Storage of Energy in Constrained Non-Equilibrium Systems

We study a quantity [Formula: see text] defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium [Formula: see text] , [Formula: see text] divided by the heat flow [Formula: see text] going out of the system. A recent study suggests that [Formula: see text] is minimized in steady states (Phys.Rev.E.99, 042118 (2019)). We evaluate this hypothesis using an ideal gas system with three methods of energy delivery: from a uniformly distributed energy source, from an external heat flow through the surface, and from an external matter flow. By introducing internal constraints into the system, we determine [Formula: see text] with and without constraints and find that [Formula: see text] is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows [Formula: see text]. In this context, we discuss natural variables for NESS, define the embedded energy (an analog of Helmholtz free energy for NESS), and provide its interpretation.