation; Richard Feynman; maintenance, repair, and operations; repairman; Nicholas Metropolis; Surely You're Joking, Mr. Feynman!; IBM; punched-card; planetary orbit; shaped-charge; implosion; competition; John von Neumann; classified information; unclassified; Harvard University; Harvard Mark I electromechanical calculator; precision; ENIAC; electronic; vacuum tube; EDVAC; stored-program computer; computer hardware; ENIAC; BRLESC-I; Wikimedia Commons; parallel computing; World War II; MANIAC I; software; Stanislaw Ulam; Monte Carlo method; uncle; gambling; Monte Carlo casino; play; profession; work; safe-cracking; safe; combination lock; John Pasta; computer science; University of Illinois at Urbana-Champaign; chess engine; chess program; Los Alamos Chess; chessboard; chess board; bishop; human; computer simulation; Fermi-Pasta-Ulam Problem; particle in a box; quantum mechanics; time evolution; temporal evolution; vibrating; nonlinear; string; normal mode; single oscillatory mode; heat; solid; thermodynamic equilibrium; equilibrium state; relaxation time; damping; amplitude; soliton; technical report; Enrico Fermi (1901-1954); fine-structure constant; cgs units; mathematician; University of Turin; Università degli Studi di Torino; Istituto Nazionale di Fisica Nucleare–Sezione Torino; Turin, Italy; University of East Anglia (Norwich, United Kingdom); Rensselaer Polytechnic Institute (Troy, New York); equipartition theorem; equipartition of energy states; acoustic dispersion; dispersive waves; thermal equilibrium; partition function; iteration; mass; linearity; nonlinear; quadratic function; spring; research; wave turbulence; irreversible process; thermodynamics; irreversible; randomness; random; orders of approximation; order; system; parameter; threshold; Yuri Lvov; author; professor; mathematical sciences; scientific visualization; resonance; manifold. |
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