  Bosonic Josephson junctions can be realized by confining ultracold gases of
bosons in multi-well traps, and studied theoretically with the $M$-site
Bose-Hubbard model. We show that canonical equilibrium states of the $M$-site
Bose-Hubbard model may be approximated by mixtures of coherent states, provided
the number of atoms is large and the total energy is comparable to $k_BT$.
Using this approximation, we study thermal fluctuations in bosonic Josephson
junctions in the mean field regime. Statistical estimates of the fluctuations
of relative phase and number, obtained by averaging over many replicates of an
experiment, can be used to estimate the temperature and the tunneling
parameter, or to test whether the experimental procedure is effectively
sampling from a canonical thermal equilibrium ensemble.
