Time Reversal Behaviour and Reciprocity Theorem of Transport-Relaxation-Equations

The coefficient matrices in the transport-relaxation-equations (moment equations of generalized thermo-hydrodynamics) depend on the polynomials used in the expansion of the underlying kinetic equation. The known behaviour under time reversal of these polynomials, Eq. (7), entails symmetries of the fore-mentioned coefficient matrices, Eqs. (12) and (16). From these symmetries a reciprocity theorem for two stationary solutions of the transport-relaxation-equations is derived, Eqs. (21 a und b) : the divergence of a certain vector field, bilinear in the two solutions, vanishes. The relaxation matrix does not appear in this form. The theorem is useful for investigating the proliferation of the Onsager symmetries from the basic differential equations into the linear algebraic relations between the few global quantities governing “discontinuous systems”. A simple example in heat conduction is worked out. A more complicated case and the role of a magnetic field are briefly considered. Equivalent with (21 a) is the kinetic reciprocity theorem (40).