Exergy ($A-A_{\mathrm{eq}}$)

For a closed system with fixed composition in an environment at temperature $T_R>0$ and pressure $p_R$, define $A=U+p_{R}V-T_{R}S$, where $U$ is internal energy, $V$ is volume and $S$ is entropy. The exergy is this availability relative to the equilibrium reference state, $A-A_{\mathrm{eq}}$. When the combined system and environment are isolated except for the specified exchanges, $\Delta S_{\mathrm{tot}}=-\Delta A/T_R$, so the Second law of thermodynamics makes $A$ decrease toward its constrained minimum. Away from equilibrium, the environmental $T_R,p_R$ must not be replaced by the system's instantaneous values.

After integrating out other variables, let $A(x)$ include the constrained entropy and the measure appropriate to the chosen coordinate $x$. Its equilibrium probability density function is proportional to $e^{-A(x)/(k_B{T}_R)}$. Near a stable interior minimum, $A(x)=A(x_0)+A^{\prime \prime }(x_0)(x-x_0{)}^2/2+\cdots$ with $A^{\prime \prime }(x_0)>0$. The resulting normal distribution has variance $k_B{T}_R/A^{\prime \prime }(x_0)$. This is the leading Gaussian fluctuation approximation; a coordinate-dependent measure contributes to the effective $A$, and higher-order terms matter near a vanishing curvature.