Abstract: The dependencies of fluxes and forces were paid close attention to the phenomenological theory of Onsager. But in such a case, it seems that the Onsager's reciprocal relations are not necessarily fulfilled. In this work, the problem of thermo-diffusion was chosen as an example and a combination of the first and the second law of thermodynamics was adopted to describe the reversible and the irreversible process. Accordingly, the Gibbs-Duhem relation was found to be followed by not only the equilibrium but also the non-equilibrium thermodynamics, i.e., the dependency of forces needs to be considered during the derivation of evolution equations. After that, a comparative study between the previous thermodynamic extremal principle (TEP) incorporating only the constraint from fluxes and the present TEP incorporating the constraints from both fluxes and forces was carried out. For the former, the well-known Dufour effect cannot be described, whereas, for the latter, both the well-known Soret effect and the Dufour effect can be predicted. Furthermore, the phenomenological equations were uniquely determined using the present TEP, and Onsager's reciprocal relations were found to be followed, thus solving the problem of the n-fold arbitrariness for the kinetic coefficients pointed out previously. The present work not only extends the TEP to non-isothermal thermodynamics but also might provide guidance for modeling dissipation systems with constraints from both fluxes and forces.

Key words: Thermodynamic extremal principle, Phenomenological theory, Onsager's reciprocal relations, Thermo-diffusion