  This paper is concerned with the long-time dynamics of the nonlinear wave
equation in one-space dimension, $$ u_{tt} - \delta^2 u_{xx} +V'(u) =0 \qquad
x\in [0,1] $$ where $\delta>0$ 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 $\delta$ 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
$\delta$, 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.
