= Sedimentation equilibrium
<Sedimentation equilibrium> occurs when the downward tendency of suspended particles is balanced by their <diffusion>, leaving a stationary concentration profile with no net particle flux. For dilute identical particles in an isothermal fluid, let $m_b=(\rho_p-\rho_f)V_p$ be their buoyant mass. Balancing the gradient of ideal <osmotic pressure> $nk_BT$ against gravity gives
$$
k_BT\frac{dn}{dz}=-m_bg n,\qquad n(z)=n(0)e^{-m_bg z/(k_BT)}.
$$
Here height increases upward, and heavy particles have $m_b>0$.
The profile is a miniature atmosphere: <thermal motion> spreads particles upward while gravity favours lower positions. The <Perrin sedimentation experiment> measured particle counts at different microscope focal heights and used this competition to infer a molecular energy scale.<Mouvement brownien et réalité moléculaire by Jean Baptiste Perrin>[1909 memoir, §§15 and 20–22, vertical grain populations and tests using independently measured grain properties]{ref} A stationary profile requires sufficiently slow background flow and controlled <temperature>; a growing deposit during ongoing settling is a different situation.
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