Onsager-Casimir reciprocal relations

TL;DR

This paper discusses the Onsager-Casimir reciprocal relations within thermodynamics, deriving conditions for reciprocal relations based on positive semi-definite quadratic forms and time-reversal invariance.

Contribution

It generalizes Onsager reciprocal relations to include Casimir relations by incorporating time-reversal symmetry considerations.

Findings

Derivation of reciprocal relations from positive semi-definite forms.

Extension of Onsager relations to Casimir relations.

Invariance of quadratic forms under time reversal.

Abstract

The Onsager reciprocal relations are established within the phenomenological framework of the thermodynamics of irreversible processes. In order to do so, the dissipated power densities associated to scalar and vectorial processes are written as positive semi-definite quadratic forms of the corresponding generalised forces, as required by the local expression of the second law in the neighbourhood of the equilibrium. The antisymmetric part of the scalar and vectorial Onsager matrices do not contribute to the dissipation, which yields the scalar and vectorial Onsager reciprocal relations. Furthermore, the positive semi-definite quadratic forms of the generalised scalar and vectorial forces are invariant under time reversal, which yields the scalar and vectorial Casimir-Onsager reciprocal relations, that are a generalisation of the Onsager reciprocal relations.