= Isothermal diffusion with position-dependent drag
{title2=$P_t=\partial_x(DP_x),\quad D=k_BT/\zeta(x)$}
With local thermal noise matched to spatially varying drag, the overdamped probability density obeys the displayed <diffusion equation>. Its equivalent <Itô diffusion> is $dX=D'(X)dt+\sqrt{2D(X)}\,dW$, since the two terms in the <Fokker-Planck equation> combine to $\partial_x(DP_x)$. Thus $d\langle X\rangle/dt=\langle D'(X)\rangle$, even with no applied force. The <drift> points toward greater <diffusivity>, but a uniform density has zero flux: the drift is balanced by the spatial variation of the noise. Holding the force-noise amplitude fixed instead would not produce this isothermal Fickian equation.
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