  We consider one-dimensional infinite chains of harmonic oscillators with
random exchanges of momenta and long-range interaction potentials which have
polynomial decay rate $|x|^{-\theta}, x \to \infty, \theta > 1$ where $x \in
\mathbb{Z}$ is the interaction range. The dynamics conserve total momentum,
total length and total energy. We prove that the systems evolve macroscopically
on superballistic space-time scale $(y \varepsilon^{-1}, t \varepsilon^{-
\frac{\theta - 1}{2}})$ when $1 < \theta < 3$, $(y \varepsilon^{-1}, t
\varepsilon^{-1} \sqrt{- \log(\varepsilon^{-1})}^{-1})$ when $\theta = 3$, and
ballistic space-time scale $(y \varepsilon^{-1}, t \varepsilon^{-1})$ when
$\theta > 3$. Combining our results and the results in [10], we show the
existence of two different space-time scales on which the systems evolve. In
addition, we prove scaling limits of recentered normal modes of superballistic
wave equations, which are analogues of Riemann invariants and capture
fluctuations around characteristics. The space-time scale is superdiffusive
when $2 < \theta \le 4$ and diffusive when $\theta > 4$.
