オンサーガーの相反定理

${\displaystyle \mathrm {d} u=T\mathrm {d} s+m\mathrm {d} r\,.}$

${\displaystyle \mathrm {d} s={\frac {1}{T}}\mathrm {d} u-{\frac {m}{T}}\mathrm {d} r\,.}$

${\displaystyle \partial _{t}u+\nabla \cdot \mathbf {J} _{u}=0\,,}$

${\displaystyle \partial _{t}r+\nabla \cdot \mathbf {J} _{r}=0\,.}$

${\displaystyle \mathbf {J} _{u}=k\,\nabla {\frac {1}{T}}}$

${\displaystyle \mathbf {J} _{r}=-k'\,\nabla {\frac {m}{T}}}$

${\displaystyle \mathbf {J} _{u}=L_{uu}\,\nabla {\frac {1}{T}}-L_{ur}\,\nabla {\frac {m}{T}}\,,}$

${\displaystyle \mathbf {J} _{r}=L_{ru}\,\nabla {\frac {1}{T}}-L_{rr}\,\nabla {\frac {m}{T}}\,.}$

${\displaystyle S=S(\mathbf {E} )\,.}$

${\displaystyle \mathrm {d} S(\mathbf {E} )=\sum _{i}{\frac {\partial S(\mathbf {E} )}{\partial E_{i}}}\mathrm {d} E_{i}\,.}$

${\displaystyle {\frac {\partial S(\lambda \mathbf {E} )}{\partial (\lambda E_{i})}}={\frac {\lambda }{\lambda }}{\frac {\partial S(\mathbf {E} )}{\partial E_{i}}}={\frac {\partial S(\mathbf {E} )}{\partial E_{i}}}.}$

${\displaystyle I_{i}:={\frac {\partial S}{\partial E_{i}}}\,.}$

${\displaystyle \mathbf {F} _{i}=-\nabla I_{i}\,.}$

${\displaystyle \partial _{t}E_{i}+\nabla \cdot \mathbf {J} _{i}=0\,.}$

${\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}\mathbf {F} _{j}\,.}$

${\displaystyle \partial _{t}E_{i}=\nabla \cdot \sum _{j}L_{ij}\,\nabla I_{j}\,.}$

${\displaystyle \sigma _{ij}={\frac {\partial E_{i}}{\partial I_{j}}}}$

${\displaystyle \sum _{j}\sigma _{ij}\,\partial _{t}I_{j}=\nabla \cdot \sum _{j}L_{ij}\,\nabla I_{j}\,.}$