  Interface equations are derived for both binary diffusive and binary fluid
systems subjected to non-equilibrium conditions, starting from the
coarse-grained (mesoscopic) models. The equations are used to describe
thermo-capillary motion of a droplet in both purely diffusive and fluid cases,
and the results are compared with numerical simulations. A mesoscopic chemical
potential shift, owing to the temperature gradient, and associated mesoscopic
corrections involved in droplet motion are elucidated.
