Thermodynamic Variational Principle Unifying Gravity and Heat Flow

Abstract

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 non-equilibrium 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'' ($g_\mathrm{eff}$), within this framework. Crucially, the sign of $g_\mathrm{eff}$ determines the stable configuration: liquid is at the bottom if $g_\mathrm{eff} > 0$, and floats above the gas if $g_\mathrm{eff} < 0$. This provides a quantitative tool for the configuration prediction under competing drives.

Figures (4)

• Figure 1: System setup exposed to gravity and heat flow.
• Figure 2: (a) Center of mass $X$ as a function of $(T_2-T_1)/L$ under fixed gravity. $X$ shows a discontinuous transition when $g_\mathrm{eff}$ changes sign. (b) Configuration of the system for each point indicated in (a). The light green regions are metastable: supercooled gas in (1) and (3) and superheated liquid in (2).
• Figure 3: (a) Thermodynamic free energy $F_g$ as a function of $mg_\mathrm{eff} L$ at fixed $(\tilde{T},V,N)$. (b) Correspondence between gravity and temperature gradient. The value of $F_g$ is kept constant along the solid line $mg_\mathrm{eff} L=\mathrm{const}$, but the local state inside the system differs.
• Figure C1: Critical radius $r_c$ for the configuration inversion of 0.5 mol of H2O as a function of the average temperature $T \equiv (T_1+T_2)/2$. The temperature difference is held constant at $T_1 - T_2 = 0.75$ K. The line separates the liquid-below-gas ($r < r_\mathrm{c}$) and gas-below-liquid ($r > r_\mathrm{c}$) regimes.