A thermodynamic derivation of the reciprocal relations

Starting with the continuity and Smoluchowski equations, we write the mass flux for a system out of equilibrium in terms of the physicochemical potential μ(g). μ(g) is a coarse-grained analog of the chemical potential in the presence of forces that drive the system out of equilibrium. The expression for flux in terms of μ(g) allows for a macroscopic derivation of the Onsager reciprocal relations for the case of transport by diffusion and drift in single or multi-component systems, without recourse to microscopic fluctuations or equations of motion. Transport coefficients for any time reversal-invariant properties now are expressed in terms of only partial molar derivatives and mobilities (diffusion coefficients). The thermodynamic derivation cannot treat time reversal.