  Motivated by compartmental analysis in engineering and biophysical systems,
we present a variational framework for the nonequilibrium thermodynamics of
systems involving both distributed and discrete (finite dimensional) subsystems
by specifically using the ideas of interconnected systems. We focus on the
process of non-isothermal diffusion and show how the resulting form of the
entropy equation naturally yields phenomenological expressions for the
diffusion and entropy fluxes between two compartments, which results in
generalized forms of Robin type interface conditions.
