Effective diffusion constant for inhomogeneous diffusion

Abstract

We consider the problem of determining analytically the effective diffusion constant Deff for diffusion in an inhomogeneous medium, described by the diffusion equation deltatP = deltax[D(x) delta xP] with a position-dependent diffusion coefficient D(x). As there is no translational invariance, two different natural definitions are possible: From the long-time behaviour of the variance, Deff = D = lim((Delta x)2)/2t. From the large-distance behaviour of the mean first passage time T(+or-x mod 0) to reach exit points at +or-x starting from the origin, Deff=D=limx to infinity x2/2T(+or-x mod 0). In general, D not=D/. We find an exact formula for D and examine a number of interesting special cases. If D(x) tends to finite limits D+or- as x to +or- infinity , then D is simply the arithmetic mean (D++D-)/2. In the important case of a periodic D(x), we find that D is the harmonic mean of D(x) in a period. We also give an argument suggesting that D=D in this case. If D(x) is piecewise constant in an arbitrary fashion, with a finite number of discontinuities, and tends to D+or- and x to +or- infinity , then D=(D++D-)/2 as before, but D=(D+D-)1/2+(1-2/pi)(D+1/2-D-1/2)2<=D (the equality obtaining only if D+=D-). Thus D is the geometric mean of D+ and D- plus a 'correction' term. We also illustrate the significant role of inhomogeneities in determining Deff with the help of a simple example involving a discrete-time random walk on a chain.