Thermodynamic Variational Principle Unifying Gravity and Heat Flow

Predicting the stable phase configuration in a liquid-gas system becomes a fundamental challenge when the stratification favored by gravity conflicts with arrangements induced by heat flow, particularly because standard equilibrium thermodynamics is insufficient in such nonequilibrium steady states. We propose a variational principle based on an extended thermodynamics, called global thermodynamics, to address this state selection problem. Our key finding is that gravity and heat flow effects are unified into a single parameter, "effective gravity" (g_{eff}), within this framework. Crucially, the sign of g_{eff} determines the stable configuration: liquid is at the bottom if g_{eff}>0, and floats above the gas if g_{eff}<0. This provides a quantitative tool for the configuration prediction under competing drives.