论文标题
热力学极限中FPUT和TODA链的绝热不变性
Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit
论文作者
论文摘要
我们考虑由$ n \ gg 1 $粒子和周期性边界条件组成的费米 - 帕斯塔 - 乌拉姆 - tsingou（fput）链，并以少量温度$β^{ - 1} $赋予相位空间。给定固定的$ {1 \ leq m \ ll n} $，我们证明，定期TODA链的第一个$ m $运动积分是fput的绝热不变式（即它们沿汉密尔顿的流量近似稳定），用于订单$β$的时间，用于大量级别的初始数据。我们还证明，谐波能的特殊线性组合是同一时间尺度上的fput的绝热不变剂，而它们始终成为TODA动力学的绝热不变性。
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 $β^{-1}$. 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 $β$, 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.
