Explanation for the Resistivity Law in Quantum Hall Systems

We consider a 2D electron system in a strong magnetic field, where the local Hall resistivity ${\rho }_{\mathrm{xy}}\left(\mathbf{r}\right)$ is a function of position and ${\rho }_{\mathrm{xx}}\left(\mathbf{r}\right)$ is small compared to ${\rho }_{\mathrm{xy}}$. Particularly if the correlations fall off slowly with distance, or if fluctuations exist on several length scales, one finds that the macroscopic longitudinal resistivity $R_{\mathrm{xx}}$ is only weakly dependent on ${\rho }_{\mathrm{xx}}$ and is approximately proportional to the magnitude of fluctuations in ${\rho }_{\mathrm{xy}}$. This may provide an explanation of the empirical law $R_{\mathrm{xx}}\propto \frac{\mathrm{Bd}{R}_{\mathrm{xy}}}{\mathrm{dB}}$ where $R_{\mathrm{xy}}$ is the Hall resistance, and $B$ is the magnetic field.