Onsager relations and extended irreversible thermodynamics

We show that when the thermodynamic fluxes are included as independent thermodynamic state variables of a generalized entropy, the original ONSAGER formulation may be directly used in the space of the fluxes. Therefore, the ONSAGER relations may be derived either in the space of the classical (slow) variables, by using a spatial FOURIER transformation, or in the space of the non-classical (fast) variables such as the physical fluxes, without need of any Fourier transform. Furthermore, we analyse the question of non-linear ONSAGER relations by studying one particular set of evolution equations of the fluxes, and considering the fluctuations of the fluxes around a non-equilibrium steady state. Comparison with kinetic theory is not completely conclusive, because of several open questions which we comment in the concluding remarks.