  We study the dynamic phase diagram of a spin model associated with the Number
Partitioning Problem, as a function of temperature and of the fraction $K/N$ of
spins allowed to flip simultaneously. The case K=1 reproduces the activated
behavior of Bouchaud's trap model, whereas the opposite limit $K=N$ can be
mapped onto the entropic trap model proposed by Barrat and M\'ezard. In the
intermediate case $1 \ll K \ll N$, the dynamics corresponds to a modified
version of the Barrat and M\'ezard model, which includes a slow (rather than
instantaneous) decorrelation at each step. A transition from an activated
regime to an entropic one is observed at temperature $T_g/2$ in agreement with
recent work on this model. Ergodicity breaking occurs for $T<T_g/2$ in the
thermodynamic limit, if $K/N \to 0$. In this temperature range, the model
exhibits a non trivial fluctuation-dissipation relation leading for $K \ll N$
to a single effective temperature equal to $T_g/2$. These results give new
insights on the relevance and limitations of the picture proposed by simple
trap models.
