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How can I derive the Einstein's relation $D=k_{b}TB$, where $D$ is the diffusion coefficient and B is the mobility coefficient, from the concept of osmotic pressure?

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There are two phenomena present

  • diffusion, which happens due to inhomogeneity in concentration. Particles "want to" go from areas of higher concentration to the lower ones. One can write this in the form of diffusion current $$J_{diff}(x) = - D \nabla \rho(x)$$ where $\rho(x)$ is the concentration. This expression is known as Fick's law but it's actually just the standard linear response to inhomogeneities.

  • drift, which is the terminal velocity particles attain due to presence of some force. E.g. the drift one can observe for balls falling in viscous liquid. One can write $$J_{drift} = \rho(x) v(x) = \rho(x) B F(x) = -B \rho(x) \nabla U(x)$$

From the requirement of equilibrium we have that $J_{diff} + J_{drift} = 0$ and from Boltzmann statistics we can obtain the concentration $\rho(x) \sim \exp(-{U(x) \over k_B T})$. Putting it all together we get $$0 = - D \nabla \rho(x) - B \rho(x) \nabla U(x) = - \nabla U(x) \rho(x) (-{D \over k_B T} + B)$$ and we can see the required relation in the last term.

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