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The diffusion coefficient relates the particle flux $J$ to the gradient in the number density (of the 'labelled' particles) $\frac{\partial \bar n}{\partial z}$ such that; $$J=-D \frac{\partial \bar n}{\partial z}$$ I have seen a number of places* give an approximate derivation of $D$. All rely on the statement that the mean number of particles travelling from above the boundary at $z=z_0$ is related to $n(z_0+\lambda)$ where $\lambda$ is the mean free path length. I cannot, however see why collisions come into such (approximate) derivations and therefore where the use of $\lambda$ is justified. Please can someone explain this to me?

*For example The mathematical theory of non-uniform gases 3rd ed by Chapman and Cowling page 102

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  • $\begingroup$ Not sure what you are asking. The book you refer to is called "Mathematical Theory" and it does, indeed, provide detailed, rigorous, derivations of transport coefficients, The page you refer is from an introductory chapter -- just keep reading. (Although the Chapman and Cowling is, admittedly, somewhat heavy going. I provide the a brief summary of the calculation here physics.stackexchange.com/questions/230380/… ) $\endgroup$ – Thomas Jan 26 '16 at 14:21
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the collision sets the time scale over which the particle can travel freely on average. Based on this picture, semiclassical, you can use the relation you quoted to calculate the Diffusion constant.

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