Classical Transport in Modulated Structures

Classical transport coefficients in a $d$-dimensional medium with a potential $V\left(\mathrm{x}\right)$ and/or conductivity $a\left(\mathrm{x}\right)$ are found to vary discontinuously as functions of the "wavelengths" of the inhomogeneities. For example, with a potential depending on one direction only, say $V\left(\mathrm{x}\right)=cos2\pi x_1+cos2\pi k{x}_1$, the effective diffusion coefficient $D\left(k\right)\mathrm{}$ has the same value $D^{*}$ for all irrational $k$, but differs from $D^{*}$ and depends on $k$ for $k$ rational. Thus $D\left(k\right)\mathrm{}$ is discontinuous at rational $k$. Moreover, $D\left(k\right)\mathrm{}$ is continuous at irrational $k$. This pathology is reflected in the time scales on which the diffusion approaches its limiting behavior.