Kinetic theory of diffusion in liquids: A hydrodynamic approximation

Diffusion in simple classical liquids is analyzed in terms of the test-particle phase-space density, with emphasis upon its long-time behavior. The Green's function of the generalized Fokker-Planck equation is used to define auxiliary quantities, in particular the transport mean path that enters solutions of the Chapman-Enskog type. Approximations for the lowest eigenvalues and eigenfunctions of the Fourier- and Laplace-transformed F.-P. operator σks are constructed, and an expansion for the resolvent operator (s + ik · v − σks)-1 proposed. With the additional assumption that branch-points on the negative real axis of s are the only singularities of the transformed F.-P. operator, a Laplace inversion is tentatively carried out, so that the general form of the solution is obtained. This is found to agree with the solution derived by hydrodynamic arguments. Only in a limited sense is the latter method equivalent to that of mode-mode coupling.