9: Onsager Relations
- Page ID
- 165981
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This Chapter gives a self-contained, step-by-step derivation showing that the Onsager Regression Hypothesis and the Onsager-Casimir Reciprocal Relations both follow rigorously from the Fluctuation-Dissipation Theorem (FDT). Starting from the Liouville equation and canonical equilibrium, we derive the key identity \(\chi(t)=-\beta \dot{C}(t) \theta(t)\) linking the linear-response function to the equilibrium auto-correlation function. We prove that a macroscopic perturbation decays with the same normalized time dependence as a spontaneous microscopic fluctuation. For the reciprocal relations we present two complementary proofs: first, via the Green-Kubo formula and time-reversal parity of currents; second, via the entropy-production formalism and the Gaussian orthogonality relation \(\left\langle A_{i} X_{j}\right\rangle=-k_{\mathrm{B}} \delta_{i j}\). Both routes arrive at \(L_{i j}=\epsilon_{i} \epsilon_{j} L_{j i}\), and the modification under an external magnetic field is derived in full generality.