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 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.

The FPU experiment

In the Summer of 1953 Fermi, Pasta, 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.

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

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 Kruskal and Zabusky in 1965 was an important step forward in nonlinear system research.

References

* E. Fermi, J. Pasta, S. Ulam, " [http://www.physics.utah.edu/~detar/phycs6720/handouts/fpu/FermiCollectedPapers1965.pdf Studies of Nonlinear Problems] ", Document LA-1940 (May 1955)