= Restricted diffusion
\b[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.<John Tanner, the return to Germelshausen, and the work that nearly never happened by Derek Jones>[Printed p.6, mica-stack account and Figure3]{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 $0\leq x\leq L$ with constant <diffusion coefficient> $D>0$:
$$
\partial_t p=D\partial_x^2p,\qquad
\partial_xp(0,t)=\partial_xp(L,t)=0.
$$
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> is
$$
\operatorname{Var}(X)=\frac1L\int_0^L(x-L/2)^2\,dx=\frac{L^2}{12}.
$$
At long time separation, the positions decorrelate. Consequently the <mean squared displacement> approaches
$$
\boxed{\lim_{\tau\to\infty}\mathbb E[(X(\tau)-X(0))^2]
=2\operatorname{Var}(X)=\frac{L^2}{6}.}
$$
At 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 $4.08\,\mu\mathrm m$.
\Image[/restricted-diffusion-interval.png]
{title=Equilibrium <restricted diffusion> saturates in a reflecting interval}
{description=Original Neumann-eigenmode calculation with uniform initial positions; time is $D\tau/L^2$ and <mean squared displacement> is divided by $L^2$. Perfectly reflecting walls give a plateau $1/6$, while free diffusion continues as $2D\tau$. These are model curves, not measurements of the mica stack.}
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.
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