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11.E.1. Derivation of the Diffusion Equation The difference between the tracer and normal particle densities is r, t nN r, t nT r, t d 3 p1 f 0 p1 h p1 , r, t d 3 p1 f1 0 h1 (11.107) which vanishes at equilibrium. Now, d 3 p1 f10 11.105 gives h1 p1 h1 3 0 d p f d p f C 1 h1 1 1 1 m r t 3 0 1 2 JD 1, C h t n0 where the diffusion current J D is defined as p p J D r, t d 3 p1 1 f 0 p1 h p1 , r, t d 3 p1 1 f1 0 h1 m m Using the same technique as for C , we can show that C Hence 2 JD C 1, h 0 t n0 (11.109) is also hermitian. (11.108) where we've used C 1 0 , which follows directly from (11.106). can write J D D By Fick's law, we (11.110) where D is the self-diffusion coefficient. D 2 t Combining (11.108,10), we have (11.111) which is a hydrodynamic equation for the self-diffusion process. To find the dispersion relation for propagating modes, we introduce the Fourier transform r, t d d 3k k, exp i k r i t 2 2 3 so that (11.111) becomes (11.113) i Dk 2 which can be satisfied for all if iDk 2 (11.114) (11.112) Thus, is imaginary so that all modes eventually die out. becomes r, t d d 3k Eq(11.112) thus k, exp i k r Dk t 2 2 2 3 The decay time for the k mode is therefore k 0 so that can be quite large. 1 . Dk 2 For long wavelength modes,