A note on representation-dependent Onsager closures for thermo-diffusive coupling

Abstract Non-isothermal diffusion can be formulated within different mixture-level approaches that account for diffusion-related transport either through an extended energy flux or through an extended entropy flux. This note compares two prominent approaches: the Gurtin approach, in which diffusion enters the energy balance through an additional diffusive energy flux, and the Müller approach, in which the classical energy balance is retained while diffusion enters through an extended entropy flux. The comparison clarifies how the way diffusion-related transport is represented in each approach and the adopted thermodynamic force basis affect a subsequent Onsager-type closure. In the selected Müller-type force basis, the diffusion-related thermodynamic force contains a temperature-gradient contribution. Consequently, a diagonal Onsager closure in this basis yields a diffusion flux with an apparent Soret-type contribution while the heat flux remains Fourier-like. In the selected Gurtin-type force basis, the corresponding diagonal closure yields uncoupled Fourier and Fick laws, whereas thermo-diffusive coupling appears through non-diagonal Onsager blocks. Thus, the apparent Soret/Dufour structure follows from how diffusion-related transport is represented in the mixture-level approach considered and from the basis of thermodynamic forces used for the Onsager closure. Although diffusion in solids provides the main motivation, the derivation does not rely on a specific constitutive law.