| Summary: | We study a quantity <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> defined as the energy U, stored in non-equilibrium steady states (NESS) over its value in equilibrium <inline-formula> <math display="inline"> <semantics> <msub> <mi>U</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo>−</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> divided by the heat flow <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mi>U</mi> </msub> </semantics> </math> </inline-formula> going out of the system. A recent study suggests that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is minimized in steady states (Phys.Rev.E.<b>99</b>, 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> with and without constraints and find that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">T</mi> </semantics> </math> </inline-formula> is the smallest for unconstrained NESS. We find that the form of the internal energy in the studied NESS follows <inline-formula> <math display="inline"> <semantics> <mrow> <mi>U</mi> <mo>=</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>∗</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mi>U</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. 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.
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