  We propose the concept of global temperature for spatially non-uniform heat
conduction systems. With this novel quantity, we present an extended framework
of thermodynamics for the whole system such that the fundamental relation of
thermodynamics holds, which we call "global thermodynamics" for heat conduction
systems. Associated with this global thermodynamics, we formulate a variational
principle for determining thermodynamic properties of the liquid-gas phase
coexistence in heat conduction, which corresponds to the natural extension of
the Maxwell construction for equilibrium systems. We quantitatively predict
that the temperature of the liquid-gas interface deviates from the equilibrium
transition temperature. This result indicates that a super-cooled gas stably
appears near the interface.
