The Law of Peaks and Mass Diffusion
Laws
Consider a system of particles with mass, assuming a constant volume. Denote the density at the point $\mathbf{x} \in \mathbb{R}^{n}$ in Euclidean space $\mathbb{R}^{n}$ by $\mathbf{u} = \mathbf{u} \left( \mathbf{x} \right)$. The flux due to diffusion $\mathbf{J} \left( \mathbf{x} \right)$ is called 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 change 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 follows 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.” In heat conduction it corresponds to Fourier’s law of heat conduction.
The second law is derived from the conservation equation and Fick’s first law. Assuming mass is conserved in the system, the time rate of change of the density 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 net amount crossing the boundary of a 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 $\mathbf{u}_{t} = \alpha \nabla^2 \mathbf{u}$.