= Solution
Let $C(x)$ be the particle <number density>, $U(x)$ the conservative <potential energy> and $F=-dU/dx$ the applied <force>. In the overdamped regime a viscous <drag coefficient> $\zeta$ gives drift <velocity> $v_d=F/\zeta$. Combining this drift with <Fick's first law> gives the number flux
$$
J=-D\frac{dC}{dx}+\frac F\zeta C.
$$
At <thermal equilibrium>, the <Boltzmann distribution> is $C_{\rm eq}\propto e^{-U/(k_{\rm B}T)}$, so $dC_{\rm eq}/dx=FC_{\rm eq}/(k_{\rm B}T)$. Setting its net flux to zero yields
$$
0=FC_{\rm eq}\left(\frac1\zeta-\frac D{k_{\rm B}T}\right),\qquad
\boxed{D=\frac{k_{\rm B}T}{\zeta}=\mu k_{\rm B}T,\quad\mu=\zeta^{-1}.}
$$
This is the <Einstein diffusion relation>. The derivation assumes a dilute equilibrium suspension, uniform <temperature>, linear drift response and a well-defined long-time <diffusion coefficient>. With a nonzero conservative <force>, the full transport equation includes drift; the force-free equation in the next section describes spreading without that drift.
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