## Appendix A  
In this appendix, calculations of equilibrium \"state 2\" are described.  
To find \(\rho_{2}\) and \(s_{2}\) values, the asymptotic expansion method with respect to the small parameter \(\varepsilon\) could be used:\[\begin{split}&\rho_{2}=\rho_{0}+\frac{\rho_{0}\epsilon^{2}\big{(}\big{<} ^{s}\big{>}^{2}\big{>}^{2}\big{>}}{2(\text{Cp})^{2}}-\frac{\rho_{0}\epsilon^{2} (\text{k}-1)\big{<}\big{<}(\text{p}^{\prime})^{2}\big{>}}{2\text{k}^{2}\big{(} \text{p}_{0}\big{)}^{2}}-\\ &-\frac{2\rho_{0}\epsilon^{2}\big{<}\text{s}^{\prime}\text{p}^{ \prime}\big{>}}{2(\text{Cp})^{2}\text{p}_{0}}-\frac{\rho_{0}\epsilon^{2} \big{<}\big{<}\big{>}^{3}\big{>}}{6(\text{Cp})^{3}}+\\ &+\frac{\rho_{0}\epsilon^{3}(\text{k}-1)(2\text{k}-1)\big{<}( \text{p}^{\prime})^{3}\big{>}}{6\text{k}^{3}\big{(}\text{p}_{0}\big{)}^{3}}+\\ &+\frac{\rho_{0}\epsilon^{3}\big{<}(\big{<}^{s}\big{>}^{2}\text{p} ^{\prime}\big{>}}{2\text{k}(\text{Cp})^{2}\text{p}_{0}}+\frac{\rho_{0}\epsilon ^{3}\big{<}\big{<}^{s}(\text{p}^{\prime})^{2}\big{>}}{2\text{k}^{2}\text{Cp} _{0}\big{)}^{2}}+\\ &+O\big{(}\epsilon^{4}\big{)},\end{split} \tag{16}\]  
\[\begin{split}&\rho_{2}s_{2}=\rho_{0}\epsilon^{2}\Bigg{(}\frac{ \big{<}s^{\prime}\text{p}^{\prime}\big{>}}{\text{kp}_{0}}-\frac{\big{<}(s^{ \prime})^{2}\big{>}}{\text{Cp}_{0}}\Bigg{)}+\\ &+\frac{\rho_{0}\epsilon^{3}}{2}\Bigg{(}\frac{\big{<}(s^{\prime} )^{3}\big{>}}{\left(\text{Cp}\right)^{2}}-\frac{(\text{k}-1)\big{<}(\text{p}^ {\prime})^{2}\text{s}^{\prime}\big{>}}{\text{k}^{2}\big{(}\text{p}_{0}\big{)} ^{2}}-\frac{2\big{<}(s^{\prime})^{2}\text{p}^{\prime}\big{>}}{\text{k}\text{Cp }_{0}}\Bigg{)}+\\ & O\big{(}\epsilon^{4}\big{)}.\end{split} \tag{17}\]  
First-order terms are eliminated according to the definitions of mean values in Equation (4). We find \(\text{p}_{2}\) using relation:  
\[\text{p}_{2}=\text{p}_{0}\bigg{(}\frac{\rho_{2}}{\rho_{0}}\bigg{)}^{\text{k}} \exp\bigg{(}\frac{(\rho_{2}s_{2})\text{k}}{\rho_{2}\text{Cp}}\bigg{)}. \tag{18}\]  
After substitution \(\text{p}_{2}\) and \((\text{p}_{2}s_{2})\) into this relation, one can get:  
\[\begin{split}&\text{p}_{2}=\text{p}_{0}-\frac{\text{p}_{0} \epsilon^{2}\text{k}\big{<}\big{<}\big{<}\big{>}^{2}\big{>}}{2(\text{Cp})^{2}} -\frac{\text{p}_{0}\epsilon^{2}(\text{k}-1)\big{<}(\text{p}^{\prime})^{2}}{2 \text{k}(\text{p}_{0})^{2}}+\\ &+\frac{\text{p}_{0}\epsilon^{3}(\text{k}-1)(2\text{k}-1)\big{<} (\text{p}^{\prime})^{3}\big{>}}{6\text{k}^{2}\big{(}\text{p}_{0}\big{)}^{3}}+ \frac{\text{p}_{0}\epsilon^{3}\text{k}\big{<}(s^{\prime})^{3}\big{>}}{3(\text{ Cp})^{3}}-\\ &-\frac{\text{p}_{0}\epsilon^{2}\big{<}(s^{\prime})^{2}\text{p}^{ \prime}\big{>}}{2(\text{Cp})^{2}\text{p}_{0}}+O\big{(}\epsilon^{4}\big{)}. \end{split} \tag{19}\]  
With pressure and entropy, we can find the energy of the system in \"state 2\":  
\[\frac{\text{E}_{2}}{\text{V}}=\frac{\text{p}_{2}}{(\text{k}-1)}. \tag{20}\]  
and the energy in \"state 1\" can also be represented as an asymptotic expansion:  
\[\begin{split}&\frac{\text{E}_{1}}{\text{V}}=\frac{\text{p}_{0}}{( \text{k}-1)}+\frac{\rho_{0}\epsilon^{2}\bigg{<}\bigg{<}\bigg{<}\bigg{<}\bigg{<} \bigg{>}^{2}\bigg{>}}{2}+\\ &+\frac{\rho_{0}\epsilon^{3}}{2}\Bigg{(}\frac{\bigg{<}\bigg{<} \bigg{>}^{2}\text{p}^{\prime}\bigg{>}}{\text{kp}_{0}}-\frac{\bigg{<}\bigg{<} \bigg{>}^{2}\text{s}^{\prime}\bigg{>}}{\text{Cp}_{0}}\Bigg{>}+O\big{(} \epsilon^{4}\big{)}.\end{split} \tag{21}\]  
According to these relations, the exergy of an ideal gas volume with fluctuations in entropy, density, and velocity may be calculated by Equation (9). The final result is presented in Equation (12).