Boltzmann equation

When electric or magnetic fields are present charge will be pushed around inside a solid. We define a probability density function $f(\vec{r},\vec{k},t)$ that decribes the probability of finding an electron at position $\vec{r}$ with a wave vector $\vec{k}$ at time $t$. The integral over this density function is the number of electrons $N$.

\begin{equation} N = \int d^3r \int d^3k f(\vec{r},\vec{k},t). \end{equation}

Since the number of electrons is conserved, the integral remains constant even if forces are applied. If the probability of finding an electron at a particular position with a particular momentum decreases, the probability must increase somewhere else. This is a statement of Liouville's theorem. The integral will remain constant if the time derivative of $f$ is zero,

\begin{equation} \frac{d}{dt} f(\vec{r},\vec{k},t) = 0. \end{equation}

Writting the total derivative,

\begin{equation} \frac{d}{dt} f(\vec{r},\vec{k},t) =\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}+\frac{\partial f}{\partial k_x}\frac{dk_x}{dt}+\frac{\partial f}{\partial k_y}\frac{dk_y}{dt}+\frac{\partial f}{\partial k_z}\frac{dk_z}{dt}+\frac{\partial f}{\partial t} = 0. \end{equation}

This can be written more compactly as,

\begin{equation} \frac{d}{dt} f(\vec{r},\vec{k},t) =\frac{d\vec{r}}{dt}\cdot\nabla_{\vec{r}}f+\frac{d\vec{k}}{dt}\cdot\nabla_{\vec{k}}f+\frac{\partial f}{\partial t} = 0. \end{equation}

Using the expression for the crystal momentum $\vec{F}_{\text{ext}}=\hbar\frac{d\vec{k}}{dt}$ and the velocity $\vec{v}=\frac{d\vec{r}}{dt}$,

\begin{equation} \frac{\partial f}{\partial t} = - \frac{1}{\hbar}\vec{F}_{\text{ext}}\cdot\nabla_{\vec{k}}f-\vec{v}\cdot\nabla_{\vec{r}}f. \end{equation}

The electrons may experience collisions that change their momentum. The collisions can be with defects in the crystal, phonons, or other electrons. These collisions do not alter the number of electrons but their momenta change suddenly. Collisions are included with a collision term.

\begin{equation} \frac{\partial f}{\partial t} = - \frac{1}{\hbar}\vec{F}_{\text{ext}}\cdot\nabla_{\vec{k}}f-\vec{v}\cdot\nabla_{\vec{r}}f + \frac{\partial f}{\partial t} \bigg\rvert_{collisions}. \end{equation}

This equation is called the Boltzmann equation.