Green-Kubo formulas for mutual diffusion coefficients in multicomponent systems

Green-Kubo relations for the mutual diffusion coefficients of an n-component system are derived from hydrodynamic equations, linear response theory, and the statistical theory of fluctuations. The mutual diffusion coefficients, which form an (n - 1)-dimensional matrix, are formulated as a product of a kinetic matrix and a thermodynamic matrix. The kinetic matrix consists of time integrals of diffusion flux correlation functions, i.e., the Onsager phenomenological coefficients. The thermodynamic matrix is determined by spatial integrals of radial distribution functions. The formulas are derived initially in the barycentric reference frame and then transformed to the volume-fixed reference frame, which is more appropriate to laboratory studies. Explicit expressions for binary and ternary mixtures are presented, from which various empirical relations between the mutual diffusion coefficients and self-diffusion coefficients are obtained using particular assumptions regarding the velocity cross-correlation functions. Assuming zero values for certain combinations of the cross-correlation functions yields the Darken or Hartley-Crank equation for binary systems and Cooper's diffusion model for ternary systems, while assigning negative values to these combinations results in Lasaga's relation, which is a special case of the so-called common force diffusion model.