= Drift-diffusion equation with a potential
{title2=$\partial_t c=\partial_x[D(\partial_xc+\beta V'c)]$}
For positive <diffusivity> $D(x)$, a <diffusion equation> with potential-driven drift has <diffusive flux> $J=-D(c_x+\beta V'c)$. On a finite interval with zero <flux> at both endpoints, the total amount is conserved. The substitution $c=e^{-\lambda t}e^{-\beta V}y$ converts the decay <eigenvalue problem> to the <Sturm-Liouville problem>
$$
-(py')'=\lambda wy,\qquad p=De^{-\beta V},\quad w=e^{-\beta V},\quad py'=0\text{ at the endpoints}.
$$
The <Rayleigh quotient> is $\lambda=\int p|y'|^2/\int w|y|^2\geq0$. Its zero <eigenfunction> is constant, giving the <stationary solution> $c_{\rm eq}=C e^{-\beta V}$. This is a spatial <Boltzmann distribution>; $D$ changes relaxation rates but cancels out of the zero-<flux> equilibrium profile. For a sufficiently regular positive $D$ and bounded $V$, the constant eigenfunction is the unique zero mode up to scale.
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