Fermi–Pasta–Ulam–Tsingou problem
In physics, the Fermi–Pasta–Ulam–Tsingou (FPUT) 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 the expected ergodic behavior.
This came as a surprise, as Enrico Fermi, certainly, expected the system to thermalize in a fairly short time.
That is, it was expected for all vibrational modes to eventually appear with equal strength, as per the equipartition theorem, or, more generally, the ergodic hypothesis.
Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of active research.
The original intent was to find a physics problem worthy of numerical simulation on the then-new MANIAC computer.
As such, it represents one of the earliest uses of digital computers in mathematical research; simultaneously, the unexpected results launched the study of nonlinear systems.
In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted 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).
Enrico 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.
In 2020, National Security Science magazine featured an article on Tsingou that included her commentary and historical reflections on the FPUT problem.
Mary Tsingou's contributions to the FPUT problem were largely ignored by the community until Thierry Dauxois (2008) published additional information regarding the development and called for the problem to be renamed to grant her attribution as well.
Fermi, Pasta, Ulam, and Tsingou simulated the vibrating string by solving the following discrete system of nearest-neighbor coupled oscillators.
FPUT used the following equations of motion: This is just Newton's second law for the j-th particle.
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.
Beginning from the "continuum form" of the lattice equations above, we first define u(x, t) to be the displacement of the string at position x and time t. We'll then want a correspondence so that
Thus one keeps the O(h2) term as well: We now make the following substitutions, motivated by the decomposition of traveling-wave solutions (of the ordinary wave equation, to which this reduces when
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.
A similar set of manipulations (and approximations) lead to the Toda lattice, which is also famous for being a completely integrable system.
It, too, has soliton solutions, the Lax pairs, and so also can be used to argue for the lack of ergodicity in the FPUT model.
[2][3] In 1966, Félix Izrailev and Boris Chirikov proposed that the system will thermalize, if a sufficient amount of initial energy is provided.
[4] The idea here is that the non-linearity changes the dispersion relation, allowing resonant interactions to take place that will bleed energy from one mode to another.
A review of such models can be found in Roberto Livi et al.[5] Yet, in 1970, Joseph Ford and Gary H. Lunsford insist that mixing can be observed even with arbitrarily small initial energies.
[6] There is a long and complex history of approaches to the problem, see Thierry Dauxois (2008) for a (partial) survey.
[7] Recent work by Miguel Onorato et al. demonstrates a very interesting route to thermalization.
[8] Rewriting the FPUT model in terms of normal modes, the non-linear term expresses itself as a three-mode interaction (using the language of statistical mechanics, this could be called a "three-phonon interaction".)
It is, however, not a resonant interaction,[9] and is thus not able to spread energy from one mode to another; it can only generate the FPUT recurrence.
Thus, the original FPUT lattice (of size 16, 32 or 64) will eventually thermalize, on a time scale of order
Generic procedures for obtaining canonical transformations that linearize away the bound modes remain a topic of active research.
However, a recent study [10] found that there are divergences in the canonical transformation used to remove the three-wave interactions due to the presence of small denominators.
These small denominators become more prominent when the lower modes are excited, and are more significant as the system size is increased.
