Fermi–Pasta–Ulam problem

In physics, the Fermi–Pasta–Ulam problem or FPU problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called Fermi–Pasta–Ulam recurrence – instead of ergodic behavior. One of the resolutions of the paradox includes the insight that many non-linear equations are exactly integrable. Another may be that ergodic behavior may depend on the initial energy of the system.

In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted numerical experiments (i.e. computer simulations) of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). 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 1955. (Enrico Fermi died in 1954 and so this technical report was published after Fermi's death.)

The FPU experiment was important both in showing the complexity of nonlinear system behavior and the value of computer simulation in analyzing systems.

In January 2008, Physics Today published additional information regarding the development of FPU.

Fermi, Pasta and Ulam (FPU) simulated the vibrating string by solving the following discrete system of nearest-neighbor coupled oscillators. We follow the explanation as given in Palais's article. Let there be N oscillators representing a string of length l with equilibrium positions ${\displaystyle p_{j}=jh,j=0,\dots ,N-1}$ where ${\displaystyle h=l/(N-1)}$ is the lattice spacing. Then the position of the jth oscillator as a function of time is ${\displaystyle X_{j}(t)=p_{j}+x_{j}(t)}$ so that ${\displaystyle x_{j}(t)}$ gives the displacement from equilibrium. FPU used the following equations of motion:

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