  We investigate the long time behavior of a wavepacket initially localized at
a single site $n_0$ in translationally invariant harmonic and anharmonic chains
with random interactions. In the harmonic case, the energy profile $ \bar{<
e_n(t)>}$ averaged on time and disorder decays for large $|n-n_0|$ as a power
law $\bar{< e_n(t)>}\approx C|n-n_0|^{-\eta}$ where $\eta=5/2$ and 3/2 for
initial displacement and momentum excitations, respectively. The prefactor $C$
depends on the probability distribution of the harmonic coupling constants and
diverges in the limit of weak disorder. As a consequence, the moments $<
m_{\nu}(t)>$ of the energy distribution averaged with respect to disorder
diverge in time as $t^{\beta(\nu)}$ for $\nu \geq 2$, where $\beta=\nu+1-\eta$
for $\nu>\eta-1$. Molecular dynamics simulations yield good agreement with
these theoretical predictions. Therefore, in this system, the second moment of
the wavepacket diverges as a function of time despite the wavepacket is not
spreading. Thus, this only criteria often considered earlier as proving the
spreading of a wave packet, cannot be considered as sufficient in any model.
The anharmonic case is investigated numerically. It is found for intermediate
disorder, that the tail of the energy profile becomes very close to those of
the harmonic case. For weak and strong disorder, our results suggest that the
crossover to the harmonic behavior occurs at much larger $|n-n_0|$ and larger
time.
