On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid

Abstract

1. In two famous papers on the Brownian motion of grains suspended in a stationary uniform fluid (liquid or gaseous) Einstein obtained, inter alia , the distribution function for the displacements of the grains during any interval t = 0 to t = τ from their positions at time t = 0. The object of this paper is to determine the distribution function for the more general case of a non-uniform fluid. The non-uniformity may refer to temperature, composition, or any other property which affects the coefficient of diffusion (D) of the grains in the fluid. The distribution function is given in 8, where it is shown how its accuracy might be experimentally tested. It contains terms additional to the one given by Einstein for the uniform case; certain of these are definitely determined, but another important term contains a coefficient that cannot be evaluated by considerations of the kind used in this paper (which depend purely on the conservation of the number of grains), but requires more detailed physical analysis; it is surmised that this coefficient vanishes in the case of Brownian particles which are large compared with the mean free path of the surrounding molecules. The main results of the paper refer to the rate of diffusion of the grains due to the non-uniformity of the fluid (9), and to the equilibrium distribution of the grains (10, 11). It is found that if their density is the same as that of the fluid, so that there is no tendency for them to settle in the lower strata, their steady distribution when the fluid is non-uniform is such that the concentration n (or number per unit volume of the fluid) is inversely proportional to D; a solution is also given for the case when the densities are not equal.