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At fixed measured count slope, $N_A^{\mathrm{sed}}\propto a^{-3}$, whereas at fixed measured diffusion coefficient, $N_A^{\mathrm{diff}}\propto a^{-1}$. First-order propagation of uncertainty follows by taking logarithmic derivatives:For independent small errors, the radius contributes $9({\sigma }_a/a{)}^2$ to the relative variance of $N_A^{\mathrm{sed}}$, compared with $({\sigma }_a/a{)}^2$ for $N_A^{\mathrm{diff}}$. Shared radius and temperature calibrations make the two estimates correlated; their agreement is therefore not two wholly independent determinations. As a finite-error example, overestimating $a$ by $5\%$ multiplies the respective answers by $1.05^{-3}\simeq 0.8638$ and $1.05^{-1}\simeq 0.9524$.

This explains the importance of the radius cross-checks in the 1909 memoir: testing the Stokes drag law through an independent radius estimate helps prevent circular inference.^[ref] Convection must also be excluded: a residual horizontal speed $u$ adds $u^2{\tau }^2$ to an uncentred mean squared displacement, making a linear fit misleading.

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.

The unnormalized autocorrelation function of a stationary velocity. For zero mean, $\chi (0)$ is the velocity variance. It is different from the normalized autocorrelation function $\chi (u)/\chi (0)$ often used in time series.

If $X(t)-X(0)={\int }_0^{t}V(s)ds$, integrating the two-time covariance over the square $[0,t{]}^2$ givesThis identity connects a mean squared displacement to the Green-Kubo formula for diffusion.