= Wall hindrance in Brownian diffusion
{c}
<Buoyancy> changes the effective gravitational <mass>; a nearby no-slip wall changes the <hydrodynamic resistance matrix> and therefore the <diffusion coefficient>. These are distinct corrections. Write the horizontal resistance as $\zeta_\parallel=6\pi\eta a/f$, with $0<f<1$ representing wall hindrance at a specified height. Applying the <Stokes–Einstein relation> with that resistance gives
$$
D_\parallel=f\frac{k_BT}{6\pi\eta a},\qquad
\frac{N_{A,\,\mathrm{inferred}}^{\rm diff}}{N_{A,\,\mathrm{true}}}=\frac1f
$$
if one mistakenly uses the bulk resistance in the inversion. A chosen $f=0.8$ would overestimate $N_A$ by $25\%$; this is a sensitivity example, not a correction measured in <Jean Baptiste Perrin>'s apparatus. Normal and parallel coefficients of the <hydrodynamic resistance matrix> need not coincide. <Stokes settling velocity> near a wall can likewise bias a radius inferred using an unbounded-fluid formula.
For a passive dilute <particle suspension> at uniform <temperature>, a change of mobility alone does not change the <Boltzmann distribution>. In the absence of additional wall forces the correct local vertical flux is $j_z=-D_z(z)[dn/dz+m_bgn/(k_BT)]$, so zero flux still gives the same <sedimentation scale height>. Walls can nevertheless invalidate that simple inference through adsorption, additional interaction energies, or sampling of inaccessible grain-centre positions. Optical position errors and exposure averaging also affect displacement readout; varying distance from the wall, observation lag and optical depth tests different possible errors.
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