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Model for heterogeneous diffusion. (English) Zbl 1467.35315

Summary: A revertible kinetic equation for Brownian particles is introduced when the turning frequency and the collision kernel are spatially heterogeneous. We derive an anisotropic diffusion equation by taking the singular limit of the kinetic equation and then balancing out its curvature effect. We see that an extra information such as the turning frequency or waiting time is also needed together with the diffusivity to model a diffusion phenomenon correctly if the environment is spatially heterogeneous. A thought experiment is introduced to test the validity of diffusion laws. We observe that diffusion models with diffusivity alone fail the test and the anisotropic model driven in the paper passes the test.

MSC:

35Q82 PDEs in connection with statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D20 Statistical mechanics of solids
35K05 Heat equation
60J60 Diffusion processes
60J65 Brownian motion
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