Criticism of Onsager's Reciprocal Relations

| Onsager's second kind of \reciprocal relations" state that (~ H) = T (? ~ H) where is the 3 3 conductance matrix, ~ H is an externally applied magnetic eld, and T means matrix transpose. This is only supposed to hold for \linear" materials for which the current density ~ J is related to the electric eld ~ E by h ~ Ji = ~ E. This is just one instance of a class of mathematically equivalent but physically diierent relations; e.g. the same relation is supposed to hold if is thermal conductivity and ~ H is angular velocity for a rotating system. We demonstrate by two counterexamples that this relation (and its isomorphs) can be dramatically false. We also have counterexamples to the Onsager relations of the rst kind (which are the special case ~ H = ~ 0). There are two aws in Onsager's reasoning and each counterexample exploits one of them. Both aws had already been thought of but I feel that they are insuuciently appreciated. If we instead deene to be the 3 3 diierential con-ductance (Jacobian) matrix = dh ~ E) where D is for the original material, but L is for a \reversed" material in which all spins and angular momenta (and consequently all magnetic dipoles), and the chirality (by a central inversion of 3-space) are reversed. In the limit j ~ Ej ! 0 (the \zero voltage limit") D(~ H) = T U (? ~ H) where U is for a \spin-reversed" material. Note that the U (or L) and D materials could be inequivalent { in which case Onsager's relation is not about one, but actually about two, materials, and hence is less useful. Apparently there have been zero intentional experimental tests (so far) of any Onsager relation in a material inequivalent to its reversal; but I will argue that a 1996 experiment by Solin, Thio, et al. quite likely disproved Onsager's relations of the second kind.