Reflecting boundary condition

A reflecting boundary condition prevents normal particle or probability flux through a wall. For drift-free scalar diffusion with $D>0$, the condition isHere $\mathbf{n}$ is the outward unit normal. The last expression is a homogeneous Neumann boundary condition. Integrating the diffusion equation and applying the divergence theorem gives conservation of total probability or material in a closed container.^[ref]

For an original example, choose units in which the interval is $0\le x\le 1$. The densitysatisfies $p_t=D{p}_{xx}$, has zero endpoint derivatives and integrates to one. With $a=0.6$, $D=0.5$ and $t=0.5$, its cosine amplitude is about $0.050883$. A nonuniform density therefore relaxes while remaining inside the interval; reflection does not hold the boundary concentration fixed.

For anisotropic diffusion, impose zero normal diffusive flux using the full diffusion tensor. At a wall with $\mathbf{n}=(1,0)$, letin consistent units. Then ${\mathbf{n}}^T\mathbf{D}\nabla C=0$, although ${\partial }_{n}C=1$. Setting the normal derivative to zero would impose a different condition. Absorption and finite permeability require their own boundary models; a reflecting wall allows continued motion within the available space.