Theoretical Evidence for Revision of Fickian First Law and New Understanding of Diffusion Problems

Abstract

Based on the divergence theorem, we reveal that the Fickian first law relevant to the diffusion flux |<i>J(t,x,y,z) </i>> in the time and space is incomplete without an integral constant |<i>J<sub>0</sub>(t) </i>> for the integral of Fickian second law. The new diffusion flux (NDF) taking it into account shows that we can systematically understand the problems of one-way diffusion, impurity diffusion and self-diffusion as a special case of the interdiffusion. Applying the NDF to the interdiffusion problem between metal plates, it is clarified that the Kirkenkall effect is caused by |<i>J<sub>0</sub>(t) </i>> and also that the interdiffusion coefficients in alloy can be easily obtained. The interdiffusion problems are reasonably solved regardless of the intrinsic diffusion conception. Thus the NDF to replace the Fickian first law is an essential equation in physics

Correlation between Diffusion Equation and Schrödinger Equation

“…As a result, we revealed that the diffusivity in (1) corresponds to the angular momentum operator L in quantum mechanics. The obtained new diffusivity will be useful for understanding of an elementary process of diffusion [3].…”

Ending of Darken Equation and Intrinsic Diffusion Concept

“…The Fickian second law is also obtained from the provability of differential equation [2]. In mathematics, the Fickian first law is obtained by integrating the Fickian second law with respect to nabla operator [3]. Therefore, it is considered that the Fickian law is only one, i.e., the Fickian second law.…”

Quantum Effect on Elementary Process of Diffusion and Collective Motion of Brown Particles

“…The Gauss theorem reveals that the diffusion region space moves as shown in the relation between Equations (20) and (21). In other words, the discussion of coordinate transformation about the diffusion equations is indispensable for investigating the diffusion phenomena meaning the relative motion between micro particles.…”

“…is mathematically valid because of ( ) { } eq 0 x J t J ∂ ∂ + = [21]. Here, it is consi-The definition of diffusivity relevant to a relative motion between micro particles indicates that the coordinate origin of diffusion equation should be usually set at a point in the diffusion region space.…”