Consider a system of charged particles confined to a volume...

Consider a system of charged particles confined to a volume $V$. The particles are in thermal equilibrium at temperature $T$ in the presence of an electric field $E$ in the $z$-direction. (a) Let $n(z)$ be the density of particles at the height $z$. Use equilibrium statistical mechanics to find the constant of proportionality between $\frac{d n}{d z}$ and $n$. (b) Suppose that the particles can be characterized by a diffusion coefficient $D$. Using the definition of $D$ find the flux $J_D$ arising from the concentration gradient obtained in (a). (c) Suppose the particles are also characterized by a mobility $\mu$ relating their drift velocity to the applied field. Find the particle flux $J_\mu$ associated with this mobility. (d) By making use of the fact that at equilibrium the particle flux must vanish, establish the Einstein relation between $\mu$ and $D$ : $$ \mu=\frac{e}{k} \frac{D}{T} . $$

Consider a system of charged particles confined to a volume $V$. The particles are in thermal equilibrium at temperature $T$ in the presence of an electric field $E$ in the $z$-direction.
(a) Let $n(z)$ be the density of particles at the height $z$. Use equilibrium statistical mechanics to find the constant of proportionality between $\frac{d n}{d z}$ and $n$.
(b) Suppose that the particles can be characterized by a diffusion coefficient $D$. Using the definition of $D$ find the flux $J_D$ arising from the concentration gradient obtained in (a).
(c) Suppose the particles are also characterized by a mobility $\mu$ relating their drift velocity to the applied field. Find the particle flux $J_\mu$ associated with this mobility.
(d) By making use of the fact that at equilibrium the particle flux must vanish, establish the Einstein relation between $\mu$ and $D$ :
$$
\mu=\frac{e}{k} \frac{D}{T} .
$$

Key Concepts

Boltzmann Distribution

This concept explains how particles distributed in a system at thermal equilibrium follow a probability law that is exponential with respect to their energy. When an external potential (such as that from an electric field) is present, the local density of particles becomes proportional to exp(?potential energy/kT), thereby establishing a link between spatial variations in density and the forces acting on the particles.

Fick's Law

Fick's law relates the diffusive flux of particles to the gradient in their concentration. It states that the flux is proportional to the negative gradient of the particle density, with the diffusion coefficient acting as the proportionality constant. This law provides the fundamental description of particle transport due purely to diffusion in the absence of other driving forces.

Mobility in an Electric Field

Mobility describes how charged particles respond to an applied electric field, quantifying the drift velocity per unit field strength. This concept is crucial when considering charged particle systems where, in addition to diffusive motion, particles experience a directed drift that contributes to the overall flux within the system.

Equilibrium Condition in Transport

At equilibrium, the net particle flux in a system must vanish even though individual contributions, such as diffusion and drift, may be non-zero. This balancing condition requires that the diffusive flux arising from concentration gradients exactly counteracts the drift flux due to external forces like an electric field, leading to a steady state distribution.

Einstein Relation

The Einstein relation is a fundamental result that connects the diffusion coefficient and the mobility of particles in a system at thermal equilibrium. It shows that mobility is proportional to diffusion, scaled by the thermal energy (typically the ratio of the charge to kT). This relation underscores the deep connection between random thermal motion (diffusion) and systematic drift in an external field.

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