  We provide an extension of a recent approach to study non-equilibrium
thermodynamics [Phys. Rev. E 81, 051130 (2010), to be denoted by I in this
work] to inhomogeneous systems by considering the latter to be composed of
quasi-independent subsystems. The system {\Sigma} along with the
(macroscopically extremely large) medium {\Sigma} forms an isolated system
{\Sigma}_0. Starting from the Gibbsian formulation of the entropy for
{\Sigma}_0, which is valid even when {\Sigma}_0 is out of equilibrium, we
derive the Gibbsian formulation of the entropy of {\Sigma}, which need not be
in equilibrium. The additivity of entropy requires quasi-independence of the
subsystems, which limits the size of various subsystems. The thermodynamic
potentials of subsystems such as the Gibbs free energy are determined by the
field parameters (temperature, pressure, etc.) of the medium even if the latter
may not exist for the subsystems. This and the requirement of
quasi-independence make our approach different from the conventional approach
due to de Groot and others. As the energy depends on the frame of reference,
the thermodynamic potentials and Gibbs fundamental relation, but not the
entropy, depend on the frame of reference. The possibility of relative motion
between subsystems described by their net linear and angular momenta gives rise
to viscous dissipation. Important consequences of internal equilibrium are
discussed. Internal variables as variables that cannot be controlled by the
observer for non-equilibrium evolution are also discussed. We finally formulate
the non-equilibrium thermodynamics of inhomogeneous systems. We also briefly
discuss the case when bodies form an isolated system without any medium to
obtain their irreversible contributions and show that this case is no different
than when bodies are in an extremely large medium.
