Fermi–Pasta–Ulam–Tsingou problem

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 ${\displaystyle \ell }$ with equilibrium positions ${\displaystyle p_j=jh,j=0,\dots ,N-1}$, where ${\displaystyle h=\ell /(N-1)}$ is the lattice spacing. Then the position of the j-th 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. FPUT used the following equations of motion:

${\displaystyle m{\ddot{x}}_j=k(x_{j+1}+x_{j-1}-2x_j)[1+\alpha (x_{j+1}-x_{j-1})].}$

This is just Newton's second law for the j-th particle. The first factor ${\displaystyle k(x_{j+1}+x_{j-1}-2x_j)}$ is just the usual Hooke's law form for the force. The factor with ${\displaystyle \alpha }$ is the nonlinear force. We can rewrite this in terms of continuum quantities by defining ${\displaystyle c=\sqrt{\kappa /\rho }}$ to be the wave speed, where ${\displaystyle \kappa =k/h}$ is the Young's modulus for the string, and ${\displaystyle \rho =m/h^3}$ is the density:

${\displaystyle {\ddot{x}}_j=\frac{c^2}{h^2}(x_{j+1}+x_{j-1}-2x_j)[1+\alpha (x_{j+1}-x_{j-1})].}$

Connection to the KdV equation

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 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(x, t) to be the displacement of the string at position x and time t. We'll then want a correspondence so that ${\displaystyle u(p_j,t)}$ is ${\displaystyle x_j(t)}$.

${\displaystyle {\ddot{x}}_j=\frac{c^2}{h^2}(x_{j+1}+x_{j-1}-2x_j)[1+\alpha (x_{j+1}-x_{j-1})].}$

We can use Taylor's theorem to rewrite the second factor for small ${\displaystyle h}$ (subscripts of u denote partial derivatives):

${\displaystyle \begin{array}{cc}\left(\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\right)& =\frac{u(x+h,t)+u(x-h,t)-2u(x,t)}{h^2}\\ & =u_{xx}(x,t)+\left(\frac{h^2}{12}\right)u_{xxxx}(x,t)+O(h^4).\end{array}}$

Similarly, the second term in the third factor is

${\displaystyle \alpha (x_{j+1}-x_{j-1})=2\alpha h{u}_x(x,t)+\left(\frac{\alpha h^3}{3}\right)u_{xxx}(x,t)+O(h^5).}$

Thus, the FPUT system is

${\displaystyle \frac{1}{c^2}{u}_{tt}-u_{xx}=(2\alpha h)u_x{u}_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}+O(\alpha h^2,h^4).}$

If one were to keep terms up to O(h) only and assume that ${\displaystyle 2\alpha h}$ approaches a limit, the resulting equation is one which develops shocks, which is not observed. Thus one keeps the O(h²) term as well:

${\displaystyle \frac{1}{c^2}{u}_{tt}-u_{xx}=(2\alpha h)u_x{u}_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}.}$

We now make the following substitutions, motivated by the decomposition of traveling-wave solutions (of the ordinary wave equation, to which this reduces when ${\displaystyle \alpha ,h}$ vanish) into left- and right-moving waves, so that we only consider a right-moving wave. Let ${\displaystyle \xi =x-ct,\tau =(\alpha h)ct,y(\xi ,\tau )=u(x,t)}$. Under this change of coordinates, the equation becomes

${\displaystyle y_{\xi \tau }-\left(\frac{\alpha h}{2}\right)y_{\tau \tau }=-y_{\xi }{y}_{\xi \xi }-\left(\frac{h}{24\alpha }\right)y_{\xi \xi \xi \xi }.}$

To take the continuum limit, assume that ${\displaystyle \alpha /h}$ tends to a constant, and ${\displaystyle \alpha ,h}$ tend to zero. If we take ${\displaystyle \delta =\underset{h\to 0}{lim}\sqrt{h/(24\alpha )}}$, then

${\displaystyle y_{\xi \tau }=-y_{\xi }{y}_{\xi \xi }-{\delta }^2{y}_{\xi \xi \xi \xi }.}$

Taking ${\displaystyle v=y_{\xi }}$ results in the KdV equation:

${\displaystyle v_{\tau }+v{v}_{\xi }+{\delta }^2{v}_{\xi \xi \xi }=0.}$

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.

Routes to thermalization

In 1966, Félix Izrailev and Boris Chirikov proposed that the system will thermalize, if a sufficient amount of initial energy is provided. 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. Yet, in 1970, Joseph Ford and Gary H. Lunsford insist that mixing can be observed even with arbitrarily small initial energies. There is a long and complex history of approaches to the problem, see Thierry Dauxois (2008) for a (partial) survey.

Recent work by Miguel Onorato et al. demonstrates a very interesting route to thermalization. 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, and is thus not able to spread energy from one mode to another; it can only generate the FPUT recurrence. The three-phonon interaction cannot thermalize the system.

A key insight, however, is that these modes are combinations of "free" and "bound" modes. That is, higher harmonics are "bound" to the fundamental, much in the same way that the higher harmonics in solutions to the KdV equation are bound to the fundamental. They do not have any dynamics of their own, and are instead phase-locked to the fundamental. Thermalization, if present, can only be among the free modes.

To obtain the free modes, a canonical transformation can be applied that removes all modes that are not free (that do not engage in resonant interactions). Doing so for the FPUT system results in oscillator modes that have a four-wave interaction (the three-wave interaction has been removed). These quartets do interact resonantly, i.e. do mix together four modes at a time. Oddly, though, when the FPUT chain has only 16, 32 or 64 nodes in it, these quartets are isolated from one-another. Any given mode belongs to only one quartet, and energy cannot bleed from one quartet to another. Continuing on to higher orders of interaction, there is a six-wave interaction that is resonant; furthermore, every mode participates in at least two different six-wave interactions. In other words, all of the modes become interconnected, and energy will transfer between all of the different modes.

The three-wave interaction is of strength ${\displaystyle 1/\alpha }$ (the same ${\displaystyle \alpha }$ as in prior sections, above). The four-wave interaction is of strength ${\displaystyle 1/{\alpha }^2}$ and the six-wave interaction is of strength ${\displaystyle 1/{\alpha }^4}$. Based on general principles from correlation of interactions (stemming from the BBGKY hierarchy) one expects the thermalization time to run as the square of the interaction. Thus, the original FPUT lattice (of size 16, 32 or 64) will eventually thermalize, on a time scale of order ${\displaystyle 1/{\alpha }^8}$: clearly, this becomes a very long time for weak interactions ${\displaystyle \alpha \ll 1}$; meanwhile, the FPUT recurrence will appear to run unabated. This particular result holds for these particular lattice sizes; the resonant four-wave or six-wave interactions for different lattice sizes may or may not mix together modes (because the Brillouin zones are of a different size, and so the combinatorics of which wave-vectors can sum to zero is altered.) Generic procedures for obtaining canonical transformations that linearize away the bound modes remain a topic of active research.

However, a recent study 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. These results also show an indication that there could be a stochasticity threshold in the ${\displaystyle \alpha }$-Fermi–Pasta–Ulam–Tsingou system.