Title
Criticism of Onsager's reciprocal relations
Author
Warren D. Smith, NECI
Abstract
Onsager's second kind of ``reciprocal relations''
%[Phys.Rev. 37 (1931) 405-426; 38 (1931) 2265-2279]
state that $\sigma (\vec{H}) = \sigma^{T} (-\vec{H})$ where $\sigma$
is the $3 \times 3$ conductance matrix, $\vec{H}$ is an externally
applied magnetic field, and $T$ means matrix transpose. This is only
supposed to hold for ``linear'' materials for which the current
density $\vec{J}$ is related to the electric field $\vec{E}$ by
$\langle \vec{J} \rangle = \sigma \vec{E}$. This is just one instance
of a class of mathematically equivalent but physically different
relations; e.g. the same relation is supposed to hold if $\sigma$ is
thermal conductivity and $\vec{H}$ 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 first kind (which are
the special case $\vec{H}=\vec{0}$). There are two flaws in Onsager's
reasoning and each counterexample exploits one of them. Both flaws
had already been thought of
%[M.Lax Symmetry Principles in Solid State and Molecular Physics,
%Wiley-Interscience 1973] %287-292
but I feel that they are insufficiently appreciated.
If we instead define $\sigma$ to be the $3\times 3$ {\em differential
conductance} (Jacobian) matrix
$\sigma = {\rm d}\langle \vec{J} \rangle/{\rm d} \vec{E}$, then
$\sigma_{D} ( \vec{H}, \vec{E} ) = \sigma_{L} ( \vec{H}, \vec{E} )$
where $\sigma_D$ is for the original material, but $\sigma_L$ 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.
%This allows nonlinear electrical materials, but
%only applies for the $3$ diagonal entries of $\sigma$.
In the limit $| \vec{E} | \to 0$ (the ``zero voltage limit'')
$\sigma_{D} ( \vec{H} ) = \sigma_{U}^{T} ( -\vec{H} )$ where
$\sigma_U$ 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.
%although those authors had not realized it.
Keywords
Onsager relation, galvanomagnetic effects, magnetoresistance,
non-ohmic resistors, nonlinear electrical materials,
chirality, errors in textbooks.