  In this paper we study the thermal equilibration of small bipartite
Bose-Hubbard systems, both quantum mechanically and in mean-field
approximation. In particular we consider small systems composed of a
single-mode "thermometer" coupled to a three-mode "bath", with no additional
environment acting on the four-mode system, and test the hypothesis that the
thermometer will thermalize if and only if the bath is chaotic. We find that
chaos in the bath alone is neither necessary nor sufficient for equilibration
in these isolated four-mode systems. The two subsystems can thermalize if the
combined system is chaotic even when neither subsystem is chaotic in isolation,
and under full quantum dynamics there is a minimum coupling strength between
the thermometer and the bath below which the system does not thermalize even if
the bath itself is chaotic. We show that the quantum coupling threshold scales
like $1/N$ (where $N$ is the total particle number), so that the classical
results are obtained in the limit $N\rightarrow \infty$.
