The Fermi–Pasta–Ulam ‘numerical experiment’: history and pedagogical perspectives

The pioneering Fermi–Pasta–Ulam (FPU) numerical experiment played a major role in the history of computer simulation because it introduced this concept for the first time. Moreover, it raised a puzzling question which was answered more than 10 years later. After an introduction to this problem, we briefly review its history and then suggest some simple numerical experiments, with the Matlab© code provided, to study various aspects of the 'FPU' problem.

[1]  Joseph Ford,et al.  Equipartition of Energy for Nonlinear Systems , 1961 .

[2]  S. Croucher,et al.  Surveys , 1965, Understanding Communication Research Methods.

[3]  M. Toda.  Vibration of a Chain with Nonlinear Interaction , 1967 .

[4]  H. Poincaré,et al.  Erratum zu: Etude des surfaces asymptotiques , 1890 .

[5]  Physics Letters , 1962, Nature.

[6]  S. Wolfram.  Statistical mechanics of cellular automata , 1983 .

[7]  Thomas P. Weissert,et al.  The genesis of simulation in dynamics: Pursuing the Fermi‐Pasta‐Ulam problem Lattice‐gas cellular automata: Simple models of complex hydrodynamics , 1998 .

[8]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[9]  N. Sait̂o,et al.  Computer Studies of Ergodicity in Coupled Oscillators with Anharmonic Interaction , 1967 .

[10]  N. Sait̂o,et al.  Long-Time Behavior of the Vibration in One-Dimensional Harmonic Lattice , 1967 .

[11]  A. Sirota,et al.  NONCOMPACT SEMISIMPLE LIE GROUPS , 1963 .

[12]  Norman J. Zabusky,et al.  Dynamics of nonlinear lattices I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior , 1967 .

[13]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[14]  David K Campbell,et al.  Introduction: The Fermi-Pasta-Ulam problem--the first fifty years. , 2005, Chaos.

[15]  K. Kaneko.  Overview of coupled map lattices. , 1992, Chaos.

[16]  Michel Peyrard,et al.  Physics of Solitons , 2006 .

[17]  Morikazu Toda,et al.  Theory Of Nonlinear Lattices , 1981 .

[18]  C. -. Lin,et al.  When is a one-dimensional lattice small? , 1999 .

[19]  Thomas P. Weissert,et al.  The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem , 1999 .

[20]  S. Ulam,et al.  Adventures of a Mathematician , 2019, Mathematics: People · Problems · Results.

[21]  D. Sholl.  Modal coupling in one-dimensional anharmonic lattices , 1990 .

[22]  Joseph Ford,et al.  The Fermi-Pasta-Ulam problem: Paradox turns discovery , 1992 .

[23]  A. Vulpiani,et al.  Relaxation to different stationary states in the Fermi-Pasta-Ulam model , 1983 .

[24]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[25]  L. Maximon,et al.  Recurrence Time of a Dynamical System , 1958 .