... is:¹

Assume interstitial diffusion, otherwise the chemical potential of the vacancy must be included.

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... simply,²

The $1/T$ gets absorbed into the the thermal conductivity for the empirical law.

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... interest.³

Just about every moderately complicated diffusion problem (i.e., for various boundary and initial conditions) has been solved previously. Carslaw and Jaeger's book on solutions to the the thermal diffusion equation and Crank's book on concentration diffusion are excellent places to start a search for known solutions.

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... times.⁴

What is meant by long times will be made explicit later. Clearly, no steady state solutions should be expected-in the strict sense-if the boundary conditions are functions of time.

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