Abstract
The correlation between the Schrodinger equation and the diffusion equation revealed that the relation of material wave is not a hypothesis but an actual one valid in a material regardless of the photon energy. Using the relations of material wave and uncertain principle, the quantum effect on elementary process of diffusion is discussed. As a result, the diffusivity is obtained as a universal expression applicable to any problem of diffusion phenomena. The Gauss theorem in theory and the Kirkendall effect in experimentation reveal the necessity of the coordinate transformation for a diffusion equation. The mathematical method for solving an interdiffusion problem of many elements system is established. The phase shift of the obtained analytical solution indicates the correlation between the solutions of each diffusion equation expressed by a fixed coordinate system and by a moving coordinate system. Based on the coordinate transformation theory, some unsolved problems of diffusion theory are reasonably solved and also some new important findings are discussed in relation to matters in the existing diffusion theory.
Highlights
The diffusion equation is one of the most fundamental and important equations in physics
The correlation between the Schrödinger equation and the diffusion equation revealed that the relation of material wave is not a hypothesis but an actual one valid in a material regardless of the photon energy
The existence of J j (t ) in Equation (21) indicates that we must take account of the difference between the diffusion equation of coordinate system set a point in the diffusion region space expressing the relative motion between micro particles and that of coordinate system set a point in the laboratory system outside the diffusion region
Summary
The diffusion equation is one of the most fundamental and important equations in physics. In relation to the collective motion of micro particles, the behavior of their concentration has been investigated by solving the diffusion equation. The elementary quantity of diffusivity and the relation of material wave unrelated to the photon energy were reasonably obtained. The actual diffusivity different from the existing theory within the category of classical mechanics and thermodynamics was obtained as a universal expression applicable to any material, regardless of the thermodynamic state. That means a phase shift of the analytical solution, which is essentially relevant to the coordinate transformation Incorporating such a shift effect into the obtained solution, the experimental results were reasonably reproduced [26]. Some new findings resulting from the coordinate transformation theory of diffusion equation and resulting from the general solutions obtained as analytical expressions are concretely and systematically discussed. The new findings obtained here will be widely useful and indispensable for analyzing the actual diffusion problems in future, just because of extremely fundamental ones
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