The Fickian second law was widely applicable not only to the analysis of various diffusion problems in material science but also to that of phenomena for the Brownian motion in other science fields, such as the behaviour of neurons in life science. It was thus one of the most dominant equations in science. In 1894, Boltzmann transformed it into an ordinary differential equation applicable to the analysis of the interdiffusion problems. Since then, however, the mathematical solutions had not yet been obtained. Here two new equations were derived which were superior, for calculation, to those of Fick and Boltzmann. Using the derived integro-differential equation, their solutions were obtained as analytical expressions in accord with the results of experimental analysis. Hereafter, the basic equations derived here will be exceedingly useful for the analysis of the non-linear problems concerning the Brownian motion in various science fields.

New Mathematical Solution for Analyzing Interdiffusion Problems. T.Okino: Materials Transactions, 2011, 52[12], 2220-7