Brownian motion of N interacting particles. II. Hydrodynamical evaluation of the diffusion tensor matrix

For two Brownian particles of arbitrary shapes, expressions in the form of infinite sums are obtained for the translational friction tensors using the method of hydrodynamic reflections. From the generalized Einstein relation which gives the diffusion tensors in terms of the friction tensors, we show that in the absence of long‐range forces among the Brownian particles, the self‐diffusion constant is of the form ${\mathrm{D\; =\; D}}_0{\mathrm{+\; D}}_1\mathrm{\prime c\hspace{0.17em}}\ln {\mathrm{c\; +\; D}}_1\mathrm{c\; +\; ···}$ , where c is the concentration of the Brownian particles. For the case of two identical spheres, the friction tensors are evaluated up to fourth order in the ratio of the radius a of the spheres to the distance between them. The diffusion tensors and the numerical value of D₁ are then evaluated in the same approximation; $D_1\prime$ is found to be zero.