Storage of Energy in Constrained Non-Equilibrium Systems

We study a quantity T defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium U 0 , Δ U = U - U 0 divided by the heat flow J U going out of the system. A recent study suggests that T 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 T with and without constraints and find that T is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows U = U 0 ∗ f ( J U ) . 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.