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 – instead of ergodic behavior.

The FPUT experiment

In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted numerical experiments of a vibrating string that included a non-linear term. 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 FPUT experiment was important both in showing the complexity of nonlinear system behavior and the value of computer simulation in analyzing systems.

Name change

The original paper names Fermi, Pasta, and Ulam as authors with an acknowledgement to Tsingou for her work in programming the MANIAC simulations. Mary Tsingou's contributions to the FPUT problem were largely ignored by the community until published additional information regarding the development and called for the problem to be renamed to grant her attribution as well.

The FPUT lattice system

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. Let there be N oscillators representing a string of length with equilibrium positions, where is the lattice spacing. Then the position of the j-th oscillator as a function of time is, so that gives the displacement from equilibrium. FPUT used the following equations of motion:
This is just Newton's second law for the j-th particle. The first factor is just the usual Hooke's law form for the force. The factor with is the nonlinear force. We can rewrite this in terms of continuum quantities by defining to be the wave speed, where is the Young's modulus for the string, and is the density:

Connection to the KdV equation

The continuum limit of the governing equations for the string is the Korteweg–de Vries equation The discovery of this relationship and of the soliton solutions of the KdV equation by Martin David Kruskal and Norman Zabusky in 1965 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. Beginning from the "continuum form" of the lattice equations above, we first define u to be the displacement of the string at position x and time t. We'll then want a correspondence so that is.
We can use Taylor's theorem to rewrite the second factor for small :
Similarly, the second term in the third factor is
Thus, the FPUT system is
If one were to keep terms up to O only and assume that approaches a limit, the resulting equation is one which develops shocks, which is not observed. Thus one keeps the O term as well:
We now make the following substitutions, motivated by the decomposition of traveling-wave solutions into left- and right-moving waves, so that we only consider a right-moving wave. Let. Under this change of coordinates, the equation becomes
To take the continuum limit, assume that tends to a constant, and tend to zero. If we take, then
Taking results in the KdV equation:
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.