Minimum Rate of Dissipation Principle and Linear Corrections to the Onsager Matrix

The Minimum Rate of Dissipation Principle (MRDP) affirms that, for time-independent boundary conditions, a thermodynamic system evolves towards a steady-state with the least possible dissipation. In this note, examples of diffusion processes of two solutes in an isothermal system are analyzed in detail. In particular, we consider the relaxation of the system when the metric tensor (i.e. the Onsager matrix) is constant and when the Onsager coefficients weakly depend on the spatial derivatives of the concentrations. We show that, to leading order, during the relaxation towards a steady-state, the system traces out a geodesic in the space of thermodynamic configurations, in accordance with the MRDP.