  The NMR technique allows one to create a non-equilibrium local polarization
and to detect its later evolution. By a change of the sign of the effective
dipolar Hamiltonian, the apparently diffusive dynamics is reverted, generating
a polarization echo (PE). This echo attenuates as a function of the time $t_R$
elapsed until the dynamics is reverted. We report a set of experiments where we
slow down the dipolar dynamics showing that it controls this irreversibility.
In cobaltocene, a cross over from Gaussian to exponential PE attenuation is
found; in ferrocene the exponential regime is not reached. We conclude that
many-body quantum systems have intrinsic instabilities, which amplify the
environment fluctuations and hinder their backward evolution.
