The Law of Peaks and Mass Diffusion
Laws
Consider a system of particles having mass, and assume the volume is constant. In Euclidean space $\mathbb{R}^{n}$, denote the density at point $\mathbf{x} \in \mathbb{R}^{n}$ by $\mathbf{u} = \mathbf{u} \left( \mathbf{x} \right)$. Call the flux due to diffusion $\mathbf{J} \left( \mathbf{x} \right)$ the diffusion flux. The following two laws relating diffusion flux and density are called Fick’s laws.
First law
The diffusion flux is proportional to the spatial variation (gradient) of the density. $$ \mathbf{J} \left( \mathbf{x} \right) = - D \nabla \mathbf{u} \left( \mathbf{x} \right) $$ Here the proportionality constant $D$ is called the mass diffusivity.
Second law
The density satisfies the diffusion equation. $$ {\frac{ \partial \mathbf{u} }{ \partial t }} = D \nabla^2 \mathbf{u} $$
Explanation
Intuitively, the first law can be summarized as “diffusion occurs in the direction of decreasing density.”
The second law is derived from the conservation equation and Fick’s first law. Assuming mass is conserved in the system, the rate of change of density with respect to time plus the divergence of the diffusion flux must sum to zero. Since $\mathbf{J}$ is the quantity passing through a unit area per unit time, the amount crossing the boundary of a finite volume is its divergence $\nabla \cdot \mathbf{J}$, and we obtain $$ {\frac{ \partial \mathbf{u} }{ \partial t }} + \nabla \cdot \mathbf{J} = 0 $$ This is called the conservation equation.
Divergence of $\nabla \mathbf{u}$: $$ \nabla \cdot \nabla \mathbf{u} = \nabla^{2} \mathbf{u} $$
Substituting Fick’s first law into the conservation equation yields the diffusion equation as follows. $$ \begin{align*} {\frac{ \partial \mathbf{u} }{ \partial t }} =& - \nabla \cdot \mathbf{J} \\ =& - \nabla \cdot \left( - D \nabla \mathbf{u} \right) \\ =& D \nabla^2 \mathbf{u} \end{align*} $$
The way mass diffusivity is defined is the same as how thermal diffusivity $\alpha$ is defined in the heat equation $\mathbf{u}_{t} = \alpha \nabla^2 \mathbf{u}$.