Consider a system of particles (each having a charge $e$ ) confined within some finite volume. The particles might, for example, be ions in a gas or ions in a solid like $\mathbf{N a C l}$. The particles are in thermal equilibrium at the temperature $T$ in the presence of an electric field $\delta$ in the $z$ direction.
(a) Denots the mean number of particles per unit volume at the position $z$ by $n(z)$. Use the results of equillbrium statistical mechanics to relats $\boldsymbol{n}(z+d z)$ to $n(z)$.
(b) Suppose that the particles can be characterized by a diffusion coefficient D. By using the definition of this coefficient, find the fiux $J_D$ (the number of particles crossing unit area per unit time in the 2 direction) due to the concentration gradient calculated in (a).
(c) Suppose that the particles are also characterized by a mobility $\mu$ relating their drift velocity to the applled field $\mathcal{E}$. Find the particle flux $J_\mu$ resulting from the drift velocity produced by the field $\mathcal{E}$.
(d) By making use of the fact that in equilibrium the net particle flux $J_D+J_\mu$ must vanish, find a relation between $D$ and $\mu$. The rasult thus obtained constitutes a very general derivation of the Einstgin relation (15-6 14).