绝热活塞周期振荡的严格结果
Rigorous Results for the Periodic Oscillation of an Adiabatic Piston
摘要：
本文研究了一个质量为$M$的重活塞作一维运动。活塞将两个气室分开，每个气室都包含在$d$维度中移动的100多个理想的单位质量气体粒子，其中$ d\geq 1$。使用平均技术，我们证明了实际的活塞运动的概率收敛到预测的时间尺度$M^ {1/2} $上的平均行为时，$M$趋于无穷大，而系统的总能量是有界的，气体粒子的数量是固定的。Neishtadt和Sinai以前指出，Anosov的平均定理应该扩展到这种情况。当$ d=1$时，气体粒子仅在一维中运动，并且我们证明了活塞的实际运动到其平均行为的收敛速率在时间尺度$M^ {1/2} $上为$\mathcal{O} (M^ {-1/2}) $。在紧集上，在所有初始条件下一致收敛。我们还研究了活塞系统时，粒子的相互作用已被平滑。在初始条件和平滑量上，向平均行为的收敛再次均匀地发生。此外，我们证明了我们的结果的推广$N$活塞分离$N+1$气室。我们还提供了一个一般性的讨论平均理论和一些以前已知的平均结果的证明。特别是，我们包括一个新的证明Anosov的平均定理光滑系统，主要是由于Dolgopyat。
英文摘要：
We study a heavy piston of mass $M$ that moves in one dimension. The piston separates two gas chambers, each of which contains finitely many ideal, unit mass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale $M^ {1/2} $ when $M$ tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed. Neishtadt and Sinai previously pointed out that an averaging theorem due to Anosov should extend to this situation. When $ d=1$, the gas particles move in just one dimension, and we prove that the rate of convergence of the actual motions of the piston to its averaged behavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. The convergence is uniform over all initial conditions in a compact set. We also investigate the piston system when the particle interactions have been smoothed. The convergence to the averaged behavior again takes place uniformly, both over initial conditions and over the amount of smoothing. In addition, we prove generalizations of our results to $N$ pistons separating $N+1$ gas chambers. We also provide a general discussion of averaging theory and the proofs of a number of previously known averaging results. In particular, we include a new proof of Anosov's averaging theorem for smooth systems that is primarily due to Dolgopyat.
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