  A model for studying atomtronic devices and circuits based on finite
temperature Bose-condensed gases is presented. The approach involves
numerically solving equations of motion for atomic populations and coherences,
derived using the Bose-Hubbard Hamiltonian and the Heisenberg picture. The
resulting cluster expansion is truncated at a level giving balance between
physics rigor and numerical demand mitigation. This approach allows parametric
studies involving time scales that cover both the rapid population dynamics
relevant to non-equilibrium state evolution, as well as the much longer time
durations typical for reaching steady-state device operation. The model is
demonstrated by studying the evolution of a Bose-condensed gas in the presence
of atom injection and extraction in a double-well potential. In this
configuration phase-locking between condensates in each well of the potential
is readily observed, and its influence on the evolution of the system is
studied.
