June 24, 2013

## 3. A solution to the FPU paradox

The FPU report was the first article on a subject that inspired many reports and many books in the last 50 years. Altough the articles do not provide a unitary explanation of the problem, one of the most exploited notions used to solve the FPU “paradox” is the metastability. This notion, introduced in 1982-83, states that at small specific energy ${\epsilon}$ (at least) two separated time scales must be considered:
case 1. in the short time scale, the system reaches a state different form the initial condition, when additional modes are excited, but the state is still far from energy equipartition. This is the state observed at low energy by Fermi;
case 2. in the long time scale, which is actually much longer than the time needed to reach the metastable state, the system finally evolves towards energy equipartition.

Averaged energy spectrum in a FPU ${\alpha+\beta}$-model with ${N=1023}$ particles, ${\alpha=-1}$, ${\beta=2}$ and specific energy ${\epsilon=10^{-4}}$.

The possible explanation of the metastability phenomenon depends on the FPU model. Precisely we have that:

case 1. in the ${\alpha+\beta}$ model the short time scale can be studied as a perturbation of the integrable Toda model, which has the potential
$\displaystyle V_T(R) = V_0(e^{\lambda r}-1-\lambda \; r),$
which for ${V_0 = \lambda^{-2}}$, ${\lambda = 2\alpha}$ coincides with the FPU potential ${V(r)}$ up to the order ${r^3}$. More explicitly, by expanding in Taylor series the Toda potential we get

$\displaystyle V_T(r) = \frac{1}{2} \; r^2 + \frac{\alpha}{3} \; r^3 + \frac{\beta_T}{4} \; r^4 + \frac{\gamma_T}{5} \; r^5 + \ldots$

where

$\displaystyle \beta_T = \frac{2}{3} \; \alpha^2, \; \; \; \gamma_T = \frac{1}{3} \; \alpha^3$
so for ${\beta \neq \beta_T}$ the perturbation is ${O(r^4)}$, while for ${\beta = \beta_T}$ the perturbation is ${O(r^5)}$.
Since the Toda model is integrable, for such a system we do not have a longer time-scale (ie, there is no possible energy equipartition); for the FPU model, instead, the equipartition does occur, and in the thermodynamic limit (ie, at fixed ${\epsilon}$ and for increasing ${N}$, a very interesting condition from the point of view of Statistical Mechanics) the equilibrium time is estimated as

$\displaystyle T_{eq} \sim \alpha^{-1/2} \; (\beta-\beta_T)^{-2} \epsilon^{-9/4} = O(\epsilon^{-9/4}).$
The previous estimate holds out of neighborhoods of ${\alpha = 0}$ and ${\beta = \beta_T}$.

case 2. In the ${\beta}$ model, on the other hand, we do not have an analogue of the Toda model, so the best integrable model we can use as an approximation during the short time scale is the chain of uncoupled oscillators. Recently it was discovered that initial conditions, namely the initial energy ditribution among normal modes, have a direct effect on the equilibrium time. It has been reported that if the frequency ${\omega_k}$ of the normal modes is bounded away form 0 then in the thermodynamic limit we get

$\displaystyle T_{eq} = O( \; e^{\epsilon^{-1/4}} \; ) ,$

while if frequencies arbitrarily close to 0 are excited we get the usual power law,

$\displaystyle T_{eq} = O(\epsilon^{-9/4}).$