Two-temperature reaction and relaxation rates

Abstract

Within the method of solving the kinetic equations for gas mixtures with internal degrees of freedom developed by the authors and based on the approximate summational invariants (ASI) concept, gas-dynamic equations for a multi-temperature model for the spatially inhomogeneous case are derived. For the two-temperature case, the expressions for the non-equilibrium reaction and relaxation rates are obtained. Special attention is drawn to corresponding thermodynamic equations. Different possibilities of introducing the gas-dynamic variables related to the internal degrees of freedom are considered. One is based on the choice of quantum numbers as the ASI, while the other is based on the choice of internal (vibrational) energy as the ASI. Limits to a one-temperature situation are considered in all the cases. For the cutoff harmonic oscillator model, explicit expressions for the reaction and relaxation rates are derived.

Appendices

Appendix 1: Thermodynamic relationships derivation

New extensive (gas-dynamic) variables are introduced according to the relations

$$\begin{aligned} \tilde{\varGamma }_\mathrm{E}=\varGamma _\mathrm{E}-\displaystyle \frac{\mathbf {p}^{2}}{2\rho },\quad \tilde{\varGamma }_{\mathbf {p}} =\varGamma _{\mathbf {p}}\equiv \mathbf {p} \quad \tilde{n}_\alpha =n_\alpha , \quad \tilde{\varGamma }_{\mathrm{a}i}=\varGamma _{\mathrm{a}i}. \end{aligned}$$

(138)

Corresponding new intensive variables are introduced by the relations

$$\begin{aligned}&\tilde{\mathbf {\gamma }}_{\tilde{\varGamma }_{\mathrm{a}i}} =\left( \displaystyle \frac{\partial \widetilde{S}}{\partial \tilde{\varGamma }_{\mathrm{a}i}}\right) ,\nonumber \\&\tilde{S}=S\left( n_{\alpha }, \mathbf {p},\varGamma _\mathrm{E},\varGamma _{\mathrm{a}i}\right) =S\left( n_{\alpha },\mathbf {p},\tilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho },\varGamma _{\mathrm{a}i}\right) . \end{aligned}$$

(139)

Providing corresponding calculation we obtain

$$\begin{aligned} \tilde{\mathbf {\gamma }}_{\mathbf {p}}\equiv & {} \left( \displaystyle \frac{\partial \widetilde{S}}{\partial \mathbf {p}}\right) _{n_\alpha ,\tilde{\varGamma }_\mathrm{E},\varGamma _{\mathrm{a}i}}\nonumber \\= & {} \left( \displaystyle \frac{\partial {S}\left( n_{\alpha },\mathbf {p},\widetilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho },\varGamma _{\mathrm{a}i}\right) }{\partial \mathbf {p}}\right) _{n_\alpha ,\tilde{\varGamma }_\mathrm{E},\varGamma _{\mathrm{a}i}} \nonumber \\= & {} \left( \displaystyle \frac{\partial S\left( n_{\alpha },\mathbf {p},\widetilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho },\varGamma _{\mathrm{a}i}\right) }{\partial \mathbf {p}}\right) _{n_\alpha ,\varGamma _\mathrm{E},\varGamma _{\mathrm{a}i}} \nonumber \\&+\left( \displaystyle \frac{\partial {S}\left( n_{\alpha },\mathbf {p},\widetilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho },\varGamma _{\mathrm{a}i}\right) }{\partial \varGamma _\mathrm{E}}\right) _{n_\alpha , \mathbf {p},\varGamma _{\mathrm{a}i}}\displaystyle \frac{\partial }{\partial \mathbf {p}}\left( \widetilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho }\right) \nonumber \\= & {} \mathbf {\gamma }_{\mathbf {p}}+\gamma _\mathrm{E}\displaystyle \frac{\partial }{\partial \mathbf {p}}\left( \widetilde{\varGamma }_\mathrm{E}+\displaystyle \frac{\mathbf {p}^2}{2\rho }\right) =\mathbf {\gamma }_{\mathbf {p}}+\gamma _\mathrm{E}\displaystyle \frac{\mathbf {p}}{\rho }=0. \end{aligned}$$

(140)

$$\begin{aligned}&\left( \displaystyle \frac{\partial \widetilde{S}}{\partial {n}_{\alpha }}\right) _{n_\beta , \mathbf {p},\tilde{\varGamma }_\mathrm{E},\varGamma _{\mathrm{a}i}}\nonumber \\&\quad =\left( \displaystyle \frac{\partial {S}\left( n_{\alpha }, \mathbf {p},\widetilde{\varGamma }_\mathrm{E}+\displaystyle \frac{\mathbf {p}^{2}}{2\rho },\varGamma _{\mathrm{a}i}\right) }{\partial n_{\alpha }}\right) _{n_\beta ,\mathbf {p},\tilde{\varGamma }_\mathrm{E},\varGamma _{\mathrm{a}i}} \nonumber \\&\qquad +\left( \displaystyle \frac{\partial {S}}{\partial \varGamma _\mathrm{E}}\right) _{n_\beta ,\mathbf {p},\varGamma _{\mathrm{a}i}} \displaystyle \frac{\partial }{\partial n_{\alpha }}\displaystyle \frac{\mathbf {p}^2}{2\rho } \nonumber \\&\quad =\gamma _\alpha -\gamma _\mathrm{E}\displaystyle \frac{\mathbf {p}^2}{2\rho ^2}m_{\alpha } =\gamma _\alpha -\gamma _\mathrm{E}\displaystyle \frac{1}{2\rho ^2} \left( -\displaystyle \frac{\rho }{\gamma _\mathrm{E}} \mathbf {\gamma }_{\mathbf {p}}\right) ^2 \nonumber \\&m_{\alpha } =\gamma _\alpha -\displaystyle \frac{m_\alpha }{2}\displaystyle \displaystyle \frac{\mathbf {\gamma }_{\mathbf {p}}^2}{\gamma _\mathrm{E}} \equiv \widetilde{\gamma }_\alpha , \end{aligned}$$

(141)

$$\begin{aligned}&\left( \frac{\partial \widetilde{S}}{\partial \widetilde{\varGamma }_\mathrm{E}}\right) _{n_\beta ,\mathbf {p},\varGamma _{\mathrm{a}i}}\nonumber \\&\quad =\left( \frac{\partial {S}\left( n_{\alpha },\mathbf {p},\widetilde{\varGamma }_\mathrm{E} +\displaystyle \frac{\mathbf {p}^2}{2\rho },\varGamma _{\mathrm{a}i}\right) }{\partial \varGamma _\mathrm{E}} \right) _{n_\beta ,\mathbf {p},\varGamma _i}\frac{\partial \varGamma _\mathrm{E}}{\partial \widetilde{\varGamma }_\mathrm{E}}\nonumber \\&\quad =\gamma _\mathrm{E}\equiv \widetilde{\gamma }_\mathrm{E}. \end{aligned}$$

(142)

Using relations (12) and $n=\sum _\alpha n_\alpha$ and expressions for $\varGamma$ we obtain

$$\begin{aligned} S= & {} n+\displaystyle \sum \limits _{i=1}^{M}\gamma _{i}\varGamma _i =\gamma _{\mathbf {p}}\cdot \mathbf {p}+\displaystyle \sum \limits _{\alpha =1}^N \left( 1+\gamma _\alpha \right) n_\alpha \nonumber \\&+\,\gamma _\mathrm{E}\left( {\tilde{\varGamma }}_\mathrm{E} +\frac{\mathbf {p}^{2}}{2\rho }\right) +\displaystyle \sum \limits _{i=N+5}^{M}\gamma _{\mathrm{a}i}\varGamma _{\mathrm{a}i} \nonumber \\= & {} -\displaystyle \frac{\gamma _\mathrm{E}}{\rho }\mathbf {p}\cdot \mathbf {p}+\displaystyle \sum \limits _{\alpha =1}^{N} \left( 1+\gamma _{\alpha }\right) n_\alpha +\gamma _\mathrm{E}\left( {\tilde{\varGamma }}_\mathrm{E} +\frac{\mathbf {p}^{2}}{2\rho }\right) \nonumber \\&+\displaystyle \sum \limits _{i=N+5}^M\gamma _{\mathrm{a}i}\varGamma _{\mathrm{a}i} \nonumber \\= & {} -\displaystyle \frac{\gamma _\mathrm{E}}{2\rho }\mathbf {p}^2+\displaystyle \sum \limits _{\alpha =1}^{N} \left( 1+\tilde{\gamma }_{\alpha }+\displaystyle \frac{m_{\alpha }}{2} \displaystyle \frac{\gamma _\mathrm{E}\mathbf {p}^{2}}{\rho ^2}\right) n_{\alpha }+\gamma _\mathrm{E}{\tilde{\varGamma }}_\mathrm{E}\nonumber \\&+\displaystyle \sum \limits _{i=N+5}^{M}\gamma _{\mathrm{a}i}\varGamma _{\mathrm{a}i} \nonumber \\= & {} \displaystyle \sum \limits _{\alpha =1}^N\left( 1+\tilde{\gamma }_\alpha \right) \tilde{n}_\alpha +\tilde{\gamma }_\mathrm{E}{\tilde{\varGamma }}_\mathrm{E}+\displaystyle \sum \limits _{i=N+5}^M\tilde{\gamma }_{\mathrm{a}i}{\tilde{\varGamma }}_{\mathrm{a}i}\nonumber \\= & {} n+\displaystyle \sum \limits _{i=4}^M{\tilde{\gamma }}_i{\tilde{\varGamma }}_i=\tilde{S}. \end{aligned}$$

(143)

$$\begin{aligned}&-\left( \displaystyle \frac{\partial \widetilde{n}}{\partial \widetilde{\gamma }_{\alpha }}\right) _{\widetilde{\gamma }_\beta , \widetilde{\mathbf {\gamma }}_{\mathbf {p}},\widetilde{\gamma }_\mathrm{E},\widetilde{\gamma }_{\mathrm{a}i}}\nonumber \\&\quad =-\left( \displaystyle \frac{\partial {n}\left( \gamma _{\alpha },\mathbf {\gamma }_{\mathbf {p}},\gamma _\mathrm{E},\gamma _{\mathrm{a}i}\right) }{\partial \gamma _{\alpha }}\right) _{\gamma _\beta ,\mathbf {\gamma }_{\mathbf {p}},\gamma _\mathrm{E},\gamma _{\mathrm{a}i}}\displaystyle \frac{\partial \gamma _{\alpha }}{\partial {\tilde{\gamma }}_\alpha }=n_{\alpha }, \end{aligned}$$

(144)

$$\begin{aligned}&-\left( \displaystyle \frac{\partial \widetilde{n}}{\partial {\widetilde{\gamma }}_\mathrm{E}}\right) _{\widetilde{\gamma }_{\alpha }, \widetilde{\mathbf {\gamma }}_{\mathbf {p}},\widetilde{\gamma }_{\mathrm{a}i}} =-\left( \displaystyle \frac{\partial \widetilde{n}}{\partial \gamma _\mathrm{E}}\right) _{\widetilde{\gamma }_{\alpha }, \widetilde{\mathbf {\gamma }}_{\mathbf {p}},\widetilde{\gamma }_{\mathrm{a}i}} \nonumber \\&\quad =-\left( \displaystyle \frac{\partial {n}}{\partial \gamma _\mathrm{E}}\right) _{\gamma _{\alpha }, \mathbf {\gamma }_{\mathbf {p}},\gamma _{\mathrm{a}i}}+\sum _{\alpha }\left( \displaystyle \frac{\partial {n}}{\partial \gamma _\alpha }\right) _{\gamma _\beta ,\mathbf {\gamma }_{\mathbf {p}}, \gamma _\mathrm{E},\gamma _{\mathrm{a}i}}\displaystyle \frac{m_{\alpha }}{2}\displaystyle \frac{\mathbf {\gamma }_{\mathbf {p}}^2}{\gamma _\mathrm{E}^2} \nonumber \\&\quad =\varGamma _\mathrm{E}-\sum _{\alpha }\displaystyle \frac{m_{\alpha }n_\alpha }{2}\displaystyle \frac{1}{\gamma _\mathrm{E}^2} \left( \displaystyle \frac{\gamma _\mathrm{E}}{\rho }\mathbf {p}\right) ^2 =\varGamma _\mathrm{E}-\displaystyle \frac{\mathbf {p}^2}{2\rho }={\widetilde{\varGamma }}_\mathrm{E}, \end{aligned}$$

(145)

$$\begin{aligned}&-\left( \frac{\partial \widetilde{n}}{\partial \widetilde{\gamma }_{\mathrm{a}i}}\right) _{\widetilde{\gamma }_{\alpha }, \widetilde{\mathbf {\gamma }}_{\mathbf {p}},\widetilde{\gamma }_\mathrm{E},\widetilde{\gamma }_{\mathrm{a}j}} =-\left( \frac{\partial \widetilde{n}}{\partial \gamma _{\mathrm{a}i}}\right) _{\widetilde{\gamma }_{\alpha }, \widetilde{\mathbf {\gamma }}_{\mathbf {p}},\widetilde{\gamma }_\mathrm{E},\widetilde{\gamma }_{\mathrm{a}j}}\nonumber \\&\quad =-\left( \frac{\partial {n}}{\partial \gamma _{\mathrm{a}i}}\right) _{\gamma _{\alpha },\mathbf {\gamma }_{\mathbf {p}}, \gamma _\mathrm{E},\gamma _{\mathrm{a}j}}=\varGamma _{\mathrm{a}i}, \end{aligned}$$

(146)

$$\begin{aligned}&-\left( \displaystyle \frac{\partial \widetilde{n}}{\partial \widetilde{\mathbf {\gamma }}_{\mathbf {p}}}\right) _{\widetilde{\gamma }_\alpha , \widetilde{\gamma }_\mathrm{E},\widetilde{\gamma }_{\mathrm{a}i}} =-\left( \displaystyle \frac{\partial \widetilde{n}}{\partial \mathbf {\gamma }_{\mathbf {p}}}\right) _{\widetilde{\gamma }_\alpha , \widetilde{\gamma }_\mathrm{E},\widetilde{\gamma }_{\mathrm{a}i}} \nonumber \\&\quad =-\left( \displaystyle \frac{\partial n}{\partial \mathbf {\gamma }_{\mathbf {p}}} \right) _{\gamma _{\alpha },\gamma _\mathrm{E},\gamma _{\mathrm{a}i}}-\sum _{\alpha }\left( \displaystyle \frac{\partial {n}}{\partial \gamma _{\alpha }}\right) _{\gamma _\beta ,\mathbf {\gamma }_{\mathbf {p}}, \gamma _\mathrm{E},\gamma _{\mathrm{a}i}}\nonumber \\&\qquad \times \, m_\alpha \displaystyle \frac{\mathbf {\gamma }_{\mathbf {p}}}{\gamma _\mathrm{E}} =\mathbf {p}+\rho \displaystyle \frac{\mathbf {\gamma }_{\mathbf {p}}}{\gamma _\mathrm{E}}=0. \end{aligned}$$

(147)

Appendix 2: Expressions for some scalar products $\langle \psi _i,\psi _j{F}^\mathrm{{qe}}\rangle$

Kinetic equation in (6), as well as equations for the gas-dynamic variables, contains terms $\langle \mathbf {v}_\alpha \psi ,F^\mathrm{{qe}}_\alpha \rangle$ with the approximate summational invariants $\psi$. Calculation of these scalar products gives

$$\begin{aligned}&\langle {\mathbf {v}_\alpha }\psi _{\mathbf {p}},F^\mathrm{{qe}}_\alpha \rangle =\langle \left( \mathbf {c_\alpha +u}\right) m_\alpha \left( \mathbf {c_\alpha +u}\right) ,F^\mathrm{{qe}}_\alpha \rangle =\mathbf {I}p+\rho \mathbf {uu},\nonumber \\&p=n/\gamma _\mathrm{E}, \end{aligned}$$

(148)

$$\begin{aligned}&\left\langle {\mathbf {v}}_\alpha \psi _\mathrm{E},F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\&\quad =\left\langle \left( \mathbf {c_\alpha +u}\right) \left( \displaystyle \frac{m_\alpha \left( \mathbf {c_\alpha +u}\right) ^2}{2} +e_\alpha \right) ,F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\&\quad =\left\langle {\mathbf {c}}_\alpha ({\mathbf {c}}_\alpha \cdot \mathbf {u}) +\mathbf {u}\left( \displaystyle \frac{m_\alpha {\mathbf {c}}_\alpha ^2}{2}+e_\alpha +\displaystyle \frac{m_\alpha \mathbf {u}^2}{2}\right) ,F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\&\quad =\mathbf {u}\left( p+\varGamma _\mathrm{E}\right) =\mathbf {u}\left( p+{\tilde{\varGamma }}_\mathrm{E}+\displaystyle \frac{\rho \mathbf {u}^2}{2}\right) . \end{aligned}$$

(149)

where $\mathbf {c_\alpha =v_\alpha -u}$ and $p=n/\gamma _\mathrm{E}$ is the pressure.

$$\begin{aligned}&\langle \psi _{\mathbf {p}}\psi _\mathrm{E},F^\mathrm{{qe}}_\alpha \rangle \nonumber \\&\quad =\left\langle {m_\alpha \mathbf {c}}_\alpha ({\mathbf {c}}_\alpha \cdot \mathbf {u}) +m_\alpha \mathbf {u}\left( \displaystyle \frac{m_\alpha {\mathbf {c}}_\alpha ^2}{2}+e_\alpha +\displaystyle \frac{m_\alpha \mathbf {u}^2}{2}\right) ,F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\&\quad =\mathbf {u}\displaystyle \sum _\alpha \rho _\alpha \left( \gamma _\mathrm{E}^{-1}+\varGamma _{\mathrm{E}\alpha }+\displaystyle \frac{m_\alpha \mathbf {u}^2}{2}\right) \nonumber \\&\quad =\mathbf {u}\left( \rho /\gamma _\mathrm{E}+\displaystyle \sum _\alpha \rho _\alpha \left( \varGamma _{\mathrm{E}\alpha } +\displaystyle \frac{m_\alpha \mathbf {u}^2}{2}\right) \right) . \end{aligned}$$

(150)

For all other scalar $\psi _i$ we have

$$\begin{aligned} \langle {\mathbf {v}}_\alpha \psi _i,F^\mathrm{{qe}}_\alpha \rangle =\langle \left( \mathbf {c_\alpha +u}\right) \psi _i,F^\mathrm{{qe}}_\alpha \rangle =\mathbf {u}\varGamma _i. \end{aligned}$$

(151)

The following relationships can be derived

$$\begin{aligned} \langle \psi _i,\psi _jF_\alpha ^\mathrm{{qe}}\rangle= & {} \displaystyle \sum _{\beta =1}^Nn_\beta \left\langle \psi _i,\delta _{\alpha \beta }\psi _j{F_\alpha ^\mathrm{{qe}}}/{n_\alpha }\right\rangle \nonumber \\= & {} -\displaystyle \sum _{\beta =1}^Nn_\beta \displaystyle \frac{\partial \varGamma _{i\beta }}{\partial \gamma _j} =\displaystyle \sum _{\beta =1}^Nn_\beta \displaystyle \frac{c_{i\beta ,j}}{\gamma _j^2} =\displaystyle \frac{nc_{i,j}}{\gamma _j^2}, \end{aligned}$$

(152)

$$\begin{aligned} \langle {\tilde{\psi }}_i,{\tilde{\psi }}_jF_\alpha ^\mathrm{{qe}}\rangle= & {} \displaystyle \sum _{\beta =1}^Nn_\beta \left( {\tilde{\varGamma }}_{i\beta }{\tilde{\varGamma }}_{j\beta } -\displaystyle \frac{\partial {\tilde{\varGamma }}_{i\beta }}{\partial {\tilde{\gamma }}_j}\right) \nonumber \\= & {} \displaystyle \sum _{\beta =1}^Nn_\beta \left( \displaystyle \frac{\tilde{c}_{i\beta ,j}}{{\tilde{\gamma }}_j^2} +{\tilde{\varGamma }}_{i\beta }{\tilde{\varGamma }}_{j\beta }\right) \nonumber \\= & {} \displaystyle \frac{n\tilde{c}_{i,j}}{{\tilde{\gamma }}_j^2} +\displaystyle \sum _{\beta =1}^Nn_\beta {\tilde{\varGamma }}_{i\beta }{\tilde{\varGamma }}_{j\beta }. \end{aligned}$$

(153)

Thus, for $j=i=\mathrm{E}$

$$\begin{aligned}&\langle {\tilde{\psi }}_\mathrm{E},{\tilde{\psi }}_\mathrm{E}{F}^\mathrm{{qe}}\rangle =\displaystyle \frac{nc_v}{k_\mathrm{B}\gamma _\mathrm{E}^2} +\displaystyle \sum _{\beta =1}^Nn_\beta {\tilde{\varGamma }}_{\mathrm{E}\beta }^2,\nonumber \\&\quad c_v=\displaystyle \sum _{\beta =1}^Nn_\beta {c_{v\beta }}/n, \end{aligned}$$

(154)

where $c_v$ is the traditional mixture specific heat and $c_{v\alpha }$ is the traditional species specific heat. Some of the expressions for the scalar products can be simplified.

Consider the scalar products that are necessary for calculating the expressions $\langle \psi _i,\psi _j{F}^\mathrm{{qe}}\rangle$, starting with $i=j=\mathrm{E}$:

$$\begin{aligned} \left\langle \psi _\mathrm{E},\psi _\mathrm{E}{F}_\alpha ^\mathrm{{qe}}\right\rangle= & {} \langle {m_\alpha {\mathbf{v}}_\alpha ^2/2+e_\alpha ^{\mathrm{(i)}}(k_\alpha )}, {m_\alpha {\mathbf{v}}_\alpha ^2/2+e_\alpha ^{\mathrm{(i)}}(k_\alpha )}{F}_\alpha ^\mathrm{{qe}}\rangle \\= & {} \langle (m_\alpha {\mathbf{v}}_\alpha ^2/2)^2,{F}_\alpha ^\mathrm{{qe}}\rangle +\langle {m}_\alpha {\mathbf{v}}_\alpha ^2,e_\alpha ^{\mathrm{(i)}}{F}_\alpha ^\mathrm{{qe}}\rangle \\&+\langle {e_\alpha ^{\mathrm{(i)}2}},{F}_\alpha ^\mathrm{{qe}}\rangle , \end{aligned}$$

where

$$\begin{aligned}&\langle (m_\alpha {\mathbf{v}}_\alpha ^2/2)^2,F_\alpha ^\mathrm{{qe}}\rangle \\&\quad =\langle (m_\alpha ({\mathbf{c}}_\alpha +{\mathbf{u}})^2/2)^2,{F}_\alpha ^\mathrm{{qe}}\rangle \\&\quad =\left\langle \left( \displaystyle \frac{m_\alpha {\mathbf{c}}_\alpha ^2}{2}\right) ^2 +\left( m_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{u}}\right) ^2\right. \\&\quad \left. +\left( \displaystyle \frac{m_\alpha {\mathbf{u}}^2}{2}\right) ^2 +m_\alpha {\mathbf{u}}^2\displaystyle \frac{m_\alpha {\mathbf{c}}_\alpha ^2}{2},F_\alpha ^\mathrm{{qe}}\right\rangle \\&\quad =\displaystyle \frac{15}{4}\displaystyle \frac{n}{\gamma _\mathrm{E}^2}+\displaystyle \frac{\rho {\mathbf{u}}^2}{\gamma _\mathrm{E}} +\displaystyle \frac{{\mathbf{u}}^4}{4}\displaystyle \sum _{\alpha =1}^Nm_\alpha ^2n_\alpha +\displaystyle \frac{3\rho {\mathbf{u}}^2}{2\gamma _\mathrm{E}},\\&\quad \langle {m}_\alpha {\mathbf{v}}_\alpha ^2,e_\alpha ^{\mathrm{(i)}}{F}_\alpha ^\mathrm{{qe}}\rangle =3\displaystyle \frac{\varGamma _{\mathrm{E}^{\mathrm{(i)}}}}{\gamma _\mathrm{E}}+{\mathbf{u}}^2\displaystyle \sum _{\alpha =1}^Nm_\alpha {n_\alpha }\varGamma _{\mathrm{E}^{\mathrm{(i)}}\alpha }. \end{aligned}$$

As a result

$$\begin{aligned} \left\langle \psi _\mathrm{E},\psi _\mathrm{E}{F}_\alpha ^\mathrm{{qe}}\right\rangle= & {} \displaystyle \frac{15}{4}\displaystyle \frac{n}{\gamma _\mathrm{E}^2}+\displaystyle \frac{\rho {\mathbf{u}}^2}{2\gamma _\mathrm{E}}\left( 5 +\displaystyle \frac{2\gamma _\mathrm{E}}{\rho }\displaystyle \sum _{\alpha =1}^Nm_\alpha {n_\alpha }\varGamma _{\mathrm{E}^{\mathrm{(i)}}\alpha }\right) \nonumber \\&+\displaystyle \frac{{\mathbf{u}}^4}{4}\displaystyle \sum _{\alpha =1}^Nm_\alpha ^2n_\alpha +3\displaystyle \frac{\varGamma _{\mathrm{E}^{\mathrm{(i)}}}}{\gamma _\mathrm{E}} +\langle {e_\alpha ^{\mathrm{(i)}2}},X_\alpha ^\mathrm{{qe}}\rangle ^\prime , \end{aligned}$$

(155)

where $X_\alpha ^\mathrm{{qe}}=\int {\mathrm{d}{\mathbf{v}}_\alpha }F_\alpha ^\mathrm{{qe}}$, $\varGamma _{\mathrm{E}^{\mathrm{(i)}}}=\langle {e_\alpha ^{\mathrm{(i)}}},F_\alpha ^\mathrm{{qe}}\rangle =\langle {e_\alpha ^{\mathrm{(i)}}},X_\alpha ^\mathrm{{qe}}\rangle ^\prime$ and $\left\langle ...,...\right\rangle ^\prime$ means that this scalar product does not contain integration over velocities but implies summation over quantum numbers and over species. For $i=j=\mathrm{a}$

$$\begin{aligned} \left\langle \psi _\mathrm{a}(k_\alpha ),\psi _\mathrm{a}(k_\alpha ){F}_\alpha ^\mathrm{{qe}}\right\rangle =\langle \psi ^2_\mathrm{a}(k_\alpha ),X_\alpha ^\mathrm{{qe}}\rangle ^\prime , \end{aligned}$$

(156)

and for crossed terms

$$\begin{aligned}&\left\langle \psi _\mathrm{a}(k_\alpha ),\psi _\mathrm{E}(k_\alpha ){F}_\alpha ^\mathrm{{qe}}\right\rangle \nonumber \\&\quad =\langle {m_\alpha {\mathbf{v}}_\alpha ^2/2+e_\alpha ^{\mathrm{(i)}} (k_\alpha )},\psi _aF_\alpha ^\mathrm{{qe}}\rangle \nonumber \\&\quad =\displaystyle \frac{3}{2}\displaystyle \frac{\varGamma _\mathrm{a}}{\gamma _\mathrm{E}} +\displaystyle \frac{{\mathbf{u}}^2}{2}\displaystyle \sum _{\alpha =1}^Nm_\alpha {n_\alpha }\varGamma _{\mathrm{a}\alpha } +\langle \psi _\mathrm{a}(k_\alpha ),e_\alpha ^{\mathrm{(i)}}X_\alpha ^\mathrm{{qe}}\rangle ^\prime \nonumber \\&\quad =\displaystyle \sum _{\alpha =1}^Nn_\alpha \varGamma _{\mathrm{E}^{\mathrm{(t)}}\alpha }\varGamma _{\mathrm{a}\alpha } +\langle \psi _\mathrm{a}(k_\alpha ),e_\alpha ^{\mathrm{(i)}}X_\alpha ^\mathrm{{qe}}\rangle ^\prime . \end{aligned}$$

(157)

For the modified set of variables

$$\begin{aligned} \langle {\tilde{\psi }}_\mathrm{E},{\tilde{\psi }}_\mathrm{E}{F}_\alpha ^\mathrm{{qe}}\rangle= & {} \langle (m_\alpha {\mathbf{c}}_\alpha ^2/2+e_\alpha ^{\mathrm{(i)}}(k_\alpha ))^2,F_\alpha ^\mathrm{{qe}}\rangle \nonumber \\= & {} \displaystyle \frac{15}{4}\displaystyle \frac{n}{\gamma _\mathrm{E}^2}+3\displaystyle \frac{\varGamma _{\mathrm{E}^{\mathrm{(i)}}}}{\gamma _\mathrm{E}} +\langle {e_\alpha ^{\mathrm{(i)}2}},X_\alpha ^\mathrm{{qe}}\rangle ^\prime , \end{aligned}$$

(158)

and

$$\begin{aligned} \langle {\tilde{\psi }}_\mathrm{a}(k_\alpha ),{\tilde{\psi }}_\mathrm{E}(k_\alpha ){F}_\alpha ^\mathrm{{qe}}\rangle= & {} \langle {m_\alpha {\mathbf{c}}_\alpha ^2}/2+e_\alpha ^{\mathrm{(i)}}(k_\alpha ),\psi _\mathrm{a}F_\alpha ^\mathrm{{qe}}\rangle \nonumber \\= & {} \displaystyle \frac{3}{2}\displaystyle \frac{\varGamma _\mathrm{a}}{\gamma _\mathrm{E}} +\langle \psi _\mathrm{a}(k_\alpha ),e_\alpha ^{\mathrm{(i)}}X_\alpha ^\mathrm{{qe}}\rangle ^\prime , \end{aligned}$$

(159)

while the expression for $\langle {\tilde{\psi }}_\mathrm{a}(k_\alpha ), {\tilde{\psi }}_\mathrm{a}(k_\alpha ){F}_\alpha ^\mathrm{{qe}}\rangle$ remains the same as in (156).

Definition (24) for $\tilde{c}_{i\alpha ,j}$ with $i=j=\mathrm{E}$, after some algebra similar to (158), leads to the following expressions

$$\begin{aligned} \tilde{c}_{\mathrm{E}\alpha ,\mathrm{E}}= & {} \gamma _\mathrm{E}^2(\langle \delta _{\alpha \beta }\tilde{e}_\alpha ^2, F^\mathrm{{qe}}_\beta /n_\beta \rangle -{\tilde{\varGamma }}_{\mathrm{E}\alpha }^2)\nonumber \\= & {} \gamma _\mathrm{E}^2\Biggl (\displaystyle \frac{15}{4\gamma _\mathrm{E}^2}+3\displaystyle \frac{\varGamma _{\mathrm{E}^{\mathrm{(i)}}\alpha }}{\gamma _\mathrm{E}} +\langle \delta _{\alpha \beta }e_\alpha ^{\mathrm{(i)}2},\displaystyle {F^\mathrm{{qe}}_\beta }/{n_\beta }\rangle \nonumber \\&-\left( \displaystyle \frac{3}{2}\displaystyle \frac{1}{\gamma _\mathrm{E}}+\varGamma _{\mathrm{E}^{\mathrm{(i)}}\alpha }\right) ^2\Biggr )\nonumber \\= & {} \frac{3}{2}+\gamma _\mathrm{E}^2(\langle \delta _{\alpha \beta }e_\alpha ^{\mathrm{(i)}2}, \displaystyle {F^\mathrm{{qe}}_\beta }/{n_\beta }\rangle -\varGamma _{\mathrm{E}^{\mathrm{(i)}}\alpha }^2)\nonumber \\= & {} \tilde{c}_{\mathrm{E}\alpha ,\mathrm{E}}^{\mathrm{(t)}}+\tilde{c}_{\mathrm{E}\alpha ,\mathrm{E}}^{\mathrm{(i)}}. \end{aligned}$$

(160)

Appendix 3: Inverting of the matrix $\partial \{\varGamma \}/\partial \{\mathrm{NV}\}$

While considering the set of new gas-dynamic variables $\{{n_\alpha },{\mathbf{p}},\gamma _\mathrm{E},\gamma _{\mathrm{a}i}\}\equiv \{\mathrm{NV}\}$ (see Sect. 3.2) calculation of the derivatives $B=\partial \{\mathrm{NV}\}/\partial \{\varGamma \}$ can be performed by inverting the matrix $A=\partial \{\varGamma \}/\partial \{\mathrm{NV}\}$: $B=A^{-1}$. Taking the A-matrix elements from  (32) the matrix $A\cdot {B}$ reads

$$\begin{aligned}&(ab)_{11\alpha \beta }=b_{11\alpha \beta }+\displaystyle \frac{m_\beta {\mathbf{p}} \cdot {\mathbf{b}}_{12\alpha }}{\rho } +\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) b_{13\alpha }, \nonumber \\&(ab)_{21x\beta }=b_{21x\beta }+\displaystyle \frac{m_\beta \left( {\mathbf{p}}\cdot {\mathbf{b}}_{22}\right) _x}{\rho } +\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) b_{23x}, \nonumber \\&(ab)_{31\beta }=b_{31\beta }+\displaystyle \frac{m_\beta {\mathbf{p}}\cdot {\mathbf{b}}_{32}}{\rho } +\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) b_{33}, \nonumber \\&(ab)_{41i\beta }=b_{41i\beta }+\displaystyle \frac{m_\beta {\mathbf{p}}\cdot {\mathbf{b}}_{42i}}{\rho } +\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) b_{43i}, \nonumber \\&(ab)_{12\beta {x}}=b_{12\beta {x}}+\displaystyle \frac{p_xb_{13\beta }}{\rho }, \quad (ab)_{22xy}=b_{22xy}+\displaystyle \frac{p_xb_{23y}}{\rho }, \nonumber \\&(ab)_{32x}=b_{32x}+\displaystyle \frac{{p_x}b_{33}}{\rho }, \quad (ab)_{42ix}=b_{42ix}+\displaystyle \frac{{p_x}b_{43i}}{\rho }, \nonumber \\&(ab)_{13\beta }=a_{33}b_{13\beta }+\displaystyle \sum _ia_{43i}b_{14i\beta },\nonumber \\&(ab)_{23x}=a_{33}b_{23x}+\displaystyle \sum _ia_{43i}b_{24xi}, \nonumber \\&(ab)_{33}=a_{33}b_{33}+\displaystyle \sum _ia_{43i}b_{34i},\nonumber \\&(ab)_{43i}=a_{33}b_{43i}+\displaystyle \sum _ja_{43j}b_{44ij}, \nonumber \\&(ab)_{14i\beta }=a_{34i}b_{13\beta }+\displaystyle \sum _ja_{44ji}b_{14\beta {j}},\nonumber \\&(ab)_{24xi}=a_{34i}b_{23x}+\displaystyle \sum _ja_{44ji}b_{24\beta {j}},\nonumber \\&(ab)_{34i}=a_{34i}b_{33}+\displaystyle \sum _ja_{44ji}b_{34\beta {j}},\nonumber \\&(ab)_{44ij}=a_{34j}b_{43i}+\displaystyle \sum _ja_{44ji}b_{44\beta {ij}}, \end{aligned}$$

(161)

where for example:

$$\begin{aligned} \displaystyle \frac{\partial \varGamma _{\mathbf{p}}}{\partial {n}_\beta }= & {} \left\langle {m}_\alpha {\mathbf{v}}_\alpha ,\displaystyle \frac{\partial {F}^\mathrm{{qe}}_\alpha }{\partial {n}_\beta }\right\rangle \nonumber \\= & {} \left\langle {m}_\alpha \left( {\mathbf{c}}_\alpha +{\mathbf{p}}/\rho \right) ,\left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } +\gamma _Em_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}\displaystyle \frac{\partial 1/\rho }{\partial {n}_\beta }\right) F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\= & {} m_\beta \displaystyle \frac{\mathbf{p}}{\rho } +\left\langle {m}_\alpha {\mathbf{c}}_\alpha , -m_\beta \gamma _\mathrm{E}\displaystyle \frac{m_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho ^2}F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\= & {} m_\beta \displaystyle \frac{\mathbf{p}}{\rho }-m_\beta \displaystyle \frac{\mathbf{p}}{\rho }=0, \end{aligned}$$

(162)

$$\begin{aligned} \displaystyle \frac{\partial \varGamma _\mathrm{E}}{\partial {n}_\beta }= & {} \left\langle {e}_\alpha ,\displaystyle \frac{\partial {F}^\mathrm{{qe}}_\alpha }{\partial {n}_\beta }\right\rangle \nonumber \\= & {} \left\langle {e}_\alpha ,\left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } +\gamma _Em_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}\displaystyle \frac{\partial 1/\rho }{\partial {n}_\beta }\right) F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\= & {} \left\langle \left( \tilde{e}_\alpha +\displaystyle \frac{m_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho } +\displaystyle \frac{m_\alpha }{2}\left( \displaystyle \frac{\mathbf{p}}{\rho }\right) ^2\right) ,\right. \nonumber \\&\left. \left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } -m_\beta \gamma _\mathrm{E}\displaystyle \frac{m_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho ^2}\right) F^\mathrm{{qe}}_\alpha \right\rangle \nonumber \\= & {} \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta \gamma _\mathrm{E}}{\rho ^3} \langle \left( m_\alpha {\mathbf{c}}_\alpha \cdot {\mathbf{p}}\right) ^2,F^\mathrm{{qe}}_\alpha \rangle \nonumber \\= & {} \varGamma _{\mathrm{E}\beta }-m_\beta ({\mathbf{p}}/\rho )^2. \end{aligned}$$

(163)

By equating matrix $A\cdot {B}$ to the unit, the following set of equations is obtained:

$$\begin{aligned}&\begin{matrix} b_{11\alpha \beta }+(\varGamma _{\mathrm{E}\beta }-{m_\beta {\mathbf{p}}^2}/{\rho ^2})b_{13\alpha }=\delta _{\alpha \beta }, \end{matrix} \end{aligned}$$

(164)

$$\begin{aligned}&\begin{matrix} b_{21x\beta }+(\varGamma _{\mathrm{E}\beta }-{m_\beta {\mathbf{p}}^2}/{\rho ^2})b_{23x}=0, \end{matrix}\end{aligned}$$

(165)

$$\begin{aligned}&\begin{matrix} b_{31\beta }+(\varGamma _{\mathrm{E}\beta }-{m_\beta {\mathbf{p}}^2}/{\rho ^2})b_{33}=0, \end{matrix}\end{aligned}$$

(166)

$$\begin{aligned}&\begin{matrix} b_{41i\beta }+(\varGamma _{\mathrm{E}\beta }-{m_\beta {\mathbf{p}}^2}/{\rho ^2})b_{43i}=0, \end{matrix}\end{aligned}$$

(167)

$$\begin{aligned}&\begin{matrix} b_{12\beta {x}}+{p_xb_{13\beta }}\rho =0, \end{matrix}\end{aligned}$$

(168)

$$\begin{aligned}&\begin{matrix} b_{22xy}+{p_xb_{23y}}\rho =\delta _{xy}, \end{matrix}\end{aligned}$$

(169)

$$\begin{aligned}&\begin{matrix} b_{32x}+{{p_x}b_{33}}/{\rho }=0, \end{matrix}\end{aligned}$$

(170)

$$\begin{aligned}&\begin{matrix} b_{42ix}+{{p_x}b_{43i}}/{\rho }=0, \end{matrix}\end{aligned}$$

(171)

$$\begin{aligned}&\begin{matrix} -{nc_{\mathrm{E,E}}}b_{13\beta }/{\gamma _\mathrm{E}^2}-\sum _i{nc_{\mathrm{a}i,\mathrm{E}}}b_{14i\beta }/{\gamma _\mathrm{E}^2}=0, \end{matrix}\end{aligned}$$

(172)

$$\begin{aligned}&\begin{matrix} -nc_{\mathrm{E,E}}b_{23x}/\gamma _\mathrm{E}^2-\displaystyle \sum _inc_{\mathrm{a}i,\mathrm{E}}b_{24xi}/\gamma _\mathrm{E}^2=0, \end{matrix}\end{aligned}$$

(173)

$$\begin{aligned}&\begin{matrix} -nc_{\mathrm{E,E}}b_{33}/\gamma _\mathrm{E}^2-\displaystyle \sum _kn\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{34k}/\gamma _\mathrm{E}^2=1, \end{matrix}\end{aligned}$$

(174)

$$\begin{aligned}&\begin{matrix} -nc_{\mathrm{E,E}}b_{43i}/\gamma _\mathrm{E}^2-\displaystyle \sum _kn\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{44ik}/\gamma _\mathrm{E}^2=0, \end{matrix}\end{aligned}$$

(175)

$$\begin{aligned}&\begin{matrix} -n\mathrm{c}_{\mathrm{E,a}i}b_{13\beta }/\gamma _{\mathrm{a}i}^2-\displaystyle \sum _jn\mathrm{c}_{\mathrm{a}j,\mathrm{a}i}b_{14\beta {j}}/\gamma _{\mathrm{a}i}^2=0, \end{matrix}\end{aligned}$$

(176)

$$\begin{aligned}&\begin{matrix} -n\mathrm{c}_{\mathrm{E,a}i}b_{23x}/\gamma _{\mathrm{a}i}^2-\displaystyle \sum _jn\mathrm{c}_{\mathrm{a}j,\mathrm{a}i}b_{24x{j}}/\gamma _{\mathrm{a}i}^2=0, \end{matrix}\end{aligned}$$

(177)

$$\begin{aligned}&\begin{matrix} -n\mathrm{c}_{\mathrm{E,a}j}b_{33}/\gamma _{\mathrm{a}j}^2-\displaystyle \sum _kn\mathrm{c}_{\mathrm{a}k,\mathrm{a}j}b_{34{k}}/\gamma _{\mathrm{a}j}^2=0, \end{matrix}\end{aligned}$$

(178)

$$\begin{aligned}&\begin{matrix} -n\mathrm{c}_{\mathrm{E,a}j}b_{43i}/\gamma _{\mathrm{a}j}^2-\displaystyle \sum _kn\mathrm{c}_{\mathrm{a}k,\mathrm{a}j}b_{44{ik}}/\gamma _{\mathrm{aj}}^2 =\delta _{ij}. \end{matrix} \end{aligned}$$

(179)

From the (168), (172) and (176) it follows

$$\begin{aligned} b_{12\beta {x}}=b_{13\beta }=b_{14\beta {i}}=0, \end{aligned}$$

(180)

and (173) and (177) give

$$\begin{aligned} b_{23x}=b_{24xi}=0. \end{aligned}$$

(181)

Thus,  (164), (169) and  (165) lead to

$$\begin{aligned} b_{11\alpha \beta }=\delta _{\alpha \beta }, \quad b_{22xy}=\delta _{xy}, \quad b_{21x\beta }=0 \end{aligned}$$

(182)

respectively. From (174) and (178) the following set of $M_\mathrm{a}$ equations for $b_{34i}$ can be obtained:

$$\begin{aligned} \displaystyle \sum _k\left( \mathrm{c}_{\mathrm{a}k,\mathrm{a}i}c_{\mathrm{E,E}}-\mathrm{c}_{\mathrm{a}k,\mathrm{E}}c_{\mathrm{E},\mathrm{a}i}\right) b_{34k}=\gamma _\mathrm{E}^2\mathrm{c}_{\mathrm{E,a}i}/ n. \end{aligned}$$

(183)

Similarly, the set of $M_\mathrm{a}\times {M_\mathrm{a}}$ equations for $b_{44ij}$ can be obtained from  (175) and (179):

$$\begin{aligned} \displaystyle \sum _k\left( \mathrm{c}_{{\mathrm{a}k,\mathrm{a}j}}c_{\mathrm{E,E}}-\mathrm{c}_{\mathrm{a}k,\mathrm{E}}\mathrm{c}_{\mathrm{E,a}{j}}\right) b_{44ik}=-\delta _{ij}\gamma _{\mathrm{a}{j}}^2c_{\mathrm{E,E}}/n, \end{aligned}$$

(184)

so that $\{b_{44ik}\}$ matrix is the inverse to the matrix $\{n(\mathrm{c}_{\mathrm{ak,E}}c_{E,\mathrm{a}j}/c_{\mathrm{E,E}}-\mathrm{c}_{\mathrm{a}k,\mathrm{a}{j}})/\gamma _{\mathrm{a}{j}}^2\}$. When compared for $i=j$,  (183) and (184) lead to $b_{44ik}/(\gamma _{\mathrm{a}i}^2c_{\mathrm{E,E}})=-b_{34k}/(\gamma _\mathrm{E}^2\mathrm{c}_{\mathrm{E,a}{i}})$. Using the relationships (26), the previous equality can be written as

$$\begin{aligned} b_{44ik}=-b_{34k}c_{\mathrm{E,E}}/c_{\mathrm{a}i,\mathrm{E}}, \end{aligned}$$

(185)

so that the matrix elements $b_{34k}$ determine several other via  (167), (171) and (175):

$$\begin{aligned}&b_{41i\beta }=-(\varGamma _{\mathrm{E}\beta } -m_\beta {p}^2/\rho ^2)\displaystyle \sum _k\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{34k}/c_{\mathrm{a}i,\mathrm{E}}, \nonumber \\&b_{42ix}=-p_xb_{43i}/\rho =-p_x\displaystyle \sum _k\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{34k}/(\rho {c}_{\mathrm{a}i,\mathrm{E}}),\nonumber \\&b_{43i}=\displaystyle \sum _k\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{34k}/c_{\mathrm{a}i,\mathrm{E}}. \end{aligned}$$

(186)

Equation (174) leads to $\sum \nolimits _k\mathrm{c}_{\mathrm{a}k,\mathrm{E}}b_{34k}=-(\gamma _\mathrm{E}^2/n+c_{\mathrm{E,E}}b_{33})$, thus, the relationships (186) can be written as

$$\begin{aligned}&b_{43i}=-\displaystyle \frac{{\gamma _\mathrm{E}^2}/n+c_{\mathrm{E,E}}b_{33}}{c_{\mathrm{a}i,\mathrm{E}}}, \nonumber \\&b_{42ix}=-\displaystyle \frac{p_x}{\rho }\displaystyle \frac{{\gamma _\mathrm{E}^2}/n+c_{\mathrm{E,E}}b_{33}}{c_{\mathrm{a}i,\mathrm{E}}}, \nonumber \\&b_{41i\beta }=\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {p}^2}{\rho ^2}\right) \displaystyle \frac{{\gamma _\mathrm{E}^2}/n+c_{\mathrm{E,E}}b_{33}}{c_{\mathrm{a}i,\mathrm{E}}}. \end{aligned}$$

(187)

Other elements can be also expressed via $b_{33}$:

$$\begin{aligned} b_{31\beta }=-\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {p}^2}{\rho ^2}\right) b_{33}, \quad b_{32x}=-p_xb_{33}/\rho . \end{aligned}$$

(188)

Appendix 4: Expressions for RHS$_\nabla$ and RHS$_{\nabla u}$

The part that contains operators $\nabla$ in the RHS of  (9) has the form

$$\begin{aligned} \mathrm{RHS}_\nabla= & {} \displaystyle \sum \limits _{i=1}^M\displaystyle \frac{\partial {F}^\mathrm{{qe}}_\alpha }{\partial \varGamma _i}\left[ {\mathbf {v}}_\alpha \cdot \nabla \varGamma _i -\nabla \cdot \left\langle {\mathbf {v}}_\alpha \psi _i,F^\mathrm{{qe}}_\alpha \right\rangle \right] \\= & {} \displaystyle \sum \limits _{i=1}^M\displaystyle \frac{\partial F^\mathrm{{qe}}_\alpha }{\partial \varGamma _i} \left[ {\mathbf {c}_\alpha }\cdot \nabla \varGamma _i-\varGamma _i\nabla \cdot {\mathbf {u}} -\nabla \cdot \left\langle {\mathbf {c}}_\alpha \psi _i,F^\mathrm{{qe}}_\alpha \right\rangle \right] . \end{aligned}$$

From  (148)–(151) it follows that the scalar products in this expression vanish, except ones corresponding to $\psi _{\mathbf {p}}$ and $\psi _\mathrm{E}$. After separating the terms with $\varGamma _{\mathbf {p}}$ and $\varGamma _\mathrm{E}$:

$$\begin{aligned} \mathrm{RHS}_{\nabla }= & {} \displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \mathbf {p}} \left( \mathbf {c}_\alpha \cdot \nabla \mathbf {p}-\mathbf {\nabla }p-\mathbf {p}\nabla \cdot \mathbf {u}\right) \\&+\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}\left( \mathbf {c} _{\alpha }\mathbf {\cdot \nabla }\varGamma _\mathrm{E}-\left( p+\varGamma _\mathrm{E}\right) \mathbf {\nabla } \cdot \mathbf {u}-\mathbf {u\cdot \nabla }p\right) \\&+\displaystyle \sum \limits _{\beta =1}^{N}\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial n_\beta }\left( \mathbf {c_{\alpha }\cdot \nabla }n_\beta -n_\beta \mathbf {\nabla }\cdot \mathbf {u}\right) \\&+\displaystyle \sum \limits _{i=N+5}^{M}\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _{i}} \left( \mathbf {c_{\alpha }\cdot \nabla }\varGamma _{i}-\varGamma _{i}\mathbf {\nabla }\cdot \mathbf {u}\right) \\= & {} \displaystyle \sum \limits _{i=1}^{M}\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _{i}}\left( \mathbf {c_{\alpha }\cdot \nabla }\varGamma _{i}-\varGamma _i \mathbf {\nabla }\cdot \mathbf {u}\right) -\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \mathbf {p}}\cdot \mathbf {\nabla }p\\&-\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}\left( p\mathbf {\nabla }\cdot \mathbf {u} +\mathbf {u\cdot \nabla }p\right) . \end{aligned}$$

where $\mathbf {p}=\rho \mathbf {u}=-\rho \gamma _{\mathbf {p}}/\gamma _\mathrm{E}$ and $p=n/\gamma _\mathrm{E}$. Terms proportional to $\nabla \varGamma$ can be brought to

$$\begin{aligned} \displaystyle \sum \limits _{i=1}^{M}\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\mathbf {c_\alpha \cdot \nabla }\varGamma _i= & {} \mathbf {c}_{\alpha }\cdot \nabla {F_\alpha }^\mathrm{{qe}}\\= & {} \displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \mathbf {u}}\mathbf {c}_\alpha :\nabla \mathbf {u} +\displaystyle \sum \limits _{i=4}^{M}\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \tilde{\gamma }_{i}}\mathbf {c_{\alpha }\cdot \nabla }{\tilde{\gamma }}_i. \end{aligned}$$

After gathering terms proportional to different gradients it can be obtained

$$\begin{aligned} \mathrm{RHS}_{\nabla }= & {} F_\alpha ^\mathrm{{qe}}{\tilde{\gamma }}_\mathrm{E}m_{\alpha }\mathbf {c}_\alpha \mathbf {c}_{\alpha }:\nabla \mathbf {u} +\displaystyle \sum \limits _{i=4}^{M} \displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \tilde{\gamma }_{i}}\mathbf {c}_{\alpha }\cdot \nabla \tilde{\gamma }_{i} \\&-\left( \displaystyle \sum \limits _{i=1}^{M}\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _{i}}\varGamma _{i}+p\displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}\right) \mathbf {\nabla }\cdot \mathbf {u}\\&-\left( \displaystyle \frac{\partial F_{\alpha }^\mathrm{{qe}}}{\partial \mathbf {p}}+\displaystyle \frac{\partial {F}_{\alpha }^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}\mathbf {u}\right) \cdot \nabla p. \end{aligned}$$

We finally get

$$\begin{aligned} \mathrm{RHS}_\nabla= & {} F_\alpha ^\mathrm{{qe}}\tilde{\gamma }_\mathrm{E}m_\alpha \left( \mathbf {c}_\alpha \mathbf {c}_\alpha -\displaystyle \frac{1}{3}c^2_\alpha \mathbf {I}\right) :\nabla \mathbf {u}\nonumber \\&+\displaystyle \sum \limits _{i=4}^M\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial {\tilde{\gamma }}_i}\mathbf {c}_\alpha \cdot \nabla {\tilde{\gamma }}_i \nonumber \\&-\left( \displaystyle \sum \limits _{i=1}^M\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i+p\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}-\displaystyle \frac{1}{3}F_{\alpha }^\mathrm{{qe}} {\tilde{\gamma }}_\mathrm{E}m_\alpha {c}^2_\alpha \right) \nabla \cdot \mathbf {u}\nonumber \\&-F_{\alpha }^\mathrm{{qe}}{\tilde{\gamma }}_\mathrm{E}\displaystyle \frac{m_\alpha }{\rho }\mathbf {c}_\alpha \cdot \nabla {p}, \end{aligned}$$

(189)

where $\mathbf {I}$ is a unit tensor. Since we are interested only in terms proportional to $\nabla \cdot \mathbf {u}$, after gathering corresponding terms, we obtain

$$\begin{aligned} \mathrm{RHS}_{\nabla \mathbf {u}}=\left( -\displaystyle \sum \limits _{i=1}^M\displaystyle \frac{\partial F^\mathrm{{qe}}_\alpha }{\partial \varGamma _i}\varGamma _i-p\displaystyle \frac{\partial {F}^\mathrm{{qe}}_\alpha }{\partial \varGamma _\mathrm{E}} +\displaystyle \frac{\gamma _\mathrm{E} m_\alpha {c}^2_\alpha }{3}F^\mathrm{{qe}}_\alpha \right) \nabla \cdot \mathbf {u}. \end{aligned}$$

(190)

Integrating expression (190) over velocities $\mathbf {v}_\alpha$ and providing summation over rotational quantum numbers, we obtain

$$\begin{aligned} \overline{\mathrm{RHS}}_{\nabla \mathbf {u}} =\left( -\displaystyle \sum \limits _{i=1}^M\displaystyle \frac{\partial X^\mathrm{{qe}}_{\alpha q}}{\partial \varGamma _i}\varGamma _i -p\displaystyle \frac{\partial X^\mathrm{{qe}}_{\alpha q}}{\partial \varGamma _\mathrm{E}} +X^\mathrm{{qe}}_{\alpha q}\right) \nabla \cdot \mathbf {u}. \end{aligned}$$

(191)

After summation over vibrational quantum numbers and over species, this expression vanishes.

Appendix 5: Expressions for RHS for equations (48) and (49)

To calculate the derivatives ${\partial {F^\mathrm{{qe}}}}/{\partial \varGamma _i}$  (31) and expressions from Appendix 3 are used. Thus, for

$$\begin{aligned}&\displaystyle \frac{\partial {F^\mathrm{{qe}}_\alpha }}{\partial \varGamma _{n_\beta }}\nonumber \\&\quad =F^\mathrm{{qe}}_\alpha \left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } -m_\beta \displaystyle \frac{m_\alpha \gamma _\mathrm{E}{\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho ^2}\right. \nonumber \\&\qquad \left. +\left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{31\beta } +\displaystyle \sum _{i=1}^{M_\mathrm{a}}\left( \varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}\right) b_{41i\beta }\right) \nonumber \\&\quad =F^\mathrm{{qe}}_\alpha \left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } -m_\beta \displaystyle \frac{m_\alpha \gamma _\mathrm{E}{\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho ^2}\right. \nonumber \\&\qquad \left. -\left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) \left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) b_{33}\right. \nonumber \\&\qquad \left. -\left( \varGamma _{\mathrm{E}\beta }-\displaystyle \frac{m_\beta {\mathbf{p}}^2}{\rho ^2}\right) \displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\right) , \end{aligned}$$

(192)

$$\begin{aligned}&\displaystyle \sum _\beta \displaystyle \frac{\partial {F^\mathrm{{qe}}_\alpha }}{\partial \varGamma _{n_\beta }}\varGamma _{n_\beta }\nonumber \\&\quad =F^\mathrm{{qe}}_\alpha \left( 1-\displaystyle \frac{m_\alpha \gamma _\mathrm{E}{\mathbf{c}}_\alpha \cdot {\mathbf{p}}}{\rho }-\left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) \left( \varGamma _\mathrm{E}-\displaystyle \frac{{\mathbf{p}}^2}{\rho }\right) b_{33}\right. \nonumber \\&\qquad \left. -\left( \varGamma _\mathrm{E}-\displaystyle \frac{{\mathbf{p}}^2}{\rho }\right) \displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\right) , \end{aligned}$$

(193)

$$\begin{aligned}&\displaystyle \frac{\partial {F^\mathrm{{qe}}_\alpha }}{\partial \varGamma _{p_x}} = F^\mathrm{{qe}}_\alpha \left( \displaystyle \frac{\gamma _\mathrm{E}m_\alpha {c}_{\alpha {x}}}{\rho }+\left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{32x}\right. \nonumber \\&\quad \quad \left. +\displaystyle \sum _{j=1}^{M_\mathrm{a}}\left( \varGamma _{\mathrm{a}j\alpha }-\psi _{\mathrm{a}j}\right) b_{42jx}\right) \nonumber \\&\quad \quad =\displaystyle \frac{F^\mathrm{{qe}}_\alpha }{\rho }\left( {\gamma _\mathrm{E}m_\alpha {c}_{\alpha {x}}} -p_x\left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{33}\right. \nonumber \\&\qquad \left. -p_x\displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\right) , \end{aligned}$$

(194)

$$\begin{aligned}&\displaystyle \frac{\partial {F^\mathrm{{qe}}_\alpha }}{\partial \varGamma _\mathrm{E}}\nonumber \\&\quad = F^\mathrm{{qe}}_\alpha \left( \left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{33} +\displaystyle \sum _{i=1}^{M_\mathrm{a}}\left( \varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}\right) b_{43i}\right) \nonumber \\&\quad =F^\mathrm{{qe}}_\alpha \left( \left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{33} +\displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\varGamma _{\mathrm{a}i\alpha }-\psi _{\mathrm{a}i}}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\right) , \end{aligned}$$

(195)

$$\begin{aligned}&\displaystyle \frac{\partial {F^\mathrm{{qe}}_\alpha }}{\partial \varGamma _{\mathrm{a}i}}\nonumber \\&\quad = F^\mathrm{{qe}}_\alpha \left( \left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha \right) b_{34i} +\displaystyle \sum _{j=1}^{M_\mathrm{a}}\left( \varGamma _{\mathrm{a}j\alpha }-\psi _{\mathrm{a}j}\right) b_{44ji}\right) \nonumber \\&\quad =F^\mathrm{{qe}}_\alpha \left( {\tilde{\varGamma }}_{\mathrm{E}\alpha }-\tilde{e}_\alpha -{c_{\mathrm{E,E}}}\displaystyle \sum _{j=1}^{M_\mathrm{a}}\displaystyle \frac{\varGamma _{\mathrm{a}j\alpha }-\psi _{\mathrm{a}j}}{c_{\mathrm{a}j,\mathrm{E}}} \right) b_{34i}, \end{aligned}$$

(196)

where i and j numerates the additional ASI, $M_\mathrm{a}$ is the number of the additional ASI, $\tilde{e}_\alpha ={m_\alpha \mathbf {c}_\alpha ^2}/2+e^i_\alpha (k_\alpha )$ and $\mathbf {c}_\alpha =\mathbf {v}_\alpha -\mathbf {p}/\rho$.

The RHS of  (48) contains the sum, where several terms vanish due to the momentum and total energy conservation [so that the corresponding terms are proportional to the scalar products with the exact summational invariants (ESI)]; therefore, it reduces to

$$\begin{aligned} \varSigma _\alpha= & {} \displaystyle \sum \limits _{i=1}^M \displaystyle \frac{\partial {F_\alpha ^\mathrm{{qe}}}}{\partial \varGamma _i}\langle \psi _i,I(F^\mathrm{{qe}})\rangle \nonumber \\= & {} \displaystyle \sum \limits _{\beta =1}^N\displaystyle \frac{\partial {F_\alpha ^\mathrm{{qe}}}}{\partial \varGamma _{n_\beta }}R_\beta ^\mathrm{{qe}} +\displaystyle \sum \limits _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\partial {F^\mathrm{{qe}}}}{\partial \varGamma _{\mathrm{a}i}}R_i^\mathrm{{qe}} \nonumber \\= & {} \left( \displaystyle \sum \limits _{\beta =1}^N \left( \displaystyle \frac{\delta _{\alpha \beta }}{n_\alpha } +\varGamma _{\mathrm{E}\beta }\left( \left( \tilde{e}_\alpha -\varGamma _{\mathrm{E}\alpha }\right) b_{33}\right. \right. \right. \nonumber \\&\left. +\displaystyle \sum _{i=1}^{M_\mathrm{a}}\left. \left. \displaystyle \frac{\psi _{\mathrm{a}i}-\varGamma _{\mathrm{a}i\alpha }}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\right) \right) R_\beta ^\mathrm{{qe}}\right. \nonumber \\&+\left( \varGamma _{\mathrm{E}\alpha }-\tilde{e}_\alpha +{c_{\mathrm{E,E}}}\displaystyle \sum _{j=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}j}-\varGamma _{\mathrm{a}j\alpha }}{c_{\mathrm{a}j,\mathrm{E}}}\right) \nonumber \\&\left. \times \displaystyle \sum \limits _{i=1}^{M_\mathrm{a}}b_{34i}R_i^\mathrm{{qe}}\right) F^\mathrm{{qe}}_\alpha , \end{aligned}$$

(197)

where $R_\beta ^\mathrm{{qe}}=\sum _{r\in {r_\beta }}(\nu _{\beta {r}}^{\,\prime }-\nu _{\beta {r}}^{\,\prime \prime })R_r^\mathrm{{qe}}$ and $R_i^\mathrm{{qe}}$ are the equilibrium reaction and relaxation rates, respectively. Summation over r means summation over reactions in which species $\beta$ participate. Because of the relationship $\sum _\beta {m_\beta }R_\beta ^\mathrm{{qe}} =\sum _{\beta ,r}{m_\beta }(\nu _{\beta {r}}^{\,\prime }-\nu _{\beta {r}}^{\,\prime \prime })R_r^\mathrm{{qe}}=0$, which follows from the mass conservation law $\sum _{\beta }{m_\beta }(\nu _{\beta {r}}^{\,\prime }-\nu _{\beta {r}}^{\,\prime \prime })=0$, terms in  (192) containing $m_\beta$ do not contribute to the expression (197). Here $\nu _{\beta {r}}^{\prime \prime (\prime )}$ are the stoichiometric coefficients of species $\beta$ in initial (final) channel of reaction r. Using representation for $R_\beta ^\mathrm{{qe}}$ via $R_r^\mathrm{{qe}}$  (197) can be written as

$$\begin{aligned} \varSigma _\alpha= & {} \bigg (\displaystyle \frac{R_\alpha ^\mathrm{{qe}}}{n_\alpha } +\displaystyle \sum \limits _{r\in {r_\alpha }}\varDelta {E_r}R^\mathrm{{qe}}_r \bigg (\left( \tilde{e}_\alpha -\varGamma _{\mathrm{E}\alpha }\right) b_{33}\nonumber \\&+\displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}i}-\varGamma _{\mathrm{a}i\alpha }}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\bigg ) \nonumber \\&+\bigg (\varGamma _{\mathrm{E}\alpha }-\tilde{e}_\alpha +{c_{\mathrm{E,E}}}\displaystyle \sum _{j=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}j}-\varGamma _{\mathrm{a}j\alpha }}{c_{\mathrm{a}j,\mathrm{E}}}\bigg )\nonumber \\&\times \sum \limits _{i=1}^{M_\mathrm{a}}b_{34i}R_i^\mathrm{{qe}}\bigg )F^\mathrm{{qe}}_\alpha , \end{aligned}$$

(198)

where $\varDelta {E_r}=\sum \nolimits _{\beta =1}^N(\nu _{\beta {r}}^{\,\prime }-\nu _{\beta {r}}^{\,\prime \prime })\varGamma _{\mathrm{E}\beta }$ is the energy transferred during the reaction r.

Equation (198) completes the expression for the RHS of  (48). For the LHS of  (48), the following expression can be derived

$$\begin{aligned}{}[J^\prime (F^\mathrm{{qe}})\varPhi _\mathrm{s}^{(0)}]_\alpha =[I^\prime (F^\mathrm{{qe}})\varPhi _\mathrm{s}^{(0)}]_\alpha -\varSigma _\alpha ^\prime \left( \varPhi _\mathrm{s}^{(0)}\right) , \end{aligned}$$

(199)

where

$$\begin{aligned} \varSigma _\alpha ^\prime= & {} \bigg (\displaystyle \frac{R_\alpha ^{\mathrm{ne}}}{n_\alpha } +\displaystyle \sum \limits _{r\in {r_\alpha }}\varDelta {E_r}R^{\mathrm{ne}}_{r} \bigg (\bigg (\tilde{e}_\alpha -\varGamma _{\mathrm{E}\alpha }\bigg )b_{33}\nonumber \\&+\displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}i}-\varGamma _{\mathrm{a}i\alpha }}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}{c_{\mathrm{a}j,\mathrm{E}}}b_{34j}\bigg ) \nonumber \\&+\bigg (\varGamma _{\mathrm{E}\alpha }-\tilde{e}_\alpha +{c_{\mathrm{E,E}}}\displaystyle \sum _{j=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}j} -\varGamma _{\mathrm{a}j\alpha }}{c_{\mathrm{a}j,\mathrm{E}}}\bigg )\nonumber \\&\times \sum \limits _{i=1}^{M_\mathrm{a}}b_{34i}R_i^{\mathrm{ne}}\bigg )F^\mathrm{{qe}}_\alpha . \end{aligned}$$

(200)

Here $R_\alpha ^{\mathrm{ne}}\equiv {R}^{(0)}_{\alpha \,\mathrm{s}}(\varPhi _\mathrm{s}^{(0)})$ and $R_i^{\mathrm{ne}}\equiv {R}^{(0)}_{\mathrm{a}{i\,\mathrm{s}}}(\varPhi _\mathrm{s}^{(0)})$ are the non-equilibrium parts of the “scalar” parts of the reaction and relaxation rates, respectively. They are the second terms in the sums of three terms in  (50) and (53), respectively. Performed calculations result in the following equation for $\varPhi ^{(0)}_{\mathrm{s}\alpha }$

$$\begin{aligned}{}[I^\prime (F^\mathrm{{qe}})\varPhi _\mathrm{s}^{(0)}]_\alpha -\varSigma _\alpha ^\prime (\varPhi _\mathrm{s}^{(0)}) =-I_\alpha (F^\mathrm{{qe}})+\varSigma _\alpha . \end{aligned}$$

(201)

Taking into account expressions (192)–(196), the following equation can be obtained

$$\begin{aligned} \sum _{i=1}^{M}\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i= & {} F_\alpha ^\mathrm{{qe}}\bigg (1+\bigg (\varGamma _{\mathrm{E}\alpha }-\tilde{e}_\alpha \nonumber \\&+\,c_{\mathrm{E,E}}\displaystyle \sum _{i=1}^{M_\mathrm{a}} \displaystyle \frac{\psi _{\mathrm{a}i}-\varGamma _{\mathrm{a}i\alpha }}{c_{\mathrm{a}i,\mathrm{E}}}\bigg ) \displaystyle \sum _{j=1}^{M_\mathrm{a}}b_{34j}\varGamma _{\mathrm{a}j}\bigg ). \end{aligned}$$

(202)

Here it should be noted that without the additional approximate summational invariants (ASI), this sum is essentially simplified:

$$\begin{aligned} \displaystyle \sum _{i=1}^{M}\displaystyle \frac{\partial {F}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i =F_\alpha ^\mathrm{{qe}}. \end{aligned}$$

(203)

Thus, for the RHS of  (49), the following expression is derived

$$\begin{aligned} \mathrm{RHS}_\mathrm{d}= & {} F_\alpha ^\mathrm{{qe}}\bigg (\displaystyle \frac{\gamma _\mathrm{E}m_\alpha {\mathbf{c}}_\alpha ^2}{3}-1 +\bigg (\tilde{e}_\alpha -\varGamma _{\mathrm{E}\alpha }\bigg )\nonumber \\&\times \,\bigg (\displaystyle \sum _{j=1}^{M_\mathrm{a}}b_{34j}\varGamma _{\mathrm{a}j}+pb_{33}\bigg ) \nonumber \\&-\,c_{\mathrm{E,E}}\displaystyle \sum _{i=1}^{M_\mathrm{a}}\displaystyle \frac{\psi _{\mathrm{a}i}-\varGamma _{\mathrm{a}i\alpha }}{c_{\mathrm{a}i,\mathrm{E}}} \displaystyle \sum _{j=1}^{M_\mathrm{a}}b_{34j}\bigg (\varGamma _{\mathrm{a}j}-\displaystyle \frac{c_{\mathrm{a}j,\mathrm{E}}}{c_{\mathrm{E,E}}}\bigg )\bigg ), \end{aligned}$$

(204)

which leads to the following equation for $\varPhi _\mathrm{d}^{(0)}$

$$\begin{aligned}{}[I^\prime (F^\mathrm{{qe}})\varPhi _\mathrm{d}^{(0)}]_\alpha -\varSigma _\alpha ^\prime (\varPhi _\mathrm{d}^{(0)})=\mathrm{RHS}_\mathrm{d}. \end{aligned}$$

(205)

Corresponding expressions for $\sum _{i=1}^{M}({\partial {X}_\alpha ^\mathrm{{qe}}}/{\partial \varGamma _i})\varGamma _i$ can be derived by performing integration over velocities and summation over rotational quantum numbers in  (205):

$$\begin{aligned} \sum _{i=1}^M\displaystyle \frac{\partial {X}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i= & {} X_\alpha ^\mathrm{{qe}}\bigg (1+\bigg (\varGamma _{\mathrm{E}^{\mathrm{(V)}}\alpha }-{e}^{\mathrm{(v)}}_\alpha \nonumber \\&-\,c_{\mathrm{E,E}}\displaystyle \sum _{j=1}^{M_\mathrm{V}}\displaystyle \frac{\varGamma _{\mathrm{V}j\alpha }-\psi _{\mathrm{V}j}}{c_{\mathrm{a}j,\mathrm{E}}}\bigg ) \displaystyle \sum _{j=1}^{M_\mathrm{a}}b_{34j}\varGamma _{\mathrm{a}j}\bigg ), \end{aligned}$$

(206)

where i numerates all the ASI and corresponding gas-dynamic variables, j numerates only the additional ASI, $M_\mathrm{V}$ is the number of the “vibrational” ASI, ${e}^{\mathrm{(v)}}_\alpha$ is the vibrational energy of species $\alpha$, $\varGamma _{\mathrm{E}^{\mathrm{(V)}}\alpha }$ is the corresponding mean energy, $\psi _{\mathrm{V}i}$ are the “vibrational” ASI, $\varGamma _{\mathrm{V}i}=\sum _\alpha \varGamma _{\mathrm{V}i\alpha }$ are the corresponding gas-dynamic variables. When one and only one additional ASI, namely “vibrational,” is considered,  (206) reduces to

$$\begin{aligned} \displaystyle \sum _{i=1}^M\displaystyle \frac{\partial {X}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i= & {} X_\alpha ^\mathrm{{qe}}\bigg (1+\bigg (\varGamma _{\mathrm{E}^{\mathrm{(V)}}\alpha }-{e}^{\mathrm{(v)}}_\alpha \nonumber \\&-\,c_{\mathrm{E,E}}\displaystyle \frac{\varGamma _{\mathrm{V}\alpha }-\psi _\mathrm{V}}{c_{\mathrm{V,E}}}\bigg ) \displaystyle \frac{\gamma _\mathrm{V}^2c_{\mathrm{V,E}}\varGamma _\mathrm{V}}{n\varDelta }\bigg ), \end{aligned}$$

(207)

where  (38) and (39) are used, the thermodynamic coefficients are $c_{\mathrm{V,E}}{=}-(\gamma ^2_\mathrm{E}/n)\partial \varGamma _\mathrm{V}/\partial \gamma _\mathrm{E}$ and $c_{\mathrm{E,V}}=-(\gamma ^2_\mathrm{V}/n)\partial \varGamma _\mathrm{E}/\partial \gamma _\mathrm{V}$, $\varDelta {=}c_{\mathrm{E,E}}c_{\mathrm{V,V}}-c_{\mathrm{V,E}}c_{\mathrm{E,V}}$. Similarly

$$\begin{aligned} \displaystyle \frac{\partial {X}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}= & {} X_\alpha ^\mathrm{{qe}}(({e}^{\mathrm{(v)}}_\alpha -\varGamma _{\mathrm{E}^{\mathrm{(V)}}\alpha })c_{\mathrm{V,V}}\nonumber \\&-\,(\psi _\mathrm{V}-\varGamma _{V\alpha })c_{\mathrm{E,V}}) \displaystyle \frac{\gamma _\mathrm{E}}{p\varDelta }, \end{aligned}$$

(208)

where p is the pressure, $p=n/\gamma _\mathrm{E}$, and thus,

$$\begin{aligned}&\displaystyle \sum _{i=1}^{M}\displaystyle \frac{\partial {X}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _i}\varGamma _i +\displaystyle \frac{\partial {X}_\alpha ^\mathrm{{qe}}}{\partial \varGamma _\mathrm{E}}p-{X}_\alpha ^\mathrm{{qe}}\nonumber \\&\quad =\displaystyle \frac{X_\alpha ^\mathrm{{qe}}}{\varDelta }\bigg (\gamma _\mathrm{E}(\varGamma _{\mathrm{E}^{\mathrm{(V)}}\alpha }-{e}^{\mathrm{(v)}}_\alpha ) \left( \gamma _\mathrm{E}c_\mathrm{EV}\displaystyle \frac{\varGamma _\mathrm{V}}{n}-c_\mathrm{VV}\right) \nonumber \\&\qquad -\left( \varGamma _{V\alpha }-\psi _\mathrm{V}\right) \left( \gamma _\mathrm{V}^2c_{\mathrm{E,E}} \displaystyle \frac{\varGamma _\mathrm{V}}{n}-c_{\mathrm{E,V}}\gamma _\mathrm{E}\right) \bigg ). \end{aligned}$$

(209)

Appendix 6: Expressions for $S(\varTheta ^{(m)})$ and $\partial {\varTheta ^{(m)}}/\partial {\varGamma _\mathrm{V}}$

From the expressions (43) and (96) for $X^\mathrm{{qe}}$, the relationship $S(n_1)=1$, and the expression

$$\begin{aligned} S(X^\mathrm{{qe}}_{1,l})= & {} X^\mathrm{{qe}}_{1,l}\left( S(n_1)/n_1+Q^{\mathrm{(V)}}_1S(1/Q^{\mathrm{(V)}}_1)\right) \\&+\,S\left( \exp \left[ -\gamma _\mathrm{E}{e}^{\mathrm{(v)}}_1(l)-\gamma _\mathrm{V}l\right] \right) n_1/{Q^{\mathrm{(V)}}_1} \\= & {} X^\mathrm{{qe}}_{1,l}\left( 1/n_1-S(\ln {Q^{\mathrm{(V)}}_1})-{e}^{\mathrm{(v)}}_1(l)S(\gamma _\mathrm{E})-lS(\gamma _\mathrm{V})\right) , \end{aligned}$$

it follows

$$\begin{aligned} S(\varTheta ^{(m)})= & {} \displaystyle \sum _{l=0}^mS(X^\mathrm{{qe}}_{1,l}) \\= & {} \displaystyle \frac{\varTheta ^{(m)}}{n_1} -\left( \left( \displaystyle \sum \limits ^m_{l=0}{e}^{\mathrm{(v)}}_1(l)X^\mathrm{{qe}}_{1,l}-\varTheta ^{(m)}\varGamma _{\mathrm{E}^{\mathrm{(V)}}1}\right) S(\gamma _\mathrm{E})\right. \\&\left. +\left( \displaystyle \sum \limits ^m_{l=0}lX^\mathrm{{qe}}_{1,l}-\varTheta ^{(m)}\displaystyle \frac{\varGamma _\mathrm{V}}{n_1}\right) S(\gamma _\mathrm{V})\right) , \end{aligned}$$

for the case when the vibrational quantum number is the ASI, and

$$\begin{aligned}&S(\varTheta ^{(m)})=\displaystyle \frac{\varTheta ^{(m)}}{n_1} -\left( \displaystyle \sum \limits ^m_{l=0}\mathrm{e}^{\mathrm{(v)}}_1(l)X^\mathrm{{qe}}_{1l} -\varTheta ^{(m)}\varGamma _{\mathrm{E}^{\mathrm{(V)}}1}\right) S(\theta _{\mathrm{E}^{\mathrm{(V)}}}),\\&\theta _{\mathrm{E}^{\mathrm{(V)}}}=\gamma _\mathrm{E}+\gamma _{\mathrm{E}^{\mathrm{(V)}}}, \end{aligned}$$

for the case when the vibrational energy is the ASI, $\varGamma _{\mathrm{E}^{\mathrm{(V)}}1}=\varGamma _{\mathrm{E}^{\mathrm{(V)}}}/n_1$. Here the equality $Q_\alpha =Q_\alpha (\gamma _\mathrm{E},\gamma _\mathrm{V})$ and the thermodynamic relationships (20) are used.

$$\begin{aligned} \displaystyle \sum \limits _{q=0}^{q_m-1}\varTheta ^{(q)}= & {} \displaystyle \sum \limits _{q=0}^{q_m}\displaystyle \sum \limits _{m=0}^qX^\mathrm{{qe}}_{1,m}-n_1\nonumber \\= & {} \displaystyle \sum \limits _{m=0}^{q_m}(q_m+1-m)X^\mathrm{{qe}}_{1,m}-n_1=q_mn_1-\varGamma _\mathrm{V}. \end{aligned}$$

(210)

Thus,

$$\begin{aligned} \displaystyle \sum \limits _{q=0}^{q_m-1}S\left( \varTheta ^{(q)}\right) =q_m,\qquad \displaystyle \sum \limits _{q=0}^{q_m-1}\displaystyle \frac{\partial \varTheta ^{(q)}}{\partial \varGamma _\mathrm{V}}=-1. \end{aligned}$$

(211)

For derivatives with respect to $\varGamma _\mathrm{V}$, similar reasoning lead to the following results. For the case when the vibrational quantum number is the ASI, we obtain

$$\begin{aligned} \displaystyle \frac{\partial \varTheta ^{(m)}}{\partial \varGamma _\mathrm{V}}= & {} -\left( \displaystyle \sum \limits ^m_{l=0}{e}^{\mathrm{(v)}}_1(l)X^\mathrm{{qe}}_{1,l} -\varTheta ^{(m)}\varGamma _{\mathrm{E}^{\mathrm{(V)}}1}\right) \displaystyle \frac{\partial \gamma _\mathrm{E}}{\partial \varGamma _\mathrm{V}}\\&-\left( \displaystyle \sum \limits ^m_{l=0}lX^\mathrm{{qe}}_{1,l} -\varTheta ^{(m)}\displaystyle \frac{\varGamma _\mathrm{V}}{n_1}\right) \displaystyle \frac{\partial \gamma _\mathrm{V}}{\partial \varGamma _\mathrm{V}}, \end{aligned}$$

and for the case when vibrational energy is the ASI, we obtain

$$\begin{aligned} \displaystyle \frac{\partial \varTheta ^{(m)}}{\partial \varGamma _\mathrm{V}}=-\left( \displaystyle \sum \limits ^m_{l=0}{e}^{\mathrm{(v)}}_1(l)X^\mathrm{{qe}}_{1l} -\varTheta ^{(m)}\varGamma _{\mathrm{E}^{\mathrm{(V)}}1}\right) \displaystyle \frac{\partial \theta _{\mathrm{E}^{\mathrm{(V)}}}}{\partial \varGamma _\mathrm{V}}. \end{aligned}$$

Appendix 7: Quasi-stationary vibrational population density

Using (71) we can write

where $\alpha _1=1$. Solving these equations over $X^{\mathrm{qs}}_{1,0}$ and $X^{\mathrm{qs}}_{1,1}$ we obtain

(212)

After substituting these relationships into expression (71), the quasi-stationary distribution for $q\ge 2$ can be represented as

To get the quasi-stationary distribution function for the case, when the vibrational energy is considered as the ASI, we use the following relationships

$$\begin{aligned}&n_1=\displaystyle \sum \limits _{n=0}^{q_m}X^{\mathrm{qs}}_{1,q} =X^{\mathrm{qs}}_{1,0}+\displaystyle \sum \limits _{q=1}^{q_m}\alpha _qX^{\mathrm{qe}}_{1,q}X^{\mathrm{qs}}_{1,1}/X^{\mathrm{qe}}_{1,1}\nonumber \\&\quad +\displaystyle \sum \limits _{q=2}^{q_m}\alpha _q\left( R_1\tilde{c}_q +R_{\mathrm{E}^{(\mathrm{V})}}\tilde{c}_{\mathrm{E}^{(\mathrm{V})}\,q}+\tilde{\tilde{c}}_q(X^{\mathrm{qs}}_1)\right) X^{\mathrm{qe}}_{1,q} \nonumber \\&=X^{\mathrm{qs}}_{1,0}{+}n_1\left( \tilde{a}_\mathrm{n}{X^{\mathrm{qs}}_{1,1}}/{X^{\mathrm{qe}}_{1,1}}+R_1\tilde{C} {+}R_{\mathrm{E}^{(\mathrm{V})}}\tilde{C}_{\mathrm{E}^{(\mathrm{V})}}{+}\tilde{\tilde{C}}(X^{\mathrm{qs}}_1)\right) . \end{aligned}$$

$$\begin{aligned} \varGamma _{\mathrm{E}^{(\mathrm{V})}}= & {} \displaystyle \sum \limits _{n=0}^{q_m}{e}^{(\mathrm{v})}_{1,q}X^{\mathrm{qs}}_{1,q} ={e}^{(\mathrm{v})}_{1,0}X^{\mathrm{qs}}_{1,0}{+}\displaystyle \sum \limits _{q=1}^{q_m}\alpha _q{e}^{(\mathrm{v})}_{1,q}X^{\mathrm{qe}}_{1,q}X^{\mathrm{qs}}_{1,1}/X^{\mathrm{qe}}_{1,1}\nonumber \\&+\displaystyle \sum \limits _{q=2}^{q_m}\alpha _q\left( R_1\tilde{c}_q +R_{\mathrm{E}^{(\mathrm{V})}}\tilde{c}_{\mathrm{E}^{(\mathrm{V})}\,q}+\tilde{\tilde{c}}_q(X^{\mathrm{qs}}_1)\right) {e}^{(\mathrm{v})}_{1,q}X^{\mathrm{qe}}_{1,q} \nonumber \\= & {} {e}^{(\mathrm{v})}_{1,0}X^{\mathrm{qs}}_{1,0}+\varGamma _{\mathrm{E}^{(\mathrm{V})}}\left( \tilde{a}_{\mathrm{E}^{(\mathrm{V})}}{X^{\mathrm{qs}}_{1,1}}/{X^{\mathrm{qe}}_{1,1}}\right. \nonumber \\&\left. +\,R_1\tilde{D}^\mathrm{(v)} +R_{\mathrm{E}^{(\mathrm{V})}}\tilde{D}^\mathrm{(v)}_{\mathrm{E}^{(\mathrm{V})}}+\tilde{\tilde{D}}^\mathrm{(v)}(X^{\mathrm{qs}}_1)\right) . \end{aligned}$$

Solving these equations over $X_{1,0}$ and $X_{1,1}$, the following expressions are obtained:

(213)

After substituting these relationships into (103), we obtain the expression (104) for the quasi-stationary distribution for $q\ge 2$.

Appendix 8: Expressions for $R_{\mathrm{VV}}$

Using the definition

and the expression (42) for the quasi-stationary distribution function , the following expressions can be derived.

Using expression (71) for and remembering that and , so that and , it can be written

where the expressions and (211) are used.

where , and .

Similarly, to we obtain

where the expressions , , , and (211) are used.

where is an arbitrary distribution function and

Appendix 9: Expressions for the harmonic oscillator model

1.1 Expressions for $\tilde{c}_q$, $\tilde{c}_{\mathrm{V}q}$ and $\tilde{\tilde{c}}_q$

The following relationships are used for getting expressions for the quasi-stationary distribution

(214)

(215)

(216)

Using (216) and (215) the expression (210), and therefore (211), can be confirmed for the specific case of the harmonic oscillator. Using relationships

$$\begin{aligned}&X^{\mathrm{qe}}_{1,q_m}=\displaystyle \frac{n_1-\varGamma _\mathrm{V}\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) }{q_m+1},\nonumber \\&\quad X^{\mathrm{qe}}_{1,0}={\mathrm{{e}}^{-\theta _\mathrm{V}}}{\left( n_1\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) +X^{\mathrm{qe}}_{1,q_m}\right) }, \end{aligned}$$

(217)

which can be obtained from (117), equations in (216) can be written as

(218)

where

Applying operators and S to both parts of (215), and using that and (217), we obtain

(219)

Thus, equations in (218) can be written as

$$\begin{aligned}&S(\varTheta ^{(m)})=\kappa _0\displaystyle \frac{\varGamma _\mathrm{V}}{n_1}\displaystyle \frac{(m+1)X^{\mathrm{qe}}_{1,m}}{n_1}\nonumber \\&\quad +\displaystyle \frac{\varTheta ^{(m)}}{n_1}\left( 1-\kappa _0\displaystyle \frac{\varGamma _\mathrm{V}}{n_1} \left( 1-\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) \displaystyle \frac{\varGamma _\mathrm{V}}{n_1}\right) \right) , \end{aligned}$$

(220)

(221)

Using expressions (121) for the vibrational transition probabilities we obtain

$$\begin{aligned}&\tilde{P}_{m+1,m}(X^{\mathrm{qe}})=(m+1)\left( P_{10}+Q_{10}^{01}\displaystyle \sum _{l=0}^{q_m-1}(l+1)X^{\mathrm{qe}}_{1,l}\right) \\&\quad =(m+1)\left( P_{10}+Q_{10}^{01}\mathrm{{e}}^{\theta _\mathrm{V}}\displaystyle \sum _{l=0}^{q_m-1}(l+1)X^{\mathrm{qe}}_{1,l+1}\right) \\&\quad =(m+1)\left( P_{10}+Q_{10}^{01}\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}\right) . \end{aligned}$$

Then the expression for $\tilde{c}_q$, defined in (71) and (72), can be represented as

$$\begin{aligned}&\tilde{c}_q=\displaystyle \frac{1}{(1+\beta _\mathrm{V})Q_{10}^{01}n_1^2} \left( \kappa _0\displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1}\right. \\&\left. \quad +\left( \displaystyle \frac{n_1}{\varGamma _\mathrm{V}} -\kappa _0\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{q_m}}{n_1}\right) \displaystyle \frac{\alpha _2^{q}\varSigma _{\alpha ,q}(\theta _\mathrm{V})}{\mathrm{{e}}^{\theta _\mathrm{V}}-1}\right) , \\&\beta _\mathrm{V}=\displaystyle \frac{P_{10}}{Q_{10}^{01}\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}}, \quad \varSigma _{\alpha ,q}(\theta _\mathrm{V})=\displaystyle \sum ^q_{m=2}\alpha ^{-m}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m},\\&\quad \tilde{c}_0=\displaystyle \frac{n_1(\tilde{D}-\tilde{C})}{X^{\mathrm{qe}}_{1,0}}, \quad \tilde{c}_1=0. \end{aligned}$$

Using (217) and that $X^{\mathrm{qe}}_{1,q_m+1}/n_1\ll 1$ and $1/q_s\ll 1$, we can write

(222)

Here and further in Appendix 9, we use sign “$\approx$” when performing such an approximation. Then

Using (216), for the harmonic oscillator we obtain

$$\begin{aligned} \begin{array}{c} \overline{\varSigma }=-\displaystyle \frac{\varSigma _{q_m}(-\theta _\mathrm{V}) +\varSigma _{q_m}(\theta _\mathrm{V})\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}}{1-\exp {(-\theta _\mathrm{V}(q_m+1))}}. \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \overline{\varSigma }^{\prime \prime }=\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}}\displaystyle \sum \limits _{m=2}^{q_m} \displaystyle \sum \limits _{q=m}^{q_m}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m}q\mathrm{{e}}^{-\theta _\mathrm{V}q}\\ \quad =\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}}\displaystyle \sum \limits _{m=2}^{q_m}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m} \left( \displaystyle \sum \limits _{q=2}^{q_m}-\displaystyle \sum \limits _{q=2}^{m-1}\right) q\mathrm{{e}}^{-\theta _\mathrm{V}q} \\ \quad =\displaystyle \sum \limits _{m=2}^{q_m}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m} \left( 1-\displaystyle \frac{\varGamma _\mathrm{V}^{(m-1)}}{\varGamma _\mathrm{V}}\right) \\ \quad =\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}}\right) }\left[ (q_m+1)\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}\varSigma _{q_m}(\theta _\mathrm{V})\right) \right. \\ \quad +\left. 1-\mathrm{{e}}^{-\theta _\mathrm{V}}-\displaystyle \frac{\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}\left( \varSigma _{q_m}(\theta _\mathrm{V})-1\right) +1 +\mathrm{{e}}^{-\theta _\mathrm{V}}\varSigma _{q_m}(-\theta _\mathrm{V})}{1-\mathrm{{e}}^{-\theta _\mathrm{V}}}\right] \\ \quad \approx (q_m+1)n_1/\varGamma _\mathrm{V}, \end{array} \end{aligned}$$

$$\begin{aligned} \varGamma _\mathrm{V}^{(m)}=X_{1,0}^{\mathrm{qe}}\left( \displaystyle \frac{\mathrm{{e}}^{-\theta _\mathrm{V}}\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}(m+1)}\right) }{\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}}\right) ^2}-\displaystyle \frac{(m+1)\mathrm{{e}}^{-\theta _\mathrm{V}(m+1)}}{1-\mathrm{{e}}^{-\theta _\mathrm{V}}}\right) . \end{aligned}$$

Using (217) and that $\displaystyle \frac{n_1}{\varGamma _\mathrm{V}} -\kappa _0\left( 1-\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) \displaystyle \frac{\varGamma _\mathrm{V}}{n_1}\right) =\kappa _0\left( q_m\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1-\displaystyle \frac{n_1}{\varGamma _\mathrm{V}}\right) +\mathrm{{e}}^{\theta _\mathrm{V}}\right)$, we derive

(223)

$$\begin{aligned}&\tilde{B}_q\approx \displaystyle \frac{(1+\beta _\mathrm{V})^{-1}\varGamma _\mathrm{V}}{n_1\left( n_1+\varGamma _\mathrm{V}\right) } \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha _2^{q-1}}{\tilde{a}}\right. \nonumber \\&\quad \times \left. \left( 1+\displaystyle \frac{n_1(1+\beta _\mathrm{V})}{n_1+\beta _\mathrm{V}(n_1+\varGamma _\mathrm{V}(1-\mathrm{{e}}^{\gamma _\mathrm{V}}))}\right) \right. \nonumber \\&\quad \left. +\displaystyle \frac{n_1+\varGamma _\mathrm{V}}{n_1} \left( \alpha _2^{q}\varSigma _{\alpha ,q}-\displaystyle \frac{\alpha _2^{q-1}}{\tilde{a}}\tilde{\varSigma }^\prime \right) \right) , \nonumber \\&\quad B_0=\displaystyle \frac{n_1(\tilde{a_\mathrm{n}}\tilde{D}-\tilde{a}\tilde{C})}{\tilde{a}X^{\mathrm{qe}}_{1,0}}, B_1=-\displaystyle \frac{\tilde{D}}{\tilde{a}}. \end{aligned}$$

(224)

$$\begin{aligned} \begin{array}{l} \tilde{c}_{\mathrm{V}q}=-\displaystyle \frac{\kappa _0}{(1+\beta _\mathrm{V})Q_{10}^{01}n_1\varGamma _\mathrm{V}}\\ \times \,\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{q_m}}{(\mathrm{{e}}^{\theta _\mathrm{V}}-1)n_1}\alpha ^{q}_2\varSigma _{\alpha ,q}\right) \\ \approx -\displaystyle \frac{\left( n_1+\varGamma _\mathrm{V}\right) ^{-1}}{(1+\beta _\mathrm{V})Q_{10}^{01}\varGamma _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{(q_m+1)\varGamma _\mathrm{V}X^{\mathrm{qe}}_{q_m}}{n^2_1}\alpha ^{q}_2\varSigma _{\alpha ,q}\right) , \\ \tilde{c}_{\mathrm{V}0}=\displaystyle \frac{n_1(\tilde{D}_\mathrm{V}-\tilde{C}_\mathrm{V})}{X^{\mathrm{qe}}_{1,0}}, \quad \tilde{c}_{\mathrm{V}1}=0. \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \tilde{C}_\mathrm{V}=-\displaystyle \frac{\kappa _0/n_1}{Q_{10}^{01}\varGamma _\mathrm{V}(1+\beta _\mathrm{V})} \left( \displaystyle \frac{\overline{\alpha }^{\,\prime }}{\alpha _2-1} -\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}\tilde{\varSigma }}{n_1(\mathrm{e}^{\theta _\mathrm{V}}-1)}\right) \\ \approx -\displaystyle \frac{\varGamma _\mathrm{V}}{Q_{10}^{01}\left( n_1+\varGamma _\mathrm{V}\right) ^2(n_1+\beta _\mathrm{V}(n_1+\varGamma _\mathrm{V}(1-{\mathrm{e}^{\gamma _\mathrm{V}}})))}. \end{array} \end{aligned}$$

$$\begin{aligned}&\tilde{D}_V=-\displaystyle \frac{\kappa _0/n_1}{Q_{10}^{01}\varGamma _\mathrm{V}(1+\beta _\mathrm{V})}\\&\quad \times \left( \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)} -\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}{\tilde{\varSigma }}^\prime }{n_1(\mathrm{{e}}^{\theta _\mathrm{V}}-1)}\right) \\&\quad \approx -\displaystyle \frac{1}{Q_{10}^{01}(1+\beta _\mathrm{V})\varGamma _\mathrm{V}\left( n_1+\varGamma _\mathrm{V}\right) }\\&\quad \times \left( 1+\displaystyle \frac{1+\beta _\mathrm{V}}{1+\beta _\mathrm{V}\left( 1-(\mathrm{{e}}^{\gamma _\mathrm{V}}-1)\varGamma _\mathrm{V}/n_1\right) }\right) . \end{aligned}$$

This leads to the following expression for $B_{\mathrm{V}q}$:

$$\begin{aligned}&B_\mathrm{V0}=\displaystyle \frac{n_1(\tilde{a_n}\tilde{D}_\mathrm{V}-\tilde{a}\tilde{C}_\mathrm{V})}{\tilde{a}X^{\mathrm{qe}}_{1,0}}, \nonumber \\ \quad&B_{\mathrm{V}1}=-\displaystyle \frac{\tilde{D}_\mathrm{V}}{\tilde{a}}, \qquad B_{\mathrm{V}q}=\displaystyle \frac{\tilde{B}_{\mathrm{V}q}}{\varGamma _\mathrm{V}Q_{10}^{01}}, \nonumber \\&\tilde{B}_{\mathrm{V}q}=-\displaystyle \frac{\kappa _0/n_1}{1+\beta _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha _2^{q-1}}{\tilde{a}}\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)}\right. \nonumber \\&\left. -\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}}{n_1(\mathrm{{e}}^{\theta _\mathrm{V}}-1)}\left( \alpha _2^{q}\varSigma _{\alpha ,q-1} -\displaystyle \frac{\alpha _2^{q-1}}{\tilde{a}}{\tilde{\varSigma }}^\prime \right) \right) . \end{aligned}$$

(225)

$$\begin{aligned}&\tilde{c}^{(0)}_q=-\displaystyle \frac{\alpha ^{q-1}_2\varGamma _\mathrm{V}}{n_1c_{\mathrm{V}0}} \displaystyle \sum _{m=1}^{q-1}\displaystyle \frac{\sum \limits _{n=1}^mnX_{1,n}^{\mathrm{qe}}-\varTheta ^{(m)}\varGamma _{\mathrm{V}1}}{\alpha ^m_2\tilde{P}_{m+1,m}(X^{\mathrm{qe}})X^{\mathrm{qe}}_{m+1}} \nonumber \\&=\displaystyle \frac{1}{(1+\beta _\mathrm{V})c_{\mathrm{V}0}Q_{10}^{01}n_1\left( e^{\theta _\mathrm{V}}-1\right) }\nonumber \\&\quad \quad \times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{(q_m+1)\alpha _2^{q}}{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}-1}\varSigma _{\alpha ,q}\right) , \nonumber \\&\varSigma _{\alpha ,q}=\displaystyle \sum _{m=2}^q\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m\alpha _2^m}, \quad \tilde{c}^{(0)}_0=\displaystyle \frac{n_1(\tilde{D}^{(0)}-\tilde{C}^{(0)})}{X^{\mathrm{qe}}_{1,0}}, \quad \nonumber \\&\tilde{c}^{(0)}_1=0,\tilde{C}^{(0)}=\displaystyle \frac{1}{(1+\beta _\mathrm{V})c_{\mathrm{V}0}Q_{10}^{01}n_1\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) }\nonumber \\&\quad \quad \times \left( \displaystyle \frac{\overline{\alpha }^{\,\prime }}{\alpha _2-1} -\displaystyle \frac{(q_m+1)\tilde{\varSigma }_{\alpha }}{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}-1}\right) , \nonumber \\&\tilde{D}^{(0)}=\displaystyle \frac{1}{(1+\beta _\mathrm{V})c_{\mathrm{V}0}Q_{10}^{01}n_1\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) }\nonumber \\&\quad \quad \times \left( \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)} -\displaystyle \frac{(q_m+1)\tilde{\varSigma }_\alpha }{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}-1}\right) . \end{aligned}$$

(226)

$$\begin{aligned} \varDelta ^{(0)}_q= & {} \displaystyle \frac{\tilde{\varDelta }^{(0)}_q}{Q_{10}^{01}\varGamma _\mathrm{V}}, \tilde{\varDelta }^{(0)}_q=\displaystyle \frac{\varGamma _\mathrm{V}}{(1+\beta _\mathrm{V})c_{\mathrm{V}0}n_1\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) }\nonumber \\&\times \,\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}}\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)}\right. \nonumber \\&\left. -\displaystyle \frac{q_m+1}{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}-1}\left( \alpha _2^{q}\varSigma _{\alpha ,q} -\displaystyle \frac{\alpha ^{q}_2}{\tilde{a}}\displaystyle \frac{\tilde{\varSigma }^\prime _{\alpha }}{\alpha _2}\right) \right) , \end{aligned}$$

(227)

where $c_{\mathrm{V}0}$ is introduced in (116).

For ${j}_{q}^{\,\prime }(X)Y=Q_{q+1,q}(X)Y_{q+1}-Q_{q,q+1}(X)Y_{q}$, for arbitrary distribution function $X_{1,q}$ it can be written

$$\begin{aligned} \begin{array}{l} {j}^{\,\prime }_m(X_1)X_1^{\mathrm{qe}}=Q_{10}^{01}(m+1)X_{1,m}^{\mathrm{qe}}\left( \displaystyle \sum \limits _{l=0}^{q_m-1}(l+1) \mathrm{{e}}^{-\theta _\mathrm{V}}X_{1,l}-\varGamma _\mathrm{V}\right) \\ \quad =Q_{10}^{01}(m+1)X_{1,m}^{\mathrm{qe}}\left( \mathrm{{e}}^{-\theta _\mathrm{V}} \left( \varGamma _\mathrm{V}-q_mX_{1,q_m}+n_1-X_{1,q_m}\right) -\varGamma _\mathrm{V}\right) \\ \quad =Q_{10}^{01}(m+1)X_{1,m}^{\mathrm{qe}}(q_m+1)\mathrm{{e}}^{-\theta _\mathrm{V}}\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) , \end{array} \end{aligned}$$

(228)

where expressions $\varGamma _\mathrm{V}=\sum _{q=1}^{q_m}qX_{1,q}$ and (217) were used. Thus,

$$\begin{aligned} \begin{array}{l} \tilde{\tilde{c}}_q(X_1)=-\displaystyle \frac{\alpha _2^{q-1}Q_{10}^{01}(q_m+1) \left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{P_{10}+Q_{10}^{01}\mathrm{e}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}} \displaystyle \sum \limits _{m=1}^{q-1}\displaystyle \frac{1}{\alpha _2^m} \\ =-\displaystyle \frac{(q_m+1)\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}(\beta _\mathrm{V}+1)}\displaystyle \frac{\alpha _2^{q-1}-1}{\alpha _2-1}, \quad \tilde{\tilde{c}}_0=\displaystyle \frac{n_1(\tilde{\tilde{D}}-\tilde{\tilde{C}})}{X^{\mathrm{qe}}_{1,0}}, \\ \tilde{\tilde{c}}_1=0, \quad \tilde{\tilde{C}}(X_1)=- \displaystyle \frac{(q_m+1)\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{\mathrm{{e}}^{\theta _\mathrm{V}}n_1\varGamma _\mathrm{V}\left( 1+\beta _\mathrm{V}\right) } \frac{\overline{\alpha }^{\,\prime }}{\alpha _2-1}, \\ \tilde{\tilde{D}}(X_1)=- \displaystyle \frac{(q_m+1)\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}\left( 1+\beta _\mathrm{V}\right) } \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)}. \end{array} \end{aligned}$$

(229)

Since $\displaystyle \sum \limits _{l=0}^{q_m}lX_{1,l} =\varGamma _\mathrm{V}$ [see (228)] was applied, the expression (229) can be used for calculating corresponding values only if $X_1$ is a distribution function and not an arbitrary function.

$$\begin{aligned} \begin{array}{l} \tilde{\tilde{c}}_q({\alpha _mX^{\mathrm{qe}}_{1,m}}/{\tilde{a}}) =\tilde{\tilde{c}}_\alpha \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1}, \\ \tilde{\tilde{c}}_\alpha =-\displaystyle \frac{(1-\alpha _2)\tilde{\varGamma }_\mathrm{V}+\alpha _2(\alpha _0\alpha _2-1)\mathrm{{e}}^{-\theta _\mathrm{V}}X_{1,0}^{\mathrm{qe}}}{(1+\beta _\mathrm{V}))\tilde{a}\alpha _2^2\varGamma _\mathrm{V}}, \\ \tilde{\tilde{C}}\left( \displaystyle \frac{\alpha _mX^{\mathrm{qe}}_{1,m}}{\tilde{a}}\right) =\tilde{\tilde{c}}_\alpha \displaystyle \frac{\varGamma _\mathrm{V}+X_{1,0}^{\mathrm{qe}}-n_1}{n_1}, \\ \tilde{\tilde{D}}\left( \displaystyle \frac{\alpha _mX^{\mathrm{qe}}_{1,m}}{\tilde{a}}\right) =\tilde{\tilde{c}}_\alpha \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)}. \end{array} \end{aligned}$$

(230)

$$\begin{aligned}&\tilde{\varGamma }_\mathrm{V}=\displaystyle \sum ^{q_m}_{m=1}m\alpha ^m_2X^{\mathrm{qe}}_{1,m} \nonumber \\&=n_1\displaystyle \frac{\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}}\right) \left( {\mathrm{{e}}^{-\tilde{\theta }_\mathrm{V}}-(q_m+1)\mathrm{{e}}^{-\tilde{\theta }_\mathrm{V}(q_m+1)}+q_m\mathrm{{e}}^{-\tilde{\theta }_\mathrm{V}(q_m+2)}}\right) }{\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}\right) \left( 1-\mathrm{{e}}^{-\tilde{\theta }_V}\right) ^2}, \nonumber \\&\tilde{\theta }_\mathrm{V}=\theta _\mathrm{V}-\ln {\alpha _2},\qquad \displaystyle \sum ^{q_m-1}_{m=0}(m+1)\alpha ^m_2X^{\mathrm{qe}}_{1,m} =\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}}\tilde{\varGamma }_\mathrm{V}}{\alpha _2}, \end{aligned}$$

(231)

$$\begin{aligned} \tilde{\varGamma }_\mathrm{V}\approx & {} \displaystyle \frac{\alpha _2\varGamma _\mathrm{V}(\mathrm{{e}}^{\theta _\mathrm{V}}-1)^2}{(\mathrm{{e}}^{\theta _\mathrm{V}}-\alpha _2)^2}\nonumber \\= & {} \displaystyle \frac{(1+\beta _\mathrm{V})(1+\beta _\mathrm{V}e^{\gamma _\mathrm{V}})\varGamma _\mathrm{V}}{\left( 1+\beta _\mathrm{V}(1+(1-\mathrm{{e}}^{\gamma _\mathrm{V}})\varGamma _\mathrm{V}/n_1)\right) ^2}\nonumber \\= & {} \varGamma _\mathrm{V}\left( 1+\beta _\mathrm{V}((\mathrm{{e}}^{\gamma _\mathrm{V}}-1)(1+2\varGamma _\mathrm{V}/n_1))+\mathrm{{O}}\left( \beta ^2_\mathrm{V}\right) \right) .\nonumber \\ \end{aligned}$$

(232)

$$\begin{aligned} \begin{array}{l} \tilde{\tilde{c}}_q(X^{\mathrm{qe}}_1)=0, \qquad \tilde{\tilde{c}}_q(mX^{\mathrm{qe}}_{1,m})=\displaystyle \frac{1}{1+\beta _\mathrm{V}}\displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1}, \\ \tilde{\tilde{C}}(mX^{\mathrm{qe}}_{1,m})=\displaystyle \frac{\varGamma _\mathrm{V}+X_{1,0}^{\mathrm{qe}}-n_1}{(1+\beta _\mathrm{V})n_1}, \\ \tilde{\tilde{D}}(mX^{\mathrm{qe}}_{1,m})=\displaystyle \frac{1}{(1+\beta _\mathrm{V})} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}(\alpha _2-1)}. \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \tilde{c}_q(X^{\mathrm{qe}}_1\varSigma )=\displaystyle \frac{n_1}{\varGamma _\mathrm{V}} \displaystyle \frac{q_m\left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}}\right) -\mathrm{{e}}^{-\theta _\mathrm{V}}(1-\mathrm{{e}}^{-\theta _\mathrm{V}q_m})}{\left( 1+\beta _\mathrm{V}\right) \left( 1-\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}\right) }\\ \displaystyle \quad \times \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} =\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,0}-n_1}{(1+\beta _\mathrm{V})\varGamma _\mathrm{V}}\displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1}\\ \equiv \tilde{\tilde{c}}_\varSigma \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} \approx \displaystyle \frac{q_mn^2_1}{\left( 1+\beta _\mathrm{V}\right) \varGamma _\mathrm{V}(n_1+\varGamma _\mathrm{V})}\displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1}, \end{array} \end{aligned}$$

$$\begin{aligned} \tilde{\tilde{C}}(X^{\mathrm{qe}}_1\varSigma )= & {} \tilde{\tilde{c}}_\varSigma \displaystyle \frac{\varGamma _\mathrm{V}+X_{1,0}^{\mathrm{qe}}-n_1}{n_1}, \\ \tilde{\tilde{D}}(X^{\mathrm{qe}}_1\varSigma )= & {} \tilde{\tilde{c}}_\varSigma \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\alpha _2-1}. \end{aligned}$$

Following expressions can be used for estimations:

$$\begin{aligned} \varSigma _q(\theta _\mathrm{V})= & {} \displaystyle \sum ^q_{m=2}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}-1}{m} \le \displaystyle \sum ^q_{m=2}\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}m}}{2}-\displaystyle \sum ^q_{m=2}\displaystyle \frac{1}{m}\\= & {} \displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}(q+1)}-\mathrm{{e}}^{2\theta _\mathrm{V}}}{2(\mathrm{{e}}^{\theta _\mathrm{V}}-1)}-\eta _q, \end{aligned}$$

and at the same time

$$\begin{aligned}&\varSigma _{q_m}(\theta _\mathrm{V})\approx \displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}}{(\mathrm{{e}}^{\theta _\mathrm{V}}-1)(q_m+1)}, \\&\quad \varSigma _{\alpha ,q_m}(\theta _\mathrm{V})\approx \displaystyle \frac{\alpha _2^{q_m+1}\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}}{(\alpha _2\mathrm{{e}}^{\theta _\mathrm{V}}-1)(q_m+1)}, \\&-\varSigma _{q_m}(-\theta _\mathrm{V})\le \eta _{q_m}-\displaystyle \frac{1}{q_m}\displaystyle \sum ^{q_m}_{m=2}\mathrm{{e}}^{-\theta _\mathrm{V}m} \approx \ln {q_m}, \\&\quad \eta _q=\displaystyle \sum ^q_{m=2}\displaystyle \frac{1}{m}\approx \ln {q}. \end{aligned}$$

$$\begin{aligned} \overline{\varSigma }(\theta _\mathrm{V})=-\displaystyle \frac{\varSigma _{q_m}(-\theta _\mathrm{V}) +\varSigma _{q_m}(\theta _\mathrm{V})\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}}{1-\exp {(-\theta _\mathrm{V}(q_m+1))}} \nonumber \\ \le \displaystyle \frac{\ln {q_m}-q_m^{-1}(\mathrm{{e}}^{\theta _\mathrm{V}}-1)^{-1}}{1-\mathrm{{e}}^{-\theta _\mathrm{V}(q_m+1)}} \approx \ln {q_m}. \end{aligned}$$

(233)

Thus, $\overline{\varSigma }(\theta _\mathrm{V})\ll \varSigma _{q_m}(\theta _\mathrm{V})$.

1.2 Expressions for $A_q$

Using expressions (228), (229), $n_1 =\displaystyle \sum \limits _{q=0}^{q_{m}}X_{1,q}$ and $\varGamma _\mathrm{V}=\displaystyle \sum \limits _{q=1}^{q_{m}}qX_{1,q}$ for arbitrary distribution $X_{1,q}$ we obtain

$$\begin{aligned} A_q(X_1)= & {} -\displaystyle \frac{(q_m+1)\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{(1+\beta _\mathrm{V})\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}}\nonumber \\&\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) . \end{aligned}$$

(234)

If $X_{\mathrm{d},q}$ is a correction to the distribution function, for which the relationships $\displaystyle \sum \limits _{q=0}^{q_{m}}X_{\mathrm{d},q}=0$ and $\displaystyle \sum \limits _{q=1}^{q_{m}}qX_{2,q}=0$ take place, the following equation is obtained

$$\begin{aligned} A_q(X_\mathrm{d})= & {} (q_m+1)\displaystyle \frac{X_{\mathrm{d},q_m}}{\varGamma _\mathrm{V}}\displaystyle \frac{\mathrm{{e}}^{-\theta _\mathrm{V}}}{1+\beta _\mathrm{V}}\nonumber \\&\times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) .\nonumber \\ \end{aligned}$$

(235)

For some other functions:

$$\begin{aligned} A_q\left( mX^{\mathrm{qe}}_{1,m}\right) =\displaystyle \frac{1}{1+\beta _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \end{aligned}$$

$$\begin{aligned} A_q\left( X^{\mathrm{qe}}_1\varSigma \right) =\tilde{\tilde{c}}_\varSigma \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \end{aligned}$$

$$\begin{aligned} A_q\left( \displaystyle \frac{\alpha _mX^{\mathrm{qe}}_{1,m}}{\tilde{a}}\right) =\tilde{\tilde{c}}_\alpha \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \end{aligned}$$

(236)

where $\tilde{\varGamma }_V$ was introduced in (231). Thus,

$$\begin{aligned} \begin{array}{l} A_q\left( \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) X^{\mathrm{qe}}_{1,m}\right) \\ \quad =-\displaystyle \frac{{\tilde{\tilde{c}}}_\alpha }{\alpha _2-1} \left( 1+\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}}{\alpha _2\varGamma _\mathrm{V}}-\tilde{a}\right) \\ \\ \qquad \times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) . \end{array} \end{aligned}$$

With $B_q$ from Eq. (223) for $A_q(BX^{\mathrm{qe}}_1)$ we obtain

$$\begin{aligned} \begin{array}{l} A_q(BX^{\mathrm{qe}}_1)=\displaystyle \frac{\kappa _0/n^2_1}{\left( 1+\beta _\mathrm{V}\right) Q^{01}_{10}} \left[ \displaystyle \frac{\alpha _2^2(1+\beta _\mathrm{V})}{\mathrm{{e}}^{\theta _\mathrm{V}}-1} \left( {\tilde{\tilde{c}}}_{\alpha \varSigma } -\displaystyle \frac{{\tilde{\varSigma }}^\prime _\alpha }{\alpha _2}{\tilde{\tilde{c}}}_\alpha \right) \right. \\ \\ \qquad \times \left( \mathrm{{e}}^{\theta _\mathrm{V}}+q_m\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1-\displaystyle \frac{n_1}{\varGamma _\mathrm{V}}\right) \right) \left. -\displaystyle \frac{{\tilde{\tilde{c}}}_\alpha }{\alpha _2-1} \left( 1+\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}}{\alpha _2\varGamma _\mathrm{V}}-\tilde{a}\right) \right] \\ \\ \quad \quad \times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \\ \quad \equiv {A}_B\displaystyle \frac{q_m(1+\delta )\kappa _0}{\left( 1+\beta _\mathrm{V}\right) Q^{01}_{10}\varGamma _\mathrm{V}n_1}\\ \\ \quad \quad \times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \quad \delta =-\displaystyle \frac{{\tilde{\varSigma }}^\prime _\alpha {\tilde{\tilde{c}}}_\alpha }{\alpha _2{\tilde{\tilde{c}}}_{\alpha \varSigma }}, \end{array} \end{aligned}$$

where $\kappa _0$ is defined in (219). At the limit $\beta _\mathrm{V}\rightarrow 0$

$$\begin{aligned} A_\mathrm{B}\approx \tilde{A}_B\rightarrow {A}_{B^\prime }=1+\mathrm{{O}}(\beta _\mathrm{V}). \end{aligned}$$

From the expression (225) it follows that

$$\begin{aligned}&A_q(B_\mathrm{V}X^{\mathrm{qe}}_1)=\displaystyle \frac{\kappa _0}{(1+\beta _\mathrm{V})^2Q^{01}_{10}\varGamma _\mathrm{V}n_1}\\&\quad \quad \times \left[ \displaystyle \frac{{\tilde{\tilde{c}}}_\alpha }{\alpha _2-1} \left( 1+\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}}{\alpha _2\varGamma _\mathrm{V}}-\tilde{a}\right) +\alpha _2^2\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}}{(e^{\theta _\mathrm{V}}-1)n_1}\right. \\&\quad \quad \times \left. \left( {\tilde{\tilde{c}}}_{\alpha \varSigma } -\displaystyle \frac{{\tilde{\varSigma }}^\prime _\alpha }{\alpha _2}{\tilde{\tilde{c}}}_\alpha \right) \right] \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \\&\equiv \displaystyle \frac{\kappa _0A_{\mathrm{BV}}(1+\delta )}{(1+\beta _\mathrm{V})^2Q^{01}_{10}\varGamma _\mathrm{V}n_1} \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) . \end{aligned}$$

Using the same approximations as previously and (117) and (222) we obtain

$$\begin{aligned} A_q(B_\mathrm{V}X^{\mathrm{qe}}_1)\approx & {} \displaystyle \frac{\tilde{A}_{\mathrm{BV}}}{(1+\beta _\mathrm{V})^2Q^{01}_{10}\varGamma _\mathrm{V}(n_1+\varGamma _\mathrm{V})}\\&\times \,\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \nonumber \\ \tilde{A}_{\mathrm{BV}}= & {} 1+O\left( \beta _\mathrm{V}\right) . \end{aligned}$$

From the expression $\varDelta ^\prime _q=\displaystyle \frac{\alpha _q}{\tilde{a}} +\displaystyle \frac{B_{\mathrm{V}q}R_{\mathrm{VV}}({\alpha }X^{\mathrm{qe}}_1/\tilde{a})}{1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)}$ and the expressions for $R_{\mathrm{VV}}$ (see further in this section) we obtain

$$\begin{aligned} \begin{array}{l} A_q(\varDelta ^\prime {X}^{\mathrm{qe}}_1)=A_{\mathrm{X}^\prime } \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \\ A_{\mathrm{X}^\prime }=\tilde{\tilde{c}}_\alpha +\displaystyle \frac{\beta _\mathrm{V}\mathrm{{e}}^{\theta _\mathrm{V}}(1-\mathrm{{e}}^{\gamma _\mathrm{V}})\kappa _0A_{\mathrm{BV}}(1+\delta )}{\alpha _2(1+\beta _\mathrm{V})(1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}})n_1} \approx \tilde{A}_{\mathrm{X}^\prime }\\ =\beta _\mathrm{V}\left( \mathrm{{e}}^{\gamma _\mathrm{V}}-1\right) \left( 1-2\displaystyle \frac{n_1^2}{(n_1+\varGamma _\mathrm{V})^2}-A_{\mathrm{BV}}(1+\delta )+O\left( \beta _\mathrm{V}\right) \right) . \end{array} \end{aligned}$$

(237)

Taking into account that $A^\prime _q(X)=A_q(X)+\displaystyle \frac{B_{\mathrm{V}q}R_{\mathrm{VV}}(A(X)X^{\mathrm{qe}}_1)}{1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)}$ we obtain

$$\begin{aligned} \begin{aligned} A_q^\prime (\varDelta ^\prime {X}^{\mathrm{qe}}_1)&=A_q(\varDelta ^\prime {X}^{\mathrm{qe}}_1) +B_{\mathrm{V}q}\displaystyle \frac{R_{\mathrm{VV}}(A_q(\varDelta ^\prime {X}^{\mathrm{qe}}_1)X^{\mathrm{qe}}_1)}{1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)} \\&=A_{\mathrm{X}^\prime } \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right. \\&\quad \left. -\displaystyle \frac{P_{10}A_\mathrm{R}\left( 1+\beta _\mathrm{V}\right) \varGamma _\mathrm{V}B_{\mathrm{V}q}}{1+\beta _\mathrm{V}\left( 1+\beta _\mathrm{V}(1-A_{\mathrm{RBV}})\right) }\right) . \end{aligned} \end{aligned}$$

Using the expression (225) for $B_{\mathrm{V}q}$, equation for $A_q^\prime$ can be written as

$$\begin{aligned}&A_q^\prime (\varDelta ^\prime {X}^{\mathrm{qe}}_1)\nonumber \\&\quad =A_{\mathrm{X}^\prime } \left( \left( 1+a^\prime _{\mathrm{X}^\prime }\right) \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \right. \nonumber \\&\quad \left. -a^\prime _{\mathrm{X}^\prime }\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}}{n_1(\mathrm{e}^{\theta _\mathrm{V}}-1)} \left( \alpha _2^{q+1}\varSigma _{\alpha ,q}-\displaystyle \frac{\alpha _2^q}{\tilde{a}}{\tilde{\varSigma }}^{\prime }\right) \right) , \nonumber \\&\quad a^\prime _{\mathrm{X}^\prime }=\displaystyle \frac{\beta _\mathrm{V}\mathrm{{e}}^{\theta _\mathrm{v}}\kappa _0A_\mathrm{R}\varGamma _\mathrm{V}/n_1}{1+\beta _\mathrm{V}\left( 1-A_{\mathrm{RBV}}\right) }. \end{aligned}$$

(238)

Thus, for corresponding approximate value we obtain

$$\begin{aligned}&A_q^\prime (\varDelta ^\prime {X}^{\mathrm{qe}}_1)\approx \displaystyle \frac{\beta _\mathrm{V}(1+\beta _\mathrm{V})(\mathrm{{e}}^{\gamma _\mathrm{V}}-1)}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}}\tilde{A}^\prime _{\mathrm{X}^\prime }\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \times \,\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \nonumber \\&\tilde{A}^\prime _{\mathrm{X}^\prime }=\tilde{A}_{\mathrm{X}^\prime }(1+\tilde{a}^\prime _{\mathrm{X}^\prime }), \quad \tilde{a}^\prime _{\mathrm{X}^\prime }=\displaystyle \frac{\beta _\mathrm{V}\tilde{A}_\mathrm{R}}{1+\beta _\mathrm{V}\left( 1-\tilde{A}_\mathrm{R}\right) }. \end{aligned}$$

(239)

From the expression $B_q^\prime =B_q+B_{\mathrm{V}q}\displaystyle \frac{R_{\mathrm{VV}}(BX^{\mathrm{qe}}_1)+q_m}{1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)}$ we obtain

$$\begin{aligned}&A_q(B^\prime {X}^{\mathrm{qe}}_1)=\displaystyle \frac{q_m(1+\delta )\kappa _0A_{\mathrm{B}^\prime }}{(1+\beta _\mathrm{V})Q^{01}_{10}\varGamma _\mathrm{V}n_1}\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \times \,\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) ,\nonumber \\&A_{\mathrm{B}^\prime }=A_\mathrm{B}+A_{\mathrm{BV}}\displaystyle \frac{1-\beta _\mathrm{V}A_{\mathrm{RB}}/(1+\beta _\mathrm{V})}{1+\beta _\mathrm{V}(1-A_{\mathrm{RBV}})}, \nonumber \\&A_{\mathrm{B}^\prime }\approx \tilde{A}_{\mathrm{B}^\prime }=\tilde{A}_\mathrm{B} +\tilde{A}_{\mathrm{BV}}\displaystyle \frac{1-\beta _\mathrm{V}\tilde{A}_\mathrm{RB}/(1+\beta _\mathrm{V})}{1+\beta _\mathrm{V}(11\tilde{A}_\mathrm{R})}, \end{aligned}$$

(240)

where $A_\mathrm{B}$ is defined above in this subsection after the expression for $A_q(BX^{\mathrm{qe}}_1)$ and $A_{\mathrm{RB}}$, $A_{\mathrm{RBV}}$ and $\tilde{A}_\mathrm{R}$ are defined in Eqs. (265), (266) and (249).

$$\begin{aligned}&A_q^\prime (B^\prime {X^{\mathrm{qe}}_1})=\displaystyle \frac{q_m\kappa _0(1+\delta )}{(1+\beta _\mathrm{V})Q^{01}_{10}\varGamma _\mathrm{V}n_1}\nonumber \\&\quad \times \left( \left( \left( 1+a^\prime _{\mathrm{X}^\prime }\right) \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \right) \right. \nonumber \\&\left. -a^\prime _{\mathrm{X}^\prime }\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}}{n_1(\mathrm{e}^{\theta _\mathrm{V}}-1)} \left( \alpha _2^{q+1}\varSigma _{\alpha ,q}-\displaystyle \frac{\alpha _2^q}{\tilde{a}}{\tilde{\varSigma }}^{\prime }\right) \right) . \end{aligned}$$

(241)

$$\begin{aligned} \begin{array}{l} A_q^{\prime }\left( \left( \displaystyle \frac{\alpha ^{m-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{m-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) X^{\mathrm{qe}}_{1,m}\right) \\ \quad =-\displaystyle \frac{\tilde{c}_\alpha }{\alpha _2-1} \quad \quad \left( 1-\beta _\mathrm{V}\left( \mathrm{{e}}^{\gamma _\mathrm{V}}-1\right) \displaystyle \frac{\tilde{a}}{\alpha _2}\right) \\ \qquad \times \left( \left( \left( 1+a^\prime _{\mathrm{X}^\prime }\right) \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \right) \right. \\ \qquad \left. -a^\prime _{\mathrm{X}^\prime }\displaystyle \frac{(q_m+1)X^{\mathrm{qe}}_{1,q_m}}{n_1(\mathrm{e}^{\theta _\mathrm{V}}-1)} \left( \alpha _2^{q+1}\varSigma _{\alpha ,q}-\displaystyle \frac{\alpha _2^q}{\tilde{a}}{\tilde{\varSigma }}^{\prime }\right) \right) , \end{array} \end{aligned}$$

$$\begin{aligned}&A_q^{\prime }\left( \left( \displaystyle \frac{\alpha ^{m-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{m-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) X^{\mathrm{qe}}_{1,m}\right) \nonumber \\&\quad \approx -\displaystyle \frac{\tilde{c}_\alpha }{\alpha _2-1} \left( 1-\beta _\mathrm{V}\left( \mathrm{{e}}^{\gamma _\mathrm{V}}-1\right) \displaystyle \frac{\tilde{a}}{\alpha _2}\right) \left( 1+\displaystyle \frac{\beta _\mathrm{V}\tilde{A}_\mathrm{R}}{1+\beta _\mathrm{V}(1-\tilde{A}_\mathrm{R})}\right) \nonumber \\&\times \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) \nonumber \\&\equiv \kappa \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) . \end{aligned}$$

(242)

The zeroth-order terms within the iterative procedure proposed at the end of the Sect. 4.2.1 read as

$$\begin{aligned}&A_q^{\prime {k}}(X^{\mathrm{qs}}_{1,(0)}) \approx \tilde{A}^\prime _{\mathrm{XB}}A_q^{\prime {k-1}}\nonumber \\&\quad \quad \times \,\left( \left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) X^{\mathrm{qe}}_{1,m}\right) \nonumber \\&\quad =\tilde{A}^\prime _{\mathrm{XB}}\kappa ^{k-1}\left( \displaystyle \frac{\alpha ^{q-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/(\varGamma _\mathrm{V}\alpha _2)-1}{\alpha _2-1}\right) , \nonumber \\&X^{\mathrm{qs}}_{1,q(0)}=\left( \varDelta ^\prime _q+R_{1(0)}B_q^\prime \right) X^{\mathrm{qe}}_{1,q}, \nonumber \\&\tilde{A}^\prime _{\mathrm{XB}}=\displaystyle \frac{\beta _\mathrm{V}(1+\beta _\mathrm{V})(\mathrm{{e}}^{\gamma _\mathrm{V}}-1)}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}}\tilde{A}^\prime _{\mathrm{X}^\prime }\nonumber \\&\quad \quad \quad \quad +\displaystyle \frac{q_mR_{1(0)}}{(1+\beta _\mathrm{V})^2Q^{01}_{10}\varGamma _\mathrm{V}(n_1+\varGamma _\mathrm{V})}, \nonumber \\&R_{1(0)}=-\displaystyle \frac{P_{\mathrm{d},q_{m}}\varDelta ^\prime _{q_m}X^{\mathrm{qe}}_{1,q_m} -P_{\mathrm{r},q_{m}}n_3n_4}{1+P_{\mathrm{d},q_m}B_{q_m}^{\prime }X^{\mathrm{qe}}_{1,q_m}}, \end{aligned}$$

(243)

where $\tilde{A}^\prime _{\mathrm{X}^\prime }$ is defined in (239). Introducing the quasi-equilibrium reaction rate, $R_1^{\mathrm{qe}}$, and the non-equilibrium factor, F, we can represent the expression for $R_{(0)}$ as

$$\begin{aligned}&R_{1(0)}=R_1^{\mathrm{qe}}F, \quad R_1^{\mathrm{qe}}=-\left( P_{\mathrm{d},q_{m}}X^{\mathrm{qe}}_{1,q_m}-P_{\mathrm{r},q_{m}}n_3n_4,\right) ,\nonumber \\&F=\displaystyle \frac{1}{1+P_{\mathrm{d},q_m}B_{q_m}^{\prime }X^{\mathrm{qe}}_{1,q_m}}. \end{aligned}$$

For small $\beta _\mathrm{V}$ expressions for $A^\prime _{\mathrm{XB}}$ can be simplified

$$\begin{aligned} \begin{array}{l} A^\prime _{\mathrm{XB}}=\beta _\mathrm{V}A^\prime _{\mathrm{XB}0}\left( 1+O\left( \beta _\mathrm{V}\right) \right) , \\ A^\prime _{\mathrm{XB}0}=\mathrm{{e}}^{\gamma _\mathrm{V}}-1+\displaystyle \frac{q_mR_{1(0)}}{P_{10}\Gamma _\mathrm{V}}. \end{array} \end{aligned}$$

(244)

1.3 Expressions for $R_{\mathrm{VV}}$

$$\begin{aligned}&R_{\mathrm{VV}}(X_1)\\&\quad =-\displaystyle \sum \limits _{q=0}^{q_m-1} \left( \tilde{j}_q(X_1^{\mathrm{qe}})X_1+\tilde{j}_q^{\,\prime }(X_1)X_1^{\mathrm{qe}} -\tilde{j}_q^{\,\prime }(X_1^{\mathrm{qe}})X_1^{\mathrm{qe}}\right) . \end{aligned}$$

$$\begin{aligned} R_{\mathrm{VV}}(X^{\mathrm{qe}}_1)= & {} \left( \mathrm{{e}}^{\gamma _\mathrm{V}}-1\right) \displaystyle \sum _{q=1}^{q_m-1}P_{q+1,q}X^{\mathrm{qe}}_{1,q}\nonumber \\= & {} \left( \mathrm{{e}}^{\gamma _\mathrm{V}}-1\right) P_{10}\varGamma _\mathrm{V}. \end{aligned}$$

(245)

$$\begin{aligned}&R_{\mathrm{VV}}((q-1)X_{1,q}^{\mathrm{qe}})\nonumber \\&\quad =-P_{10}\left( \left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) \displaystyle \sum \limits _{q=0}^{q_m-1}(q+1)qX_{1,q+1}^{\mathrm{qe}}\right. \nonumber \\&\quad \quad \left. +\mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \sum \limits _{q=0}^{q_m-1}(q+1)X_{1,q+1}^{\mathrm{qe}}\right) \nonumber \\&\quad =-P_{10}\left( \left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) (\varGamma ^\prime _V-\varGamma _\mathrm{V})+\mathrm{{e}}^{\gamma _\mathrm{V}}\varGamma _\mathrm{V}\right) . \end{aligned}$$

(246)

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( \displaystyle \frac{\alpha ^{m}_2X^{\mathrm{qe}}_{1,m}}{\tilde{a}}\right) =P_{10}\displaystyle \frac{(\mathrm{{e}}^{\gamma _\mathrm{V}}-1)\tilde{\varGamma }_\mathrm{V}}{\tilde{a}\left( 1+\beta _\mathrm{V}\mathrm{{e}}^{{\gamma _\mathrm{V}}}\right) }. \end{array} \end{aligned}$$

(247)

From (246) and (247) it follows

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( \left( \displaystyle \frac{\alpha ^{m-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{m-1}_2}{\tilde{a}}\displaystyle \frac{\tilde{\varGamma }_\mathrm{V}/\alpha _2-\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}}\right) X^{\mathrm{qe}}_{1,m}\right) \\ =-\displaystyle \frac{P_{10}\varGamma _\mathrm{V}}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}}\left[ \mathrm{{e}}^{\gamma _\mathrm{V}} +\beta _\mathrm{V}^{-1}\alpha ^{-1}_2\left( 1-\displaystyle \frac{\tilde{a}}{\alpha _2}\right) \right] \equiv -P_{10}\varGamma _\mathrm{V}A_\mathrm{R}, \end{array} \end{aligned}$$

(248)

and

$$\begin{aligned} \varGamma ^\prime _\mathrm{V}=\displaystyle \sum \limits _{q=1}^{q_m}q^2X_{1,q}^{\mathrm{qe}} \approx \displaystyle \frac{\varGamma _\mathrm{V}(n_1+\varGamma _\mathrm{V})}{n_1} \approx \displaystyle \frac{n_1\mathrm{{e}}^{\theta _\mathrm{V}}}{(\mathrm{{e}}^{\theta _\mathrm{V}}-1)^2}, \end{aligned}$$

$$\begin{aligned} \begin{array}{l} A_\mathrm{R}\approx \tilde{A}_\mathrm{R}=\mathrm{e}^{\gamma _\mathrm{V}}+(1-\mathrm{{e}}^{\gamma _\mathrm{V}})\tilde{a}_\mathrm{R}, \\ \tilde{a}_\mathrm{R}=\displaystyle \frac{\alpha _2^{-1}}{\mathrm{{e}}^{\theta _\mathrm{V}}-\alpha _2} =\displaystyle \frac{\varGamma _\mathrm{V}}{n_1}(1+\mathrm{{O}}(\beta _\mathrm{V})), \end{array} \end{aligned}$$

(249)

where (232) is used for $\tilde{\varGamma }_\mathrm{V}$. Taking into account (234) this results in

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( A\left( X_1\right) X^{\mathrm{qe}}_1\right) =\displaystyle \frac{(q_m+1)P_{10}\left( X_{1,q_m}^{\mathrm{qe}}-X_{1,q_m}\right) }{\mathrm{{e}}^{\theta _\mathrm{V}}(1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}})} \\ \times \left( \mathrm{{e}}^{\gamma _\mathrm{V}}+\displaystyle \frac{1-{\tilde{a}}/{\alpha _2}}{\beta _\mathrm{V}\alpha _2}\right) \equiv -\displaystyle \frac{(q_m+1)P_{10}A_\mathrm{R}}{(1+\beta _\mathrm{V})\mathrm{{e}}^{\theta _\mathrm{V}}}\left( X_{1,q_m}-X^{\mathrm{qe}}_{1,q_m}\right) . \end{array} \end{aligned}$$

(250)

Similarly, for (235) the following expression is obtained

$$\begin{aligned} R_{\mathrm{VV}}\left( A(X_\mathrm{d})X^{\mathrm{qe}}_1\right) =-\displaystyle \frac{(q_m+1)P_{10}A_\mathrm{R}}{(1+\beta _\mathrm{V})\mathrm{{e}}^{\theta _\mathrm{V}}}X_{\mathrm{d},q_m}. \end{aligned}$$

(251)

Substitution of the expression (124) for $X^{\mathrm{qs}}_{1,q}$ into (250) leads to the expression:

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( A\left( X^{\mathrm{qs}}_1\right) X^{\mathrm{qe}}_1\right) =-Q^{01}_{10}\varGamma _\mathrm{V}G\left( 1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)\right) \\ \times \left( \varDelta ^\prime _{q_m}-1+A_{q_m}(X^{\mathrm{qs}}_1)+R_1B^\prime _{q_m}\right) , \\ G=\displaystyle \frac{(q_m+1)\beta _\mathrm{V}A_\mathrm{R}X^{\mathrm{qe}}_{1,q_m}}{(1+\beta _\mathrm{V})(1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1))+(q_m+1)\beta _\mathrm{V}A_\mathrm{R}\tilde{B}_{\mathrm{V}q_m}X^{\mathrm{qe}}_{1,q_m}}, \end{array} \end{aligned}$$

(252)

where $\tilde{B}_{\mathrm{V}q}$ is defined in (225), so that the distribution $X^{\mathrm{qs}}_{1,q}$ can now be written as

$$\begin{aligned} X^{\mathrm{qs}}_{1,q}= & {} \left( \varDelta ^\prime _{q}+A_{q}(X^{\mathrm{qs}}_1)+R_1B^\prime _{q} -G\tilde{B}_{\mathrm{V}q}\right. \nonumber \\&\times \left. \left( \varDelta ^\prime _{q_m}-1+A_{q_m}(X^{\mathrm{qs}}_1)+R_1B^\prime _{q_m}\right) \right) X^{\mathrm{qe}}_{1,q}. \end{aligned}$$

(253)

After substituting this expression into (234) for $A_{q}(X^{\mathrm{qs}}_{1})$, taking $q=q_m$ and resolving the resulting equation over $A_{q_m}(X^{\mathrm{qs}}_{1})$, the following expression can be obtained

$$\begin{aligned} \begin{array}{l} A_{q_m}(X^{\mathrm{qs}}_{1})=-\displaystyle \frac{Q}{Q+1}\left( \varDelta ^\prime _{q_m}-1+R_1B^\prime _{q_m}\right) , \\ Q=\displaystyle \frac{(q_m+1)(G\tilde{B}_{\mathrm{V}q_m}-1)}{(1+\beta _\mathrm{V})\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}}\\ \quad \times \left( \displaystyle \frac{\alpha ^{q_m-1}_2-1}{\alpha _2-1} -\displaystyle \frac{\alpha ^{q_m-1}_2}{\tilde{a}} \displaystyle \frac{\tilde{a}/\alpha _2-1}{\alpha _2-1}\right) X^{\mathrm{qe}}_{1,q_m}. \end{array} \end{aligned}$$

(254)

After substituting this expression into (252), $R_{\mathrm{VV}}(A\left( X^{\mathrm{qs}}_1\right) X^{\mathrm{qe}}_1)$ reads as

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( A\left( X^{\mathrm{qs}}_1\right) X^{\mathrm{qe}}_1\right) =-\displaystyle \frac{Q^{01}_{10}\varGamma _\mathrm{V}G}{Q+1}\left( 1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)\right) \\ \times \left( \varDelta ^\prime _{q_m}-1+R_1B^\prime _{q_m}\right) . \end{array} \end{aligned}$$

(255)

It results with the expression

$$\begin{aligned} \begin{array}{c} A^\prime _{q_m}(X^{\mathrm{qs}}_{1})=-\displaystyle \frac{Q+G\tilde{B}_{\mathrm{V}q_m}}{Q+1}\left( \varDelta ^\prime _{q_m}-1+R_1B^\prime _{q_m}\right) . \end{array} \end{aligned}$$

(256)

Similar algebra leads to the equations

$$\begin{aligned}&R_{\mathrm{VV}}\left( A\left( X^{\mathrm{qs}(0)}_\mathrm{d}\right) X^{\mathrm{qe}}_1\right) =-Q^{01}_{10}\varGamma _\mathrm{V}G\left( 1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)\right) \nonumber \\&\quad \times \left( \varDelta ^{(0)\prime }_{q_m}+A_{q_m}(X^{\mathrm{qs}(0)}_\mathrm{d})+R_\mathrm{d}^{(0)}B^\prime _{q_m}\right) , \end{aligned}$$

(257)

$$\begin{aligned} \begin{array}{l} X^{\mathrm{qs}(0)}_{\mathrm{d},q}=\left( \varDelta ^{(0)\prime }_{q}+A_{q}(X^{\mathrm{qs}(0)}_\mathrm{d})+R_\mathrm{d}^{(0)}B^\prime _{q}\right. \\ \left. -G\tilde{B}_{\mathrm{V}q}\left( \varDelta ^{(0)\prime }_{q_m}+A_{q_m}(X^{\mathrm{qs}(0)}_\mathrm{d}) +R_\mathrm{d}^{(0)}B^\prime _{q_m}\right) \right) X^{\mathrm{qe}}_{1,q}, \end{array} \end{aligned}$$

(258)

$$\begin{aligned} \begin{array}{l} A_{q_m}(X^{\mathrm{qs}(0)}_\mathrm{d})=-\displaystyle \frac{Q}{Q+1}\left( \varDelta ^{(0)\prime }_{q_m}+R_\mathrm{d}^{(0)}B^\prime _{q_m}\right) , \end{array} \end{aligned}$$

(259)

$$\begin{aligned} \begin{array}{l} A^\prime _{q_m}(X^{\mathrm{qs}(0)}_\mathrm{d})=-\displaystyle \frac{Q+G\tilde{B}_{\mathrm{V}q_m}}{Q+1} \left( \varDelta ^{(0)\prime }_{q_m}+R_\mathrm{d}^{(0)}B^\prime _{q_m}\right) , \end{array} \end{aligned}$$

(260)

where $\varDelta ^{(0)}$ and $A_q(X^{\mathrm{qs}}_\mathrm{d})$ are determined in (227) and (235) respectively. Thus,

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}\left( A\left( X^{\mathrm{qs}(0)}_\mathrm{d}\right) X^{\mathrm{qe}}_1\right) \\ \quad =-\displaystyle \frac{Q^{01}_{10}\varGamma _\mathrm{V}G}{Q+1}\left( 1-R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)\right) \left( \varDelta ^{(0)\prime }_{q_m}+R^{(0)}_\mathrm{d}B^\prime _{q_m}\right) . \end{array} \end{aligned}$$

(261)

For estimations, the following expressions can be useful:

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}(X^{\mathrm{qe}}_{1,q}\alpha _2^{q+1}\varSigma (\theta _\mathrm{V})) =-{\alpha _2}{P_{10}\varGamma _\mathrm{V}} \left( \displaystyle \frac{1-\mathrm{{e}}^{\gamma _\mathrm{V}}}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}^\prime }{\varGamma _\mathrm{V}}\right. \\ \quad \left. +\mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q_m}_2-1}{\alpha _2-1} -\displaystyle \frac{1-\mathrm{{e}}^{-q_m\tilde{\theta }_\mathrm{V}}}{\mathrm{{e}}^{\theta _\mathrm{V}}-\alpha _2}\right) \right) . \end{array} \end{aligned}$$

$$\begin{aligned} \tilde{\varGamma }_\mathrm{V}^\prime= & {} \displaystyle \sum _{q=1}^{q_m}q\alpha _2^q\varSigma _{\alpha ,q}X^{\mathrm{qe}}_{1,q} \approx \varGamma _\mathrm{V}q_m\displaystyle \frac{(\mathrm{{e}}^{\theta _\mathrm{V}}-1)^2}{\alpha _2({\alpha _2\mathrm {e}}^{\theta _\mathrm{V}}-1)}, \nonumber \\&\tilde{\theta }_\mathrm{V}=\theta _\mathrm{V}-\ln {\alpha _2}. \end{aligned}$$

(262)

$$\begin{aligned}&R_{\mathrm{VV}}(BX^{\mathrm{qe}}_1)\nonumber \\&\quad =-\displaystyle \frac{\beta _\mathrm{V}\kappa _0\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma ^2_\mathrm{V}}{\left( 1+\beta _\mathrm{V}\right) n^2_1}\Biggl [A_\mathrm{R}+\displaystyle \frac{q_m\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1-{n_1}/{\varGamma _\mathrm{V}}\right) +\mathrm{{e}}^{\theta _\mathrm{V}}}{\alpha ^{-1}_2(\mathrm{{e}}^{\theta _\mathrm{V}}-1)} \nonumber \\&\quad \quad \times \left( \mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q_m}_2-1}{\alpha _2-1} -\displaystyle \frac{1-\mathrm{{e}}^{-q_m\tilde{\theta }_\mathrm{V}}}{\mathrm{{e}}^{\theta _\mathrm{V}}-\alpha _2}\right) \right. \nonumber \\&\quad \quad \left. +\displaystyle \frac{1-\mathrm{{e}}^{\gamma _\mathrm{V}}}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}} \left( \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}^\prime }{\varGamma _\mathrm{V}} -\displaystyle \frac{\tilde{\varSigma }^\prime }{\alpha _2}\right) \right) \Biggr ], \end{aligned}$$

(263)

where $\kappa _0$ is introduced in (219). From the assumption that $q_m\gg 1$ it follows

$$\begin{aligned} R_{\mathrm{VV}}(BX^{\mathrm{qe}}_1)\approx & {} -\displaystyle \frac{\beta _\mathrm{V}}{1+\beta _\mathrm{V}} \left[ \mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \frac{\alpha ^{q_m}_2-1}{\alpha _2-1} -{\beta _\mathrm{V}\alpha _2\left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) ^2}\right. \nonumber \\&\left. \displaystyle \frac{1+\varGamma _\mathrm{V}/n_1}{(1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}})^2} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}^\prime }{\varGamma _\mathrm{V}}\right] \nonumber \\= & {} -\beta _\mathrm{V}q_m\left( \mathrm{{e}}^{\gamma _\mathrm{V}} +\mathrm{{O}}\left( \beta _\mathrm{V}\right) \right) , \end{aligned}$$

(264)

One more representation for $R_{\mathrm{VV}}(BX^{\mathrm{qe}}_1)$ in (263) reads as

$$\begin{aligned} R_{\mathrm{VV}}(BX^{\mathrm{qe}}_1) \equiv -\displaystyle \frac{\beta _\mathrm{V}}{1+\beta _\mathrm{V}}q_mA_{\mathrm{RB}}, \end{aligned}$$

(265)

$$\begin{aligned} \begin{array}{l} A_{\mathrm{RB}}\approx \tilde{A}_\mathrm{RB}\\ =q_m^{-1}\left[ \mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \frac{\alpha ^{q_m}_2-1}{\alpha _2-1} -{\beta _\mathrm{V}\alpha _2\left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) ^2} \displaystyle \frac{1+\varGamma _\mathrm{V}/n_1}{(1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}})^2} \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}^\prime }{\varGamma _\mathrm{V}}\right] \\ =\mathrm{{e}}^{\gamma _\mathrm{V}} +\mathrm{{O}}\left( \beta _\mathrm{V}\right) . \end{array} \end{aligned}$$

$$\begin{aligned}&R_{\mathrm{VV}}(B_\mathrm{V}X^{\mathrm{qe}}_1)=\displaystyle \frac{\beta _\mathrm{V}\kappa _0\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}}{\left( 1+\beta _\mathrm{V}\right) n_1} \left[ A_\mathrm{R}-\displaystyle \frac{\mathrm{{e}}^{\theta _\mathrm{V}}-1-{n_1}/{\varGamma _\mathrm{V}}}{\alpha _2^{-2}(\mathrm{{e}}^{\theta _\mathrm{V}}-1)}\right. \nonumber \\&\left. \times \left( \mathrm{{e}}^{\gamma _\mathrm{V}}\displaystyle \frac{X^{\mathrm{qe}}_{1,0}}{\varGamma _\mathrm{V}} \left( \displaystyle \frac{\alpha ^{q_m}_2-1}{\alpha _2-1} -\displaystyle \frac{1-\mathrm{{e}}^{-q_m\tilde{\theta }_\mathrm{V}}}{\mathrm{{e}}^{\theta _\mathrm{V}}-\alpha _2}\right) \right. \right. \nonumber \\&\left. \left. +\displaystyle \frac{1-\mathrm{{e}}^{\gamma _\mathrm{V}}}{1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}} \left( \displaystyle \frac{\tilde{\varGamma }_\mathrm{V}^\prime }{\varGamma _\mathrm{V}}-\displaystyle \frac{\tilde{\varSigma }^\prime }{\alpha _2}\right) \right) \right] \nonumber \\&\equiv \displaystyle \frac{\beta _\mathrm{V}}{1+\beta _\mathrm{V}}A_{\mathrm{RBV}} {\approx }\displaystyle \frac{\beta _\mathrm{V}}{1+\beta _\mathrm{V}}\tilde{A}_\mathrm{R}, \end{aligned}$$

(266)

where $\tilde{A}_\mathrm{R}$ is introduced in (249).

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}(\varDelta ^{(0)}X^{\mathrm{qe}}_1)=\displaystyle \frac{\beta _\mathrm{V}\varGamma ^2_VA_{\mathrm{R}\Delta }}{(1+\beta _\mathrm{V})n_1c_{\mathrm{V}0}\left( \mathrm{{e}}^{\theta _\mathrm{V}}-1\right) }, \\ A_{R\Delta }=A_\mathrm{R}+\displaystyle \frac{q_m+1}{\mathrm{{e}}^{\theta _\mathrm{V}(q_m+1)}-1} R_{\mathrm{VV}}\left( \alpha _2^{q+1}\varSigma _{\alpha ,q} -\displaystyle \frac{\alpha _2^{q}}{\tilde{a}}\tilde{\varSigma }_\alpha \right) \approx \tilde{A}_\mathrm{R}. \\ \end{array} \end{aligned}$$

(267)

where $\varDelta ^{(0)}_q$ is introduced in (227).

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}(A(BX^{\mathrm{qe}}_1)X^{\mathrm{qe}}_1) =-\displaystyle \frac{\beta _\mathrm{V}q_m\kappa _0\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}A_\mathrm{B}A_\mathrm{R}(1+\delta )}{(1+\beta _\mathrm{V})^2n_1} \\ \approx -\displaystyle \frac{q_m\beta _\mathrm{V}(1+\delta )}{(1+\beta _\mathrm{V})^2}\tilde{A}_\mathrm{B}\tilde{A}_\mathrm{R}\\ =-\displaystyle \frac{q_m\beta _\mathrm{V}n_1}{n_1+\varGamma _\mathrm{V}}(1+\delta )\left( \mathrm{{e}}^{\gamma _\mathrm{V}} +\left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) \displaystyle \frac{n_1+\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}} +\mathrm{{O}}\left( \beta _\mathrm{V}\right) \right) , \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}(A(B_\mathrm{V}X^{\mathrm{qe}}_1)X^{\mathrm{qe}}_1) =\displaystyle \frac{\beta _\mathrm{V}\kappa _0\mathrm{{e}}^{\theta _\mathrm{V}}\varGamma _\mathrm{V}}{(1+\beta _\mathrm{V})^2n_1}A_{\mathrm{BV}}A_\mathrm{R}(1+\delta ) \\ \approx \displaystyle \frac{\beta _\mathrm{V}}{(1+\beta _\mathrm{V})^2}\tilde{A}_{\mathrm{BV}}\tilde{A}_\mathrm{R}(1+\delta )\\ =\beta _\mathrm{V}(1+\delta )\left( \mathrm{{e}}^{\gamma _\mathrm{V}} +\left( 1-\mathrm{{e}}^{\gamma _\mathrm{V}}\right) \displaystyle \frac{n_1+\varGamma _\mathrm{V}}{\varGamma _\mathrm{V}} +\mathrm{{O}}\left( \beta _\mathrm{V}\right) \right) . \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} R_{\mathrm{VV}}(A(\varDelta ^\prime {X}^{\mathrm{qe}}_1)X^{\mathrm{qe}}_1)\\ \quad =-P_{10}A_\mathrm{R}A_{\mathrm{X}^\prime }\varGamma _\mathrm{V} \displaystyle \frac{\beta _\mathrm{V}(1+\beta _\mathrm{V})(1-\mathrm{{e}}^{\gamma _\mathrm{V}})n_1^2(3n_1+2\varGamma _\mathrm{V})}{\left( 1+\beta _\mathrm{V}\mathrm{{e}}^{\gamma _\mathrm{V}}\right) (n_1+\varGamma _\mathrm{V})^3}, \end{array} \end{aligned}$$

(268)

where $A_{X'}$ is defined in (237), and $A_\mathrm{R}$ in (248).