Diffusion of a pointlike particle in a thin layer of varying thickness $h(x,y)$ bounded by reflecting boundaries is considered. Using the mapping technique derived for diffusion in a nonhomogeneous quasi-one-dimensional channel, we derive an effective two-dimensional Fick-Jacobs equation governing evolution of the marginal density $p(x,y,t)$ of the planar coordinates $x$ and $y$. The equation is modified by the space-dependent effective diffusion coefficient $\mathbb{D}(x,y)$, which is a 2×2 tensor introducing anisotropy of diffusion, depending on the shaping function $h(x,y)$. Biasing of the diffusional flux is demonstrated on a layer bounded by the sinusoidally corrugated upper wall.

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• Brownian motion
• Classical statistical mechanics
• Diffusion
• Nonequilibrium statistical mechanics
• Brownian ratchet
• Colloids
• Coarse graining
• Fokker–Planck equation