Question #42717

In a modified Drude model for a metal, the instantaneous collision frequency $\nu$ (in $s^{-1}$) of electrons with the lattice is a stochastic variable due to local thermal fluctuations. The collision frequency has a mean value $\nu_0$ and a standard deviation $\Delta \nu$, such that the relative fluctuation is $\delta = \frac{\Delta \nu}{\nu_0} \ll 1$. Assume the fluctuations are slow enough that the electron drift velocity continuously reaches its steady-state value $v_d = \frac{eE}{m\nu}$ under a constant applied electric field $E$ (in $V \cdot m^{-1}$). The macroscopic effective conductivity $\sigma_{\text{eff}}$ is defined as the ensemble average of the instantaneous conductivity $\sigma$. Let $\sigma_0 = \frac{ne^2}{m\nu_0}$, where $n$ is the electron number density (in $m^{-3}$), $e$ is the elementary charge (in $C$), and $m$ is the electron mass (in $kg$). Which of the following expressions correctly gives the effective conductivity $\sigma_{\text{eff}}$ (up to order $\delta^2$) and the relative standard deviation of the conductivity $\frac{\Delta \sigma}{\sigma_{\text{eff}}}$ (to leading order in $\delta$)?