热力学极限下FPU的一个平均定理
An averaging theorem for FPU in the thermodynamic limit
摘要：
考虑由$N\gg 1$粒子组成的FPU链，并赋予相空间对应于小温度$β^{-1}$的Gibbs测度。给定固定的$K<N$，我们构造其能量是绝热不变量的正常模式的$K$包（即，对于大测度集合中的初始数据，对于阶数为$β^{1-a}$，$a>0$的时间近似恒定。此外，每个分组的能量的时间自相关函数对于$β$阶的时间不显著衰减。对包的形状的限制是非常温和的。所有的估计在粒子数$N$上是一致的，因此保持在热力学极限$N\to\infty$，$β>0$内。
英文摘要：
Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $β^{-1}$. Given a fixed $K<N$, we construct $K$ packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order $β^{1-a}$, $a>0$) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order $β$. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number $N$ of particles and thus hold in the thermodynamic limit $N\to\infty$, $β>0$.
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