非线性Wave Equation的亚稳性：过渡态理论的启示
Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory
摘要：
本文研究一维空间中非线性波动方程$$ u_{tt} - δ^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$的长时间动力学，其中$δ>0$是一个参数，$V(u)$是一个自下而上有界的势函数，它至少象$u^2$那样增长.这个方程的无限能量解保持了一个自然的吉布斯不变测度，并且当势是双阱时，例如当$V(u) = \tfrac14(1-u^2)^2$时，存在一个区域，使得系统相空间中的两个小的不相交集合集中了这个测度的大部分质量。这表明非线性波动方程的解在这些集合上是亚稳定的，在这个意义上，它们在这些集合中花费很长时间，很少在它们之间过渡。在这里，我们量化这一现象，通过计算准确地通过过渡态理论（TST）的平均频率，在该非线性波动方程的解决方案与初始条件从其不变测度交叉的分割面之间的亚稳集。数值结果表明，当参数$δ$足够小时，非线性波动方程的动力学是遍历的，并且关于Gibbs不变测度是快速混合的.这是一种非线性波动方程动力学表现出亚稳态行为的状态，这种亚稳态行为与在随机波动方程中观察到的行为没有根本的不同，在随机波动方程中，随机噪声和阻尼项被添加到方程中。然而，对于较大的$δ$，动态或者停止遍历，或者其混合时间变得大于TST频率的倒数，这表明亚稳态集之间的连续过渡是相关的，粗粒度到马尔可夫链失败。
英文摘要：
This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - δ^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $δ>0$ is a parameter and $V(u)$ is a potential bounded from below and growing at least like $u^2$ as $|u|\to\infty$. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure and when the potential is double-welled, for example when $V(u) = \tfrac14(1-u^2)^2$, there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here we quantify this phenomenon by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter $δ$ in small enough. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger $δ$, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.
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