The apparent contradiction of the results of the Fermi-Pasta-Ulam experiment conducted in and with the hypothesis that essentially any nonlinearity would lead to a system exhibiting ergodic behaviour has become known as the Fermi-Pasta-Ulam Problem. The contributions comprise studies of one-dimensional chains, descriptions of numerical methods, heuristic theories, addressing the "long standing and controversial problem of distinguishing chaos from noise in signal analysis," metastability, the relation of the FPU motions with the integrable equations, approaches using methods of perturbation theory and the proof of the applicability of KAM theory in FPU chains with energy very close to a minimum. For the convenience of the reader the original work of FPU is reprinted in an appendix. The order of the contributions reflects the aim of leading the interested but inexperienced reader through gradual understanding, starting from general analysis, and proceeding towards more specialized topics. Skip to main content Skip to table of contents.

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In physics , the Fermi—Pasta—Ulam—Tsingou problem or formerly the Fermi—Pasta—Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior — called Fermi—Pasta—Ulam—Tsingou recurrence or Fermi—Pasta—Ulam recurrence — instead of ergodic behavior. They found that the behavior of the system was quite different from what intuition would have led them to expect.

Fermi thought that after many iterations, the system would exhibit thermalization , an ergodic behavior in which the influence of the initial modes of vibration fade and the system becomes more or less random with all modes excited more or less equally. Instead, the system exhibited a very complicated quasi-periodic behavior. They published their results in a Los Alamos technical report in Enrico Fermi died in , and so this technical report was published after Fermi's death.

The FPUT experiment was important both in showing the complexity of nonlinear system behavior and the value of computer simulation in analyzing systems. Fermi, Pasta, Ulam, and Tsingou simulated the vibrating string by solving the following discrete system of nearest-neighbor coupled oscillators. We follow the explanation as given in Richard Palais 's article. FPUT used the following equations of motion:. Note: this equation is not equivalent to the classical one given in the French version of the article.

This is just Newton's second law for the j -th particle. The continuum limit of the governing equations for the string with the quadratic force term is the Korteweg—de Vries equation KdV equation.

The discovery of this relationship and of the soliton solutions of the KdV equation by Martin David Kruskal and Norman Zabusky in was an important step forward in nonlinear system research. We reproduce below a derivation of this limit, which is rather tricky, as found in Palais's article.

Thus one keeps the O h 2 term as well:. Under this change of coordinates, the equation becomes. Zabusky and Kruskal argued that it was the fact that soliton solutions of the KdV equation can pass through one another without affecting the asymptotic shapes that explained the quasi-periodicity of the waves in the FPUT experiment.

In short, thermalization could not occur because of a certain "soliton symmetry" in the system, which broke ergodicity. From Wikipedia, the free encyclopedia. Chaos theory.

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## Fermi–Pasta–Ulam–Tsingou problem

In physics , the Fermi—Pasta—Ulam—Tsingou problem or formerly the Fermi—Pasta—Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior — called Fermi—Pasta—Ulam—Tsingou recurrence or Fermi—Pasta—Ulam recurrence — instead of ergodic behavior. They found that the behavior of the system was quite different from what intuition would have led them to expect. Fermi thought that after many iterations, the system would exhibit thermalization , an ergodic behavior in which the influence of the initial modes of vibration fade and the system becomes more or less random with all modes excited more or less equally. Instead, the system exhibited a very complicated quasi-periodic behavior. They published their results in a Los Alamos technical report in Enrico Fermi died in , and so this technical report was published after Fermi's death.

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