Restricted diffusion

Restricted diffusion is diffusion whose displacement statistics are constrained by boundaries that particles cannot cross. A reflecting boundary condition models an impermeable wall: particles continue moving, but there is no normal diffusive flux through it. Finite permeability would require a different boundary model.

A physical demonstration uses a liquid confined between solid lamellae. In his first-person account, John E. Tanner described constructing 99 mica layers separated by aluminium-foil strips, removing wrinkles and burrs to obtain regular spacing; Edward O. Stejskal inspected the stack with a stereomicroscope. Pulsed magnetic-field gradients encode displacements in the nuclear magnetic resonance signal, allowing motion normal to the barriers to be compared with less confined directions.^[ref] This is a material confinement experiment, not a change in the liquid's chemistry.

For an original one-dimensional model, let particles diffuse in $0\le x\le L$ with constant diffusion coefficient $D>0$:The wall conditions are homogeneous Neumann boundary conditions for the diffusion equation. Start in equilibrium, so $X(0)$ has the uniform distribution on $[0,L]$. Its variance isAt long time separation, the positions decorrelate. Consequently the mean squared displacement approachesAt short times relative to $L^2/D$, most displacements have not encountered a wall, and the leading scale is the free-diffusion result $2D\tau$. For a hypothetical gap $L=10\mu \mathrm{m}$, the long-time plateau is $100/6\mu {\mathrm{m}}^2$ and its root is about $4.08\mu \mathrm{m}$.

The initial condition matters: fixing $X(0)=L/2$ instead gives the long-time value $L^2/12$. Likewise, echo attenuation in a confined sample must not automatically be converted into a free-diffusion coefficient. Confinement geometry, timing and the selected boundary model all affect the inference.

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.