  We analyze the coherence properties of neutron wave packets, after they have
interacted with a phase shifter undergoing different kinds of statistical
fluctuations. We give a quantitative (and operational) definition of
decoherence and compare it to the standard deviation of the distribution of the
phase shifts. We find that in some cases the neutron ensemble is more coherent,
even though it has interacted with a wider (i.e. more disordered) distribution
of shifts. This feature is independent of the particular definition of
decoherence: this is shown by proposing and discussing an alternative
definition, based on the Wigner function, that displays a similar behavior. We
briefly discuss the notion of entropy of the shifts and find that, in general,
it does not correspond to that of decoherence of the neutron.
