We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N \gg 1$ particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $\beta^{-1}$. Given a fixed ${1\leq m \ll N}$, we prove that the first $m$ integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $\beta$, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

中文翻译：

---

热力学极限下 FPUT 和 Toda 链的绝热不变量

我们考虑由$N \gg 1$ 粒子和周期性边界条件组成的Fermi-Pasta-Ulam-Tsingou (FPUT) 链，并赋予相空间在小温度$\beta^{-1}$ 下的吉布斯测度。给定一个固定的 ${1\leq m \ll N}$，我们证明周期 Toda 链运动的第一个 $m$ 积分是 FPUT 的绝热不变量（即它们沿着 FPUT 的哈密顿流近似恒定）对于订单 $\beta$ 的时间，对于一组大度量中的初始数据。我们还证明了谐波能量的特殊线性组合是 FPUT 在同一时间尺度上的绝热不变量，而对于 Toda 动力学，它们在所有时间都成为绝热不变量。