  We propose a stochastic order parameter equation for describing phase
coexistence in steady heat conduction near equilibrium. By analyzing the
stochastic dynamics with a non-equilibrium adiabatic boundary condition, where
total energy is conserved over time, we derive a variational principle that
determines thermodynamic properties in non-equilibrium steady states. The
resulting variational principle indicates that the temperature of the interface
between the ordered region and the disordered region becomes greater (less)
than the equilibrium transition temperature in the linear response regime when
the thermal conductivity in the ordered region is less (greater) than that in
the disordered region. This means that a super-heated ordered (super-cooled
disordered) state appears near the interface, which was predicted by an
extended framework of thermodynamics proposed in [N. Nakagawa and S.-i. Sasa,
Liquid-gas transitions in steady heat conduction, Phys. Rev. Lett. {\bf 119},
260602, (2017).]
