Local Resistivities
We need to relate the resistivities \(r^{e}\) from Eq. 8.23 to the resistivities \(r^{\,\prime}\) from Eq. 8.29. This is done by comparing the coefficients at the same fluxes in these equations. To do this we need to translate the set of fluxes used in Eq. 8.29
\(\{J_{q}^{\,\prime},\,J_{1}, \ldots, J_{n-1}\}\), to the set of fluxes used in Eq. 8.23, \(\{J_{e},\,J_{\xi_{1}}, \ldots, J_{\xi_{n}}\}\). This is done with the help of the relation
$$ \begin{aligned} & J_{i} = {J_{\xi}}_{i} - \xi_{i}\sum_{k=1}^{n}{J_{\xi_{k}}}\\ & J_{q}^{\,\prime}= J_{e} - \sum_{k=1}^{n}{\tilde{h}_{k}\,J_{\xi_{k}}}\\ \end{aligned} $$
(8.40)
Substituting \(J_{q}^{\,\prime}\) and \(J_{i}\) into the first of Eq. 8.29 we obtain
$$ \nabla_{\perp}\frac{1}{T} = r^{\,\prime}_{qq}\,J_{e} + \sum\limits_{i=1}^{n-1}{J_{\xi}}_{i}\left( r^{\,\prime}_{qi} - r^{\,\prime}_{qq}\,\tilde{h}_{i} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}}\right) - J_{\xi_{n}}\left(r^{\,\prime}_{qq}\,\tilde{h}_{n} + \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}}\right) $$
(8.41)
Comparing it with the first of Eq. 8.23 we obtain
$$ \begin{aligned} & r^{e}_{qq} = r^{\,\prime}_{qq} \\ & r^{e}_{qi} = - r^{\,\prime}_{qq}\,\tilde{h}_{i} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}} + r^{\,\prime}_{qi},\quad i=\overline{1,n-1} \\ & r^{e}_{qn} = - r^{\,\prime}_{qq}\,\tilde{h}_{n} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}} \\ \end{aligned} $$
(8.42)
which are the first three equations of Eq. 8.32.
In order to obtain the remaining relations we consider the second of Eq. 8.23, which gives
$$ \begin{aligned}
-\sum\limits_{j=1}^{n}{\xi_{j}\,\nabla_{\perp}\frac{\tilde{\mu}_{j}}{T}}
&=J_{e}\sum_{j=1}^{n}{r^{e}_{jq}\,\xi_{j}} +
\sum\limits_{i=1}^{n}{J_{\xi}}_{i}\sum_{j=1}^{n}{r^{e}_{ji}\,\xi_{j}}
\\ \left(-\nabla_{\perp}\frac{\tilde{\mu}_{j}}{T}\right) -
\left(-\nabla_{\perp}\frac{\tilde{\mu}_{n}}{T}\right) &=
J_{e}(r^{e}_{jq}-r^{e}_{nq}) +
\sum\limits_{i=1}^{n}{J_{\xi}}_{i}(r^{e}_{ji}-r^{e}_{ni}),\quad
j=\overline{1,n-1} \end{aligned}$$
(8.43)
Furthermore, we use Eq. 8.8. Together with the second of Eq. 8.29 it gives
$$ \begin{aligned} & -\sum\limits_{i=1}^{n}{\xi_{i}\,\nabla_{\perp}\frac{\tilde{\mu}_{i}}{T}} = -\sum\limits_{i=1}^{n}{\xi_{i}\,\tilde{h}_{i}\,\nabla_{\perp}\frac{1}{T}}\\ & -\nabla_{\perp}\frac{\psi_{j}}{T} = -\eta_{j}\,\nabla_{\perp}\frac{1}{T} + r^{\,\prime}_{jq}\,J_{q}^{\,\prime} + \sum\limits_{i=1}^{n-1}{r^{\,\prime}_{ji}\,J_{i}} \end{aligned} $$
(8.44)
Substituting \(\nabla_{\perp}(1/T)\) from Eq. 8.41 and \(J_{q}^{\,\prime}\) and \(J_{i}\) from Eq. 8.40 we obtain the left-hand side of Eq. 8.44 expressed in terms of the fluxes \(J_{e}\) and \({J_{\xi}}_{i}\) and the resistivities \(r^{\,\prime}\). Comparing the result with Eq. 8.43 we obtain the following equations sets
$$ \begin{aligned} & \sum\limits_{k=1}^{n}{r^{e}_{kq}\,\xi_{k}} = - r^{\,\prime}_{qq}\sum\limits_{k=1}^{n}{\xi_{k}\,\tilde{h}_{k}}\\ & r^{e}_{jq} - r^{e}_{nq} = - r^{\,\prime}_{qq}\,\eta_{j} + r^{\,\prime}_{jq}, \quad j=\overline{1,n-1} \\ \end{aligned} $$
(8.45a)
$$ \begin{aligned} \sum\limits_{k=1}^{n}{r^{e}_{kj}\,\xi_{k}}
&= \left(r^{\,\prime}_{qq}\,\tilde{h}_{i} +
\sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}} -
r^{\,\prime}_{qi}\right)
\sum\limits_{k=1}^{n}{\xi_{k}\,\tilde{h}_{k}}\\ r^{e}_{ji} -
r^{e}_{ni} &= \left(r^{\,\prime}_{qq}\,\tilde{h}_{i} +
\sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\,\xi_{k}} -
r^{\,\prime}_{qi}\right)\eta_{j}\\ \quad -
\sum\limits_{k=1}^{n-1}{r^{\,\prime}_{jk}\,\xi_{k}} -
r^{\,\prime}_{jq}\,\tilde{h}_{i} + r^{\,\prime}_{ji}, \quad
j,i=\overline{1,n-1}\\ \end{aligned} $$
(8.45b)
$$ \begin{aligned} & \sum\limits_{k=1}^{n}{r^{e}_{kn}\xi_{k}} = \left(r^{\,\prime}_{qq}\,\tilde{h}_{n} + \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\xi_{k}}\right) \sum\limits_{k=1}^{n}{\xi_{k}\,\tilde{h}_{k}}\\ & r^{e}_{jn} - r^{e}_{nn} = \left(r^{\,\prime}_{qq}\,\tilde{h}_{n} + \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{qk}\xi_{k}}\right)\eta_{j} \\ &\quad -\sum\limits_{k=1}^{n-1}{r^{\,\prime}_{jk}\xi_{k}} - r^{\,\prime}_{jq}\,\tilde{h}_{n}, \quad j=\overline{1,n-1}\\ \end{aligned} $$
(8.45c)
solving which we obtain the relations (8.46) between the remaining resistivities.
As one can confirm the symmetry of the \(r^{\,\prime}\)-matrix leads to the symmetry of the \(r^{e}\)-matrix and vice versa. We therefore do not give the expressions for \(r^{e}_{jq},r^{e}_{nq}\) and \(r^{e}_{jn}\) in Eq. 8.32.
The relations (8.33) between the \(r^{e}\)- and \(r\)-resistivities are derived in the similar manner.
$$ \begin{aligned} & r^{e}_{jq} = -r^{\,\prime}_{qq}\,\tilde{h}_{j} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{kq}\,\xi_{k}} + r^{\,\prime}_{jq},\quad j=\overline{1,n-1}\\ & r^{e}_{nq} = -r^{\,\prime}_{qq}\,\tilde{h}_{n} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{kq}\,\xi_{k}} \\ & r^{e}_{ji} = r^{\,\prime}_{qq}\,\tilde{h}_{j}\tilde{h}_{i} + \sum\limits_{k=1}^{n-1}{\xi_{k}\left(r^{\,\prime}_{kq}\,\tilde{h}_{i}+r^{\,\prime}_{qk}\,\tilde{h}_{j}\right)} -\left(r^{\,\prime}_{jq}\,\tilde{h}_{i}+r^{\,\prime}_{qi}\,\tilde{h}_{j}\right) \\ &\quad + \sum\limits_{k=1}^{n-1}\sum\limits_{l=1}^{n-1}{r^{\,\prime}_{kl}\,\xi_{k}\xi_{l}} - \sum\limits_{k=1}^{n-1}{\xi_{k}\left(r^{\,\prime}_{ki} + r^{\,\prime}_{jk}\right)} + r^{\,\prime}_{ji} , \quad j,i=\overline{1,n-1}\\ & r^{e}_{jn} = r^{\,\prime}_{qq}\,\tilde{h}_{j}\tilde{h}_{n} + \sum\limits_{k=1}^{n-1}{\xi_{k}\left(r^{\,\prime}_{kq}\,\tilde{h}_{n}+r^{\,\prime}_{qk}\,\tilde{h}_{j}\right)} - r^{\,\prime}_{jq}\,\tilde{h}_{n} \\ &\quad + \sum\limits_{k=1}^{n-1}\sum\limits_{l=1}^{n-1}{r^{\,\prime}_{kl}\,\xi_{k}\xi_{l}} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{jk}\,\xi_{k}}, \quad j=\overline{1,n-1}\\ & r^{e}_{ni} = r^{\,\prime}_{qq}\,\tilde{h}_{n}\tilde{h}_{i} + \sum\limits_{k=1}^{n-1}{\xi_{k}\left(r^{\,\prime}_{kq}\,\tilde{h}_{i}+r^{\,\prime}_{qk}\,\tilde{h}_{n}\right)} - r^{\,\prime}_{qi}\,\tilde{h}_{n} \\ &\quad + \sum\limits_{k=1}^{n-1}\sum\limits_{l=1}^{n-1}{r^{\,\prime}_{kl}\,\xi_{k}\xi_{l}} - \sum\limits_{k=1}^{n-1}{r^{\,\prime}_{ki}\,\xi_{k}}, \quad i=\overline{1,n-1}\\ & r^{e}_{nn} = r^{\,\prime}_{qq}\,\tilde{h}_{n}^{2} + \tilde{h}_{n}\sum\limits_{k=1}^{n-1}\,{\xi_{k}\left(r^{\,\prime}_{kq}+r^{\,\prime}_{qk}\right)} + \sum\limits_{k=1}^{n-1}\sum\limits_{l=1}^{n-1}{r^{\,\prime}_{kl}\,\xi_{k}\xi_{l}} \\ \end{aligned} $$
(8.46)
