The approximate nature of the Onsager-Casimir reciprocal relations

We study thermodynamic systems near equilibrium described by both slow and fast variables. A reduced relaxation matrix for the slow variables can be obtained from the full relaxation matrix by a systematic elimination of the fast variables. When the full relaxation matrix possesses Onsager-Casimir symmetry, the reduced matrix has contributions that violate this symmetry. For a sensible choice of the variables they are in general of second order in the time-scale ratio relative to the dominant terms. The occurrence of such a symmetry violation is demonstrated for a simple example. The symmetry violation is related to initial slip in the correlation functions on the reduced level of description. We discuss a way to salvage the Onsager-Casimir relations by an appropriate modification of the thermodynamic potential, or rather of an associated function similar to the Lagrangian in mechanics.