Flux and storage of energy in non-equilibrium, stationary states

Systems kept out of equilibrium in stationary states by an external source of energy store an energy $\Delta U=U-U_0$. $U_0$ is the internal energy at equilibrium state, obtained after the shutdown of energy input. We determine $\Delta U$ for two model systems: ideal gas and Lennard-Jones fluid. $\Delta U$ depends not only on the total energy flux, $J_U$, but also on the mode of energy transfer into the system. We use three different modes of energy transfer where: the energy flux per unit volume is (i) constant; (ii) proportional to the local temperature (iii) proportional to the local density. We show that $\Delta U /J_U=\tau$ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. $\tau$ is the characteristic time scale of energy outflow from the system immediately after the shutdown of energy flux. We prove that $\tau$ is minimized in stable states of the Rayleigh-Benard cell.