Onsager reciprocal relations

Remarks

The proof uses microscopic reversibility: the equilibrium correlation $\langle \alpha_i(0)\,\alpha_k(\tau) \rangle$ equals $\langle \alpha_k(0)\,\alpha_i(\tau) \rangle$. Differentiating in $\tau$ and inserting the linear regression law, together with the equilibrium identity $\langle \alpha_i X_k \rangle = -k_B \delta_{ik}$, gives $L_{ki} = L_{ik}$. Example: in an anisotropic crystal the heat flux is $J_q = -K \nabla T$ for a conductivity tensor $K$, and the relations give $K_{ij} = K_{ji}$, which was confirmed by experiment. The relations also link the Seebeck and Peltier effects and the Soret and Dufour effects. For variables that are odd under time reversal, such as momenta, some relations change sign: $L_{ik} = -L_{ki}$ when exactly one of $\alpha_i, \alpha_k$ is odd.

Going deeper

For a student

The relations reduce the number of independent transport coefficients from $n^2$ to $n(n+1)/2$. Together with the entropy production $\sigma = \sum_{i,k} L_{ik} X_i X_k \ge 0$, they make $L$ a symmetric positive semidefinite matrix. They are the main structural result of linear irreversible thermodynamics, and they underlie the Kelvin relations of thermoelectricity and Prigogine's minimum entropy production principle. They hold only in the linear regime near equilibrium.