8.3.1 Continuity Equation

The physical quantity associated with the zeroth order moment is the carrier density $n(x)$:

$$n(x) = \int \frac{dk}{2 \pi} f(x,k) .$$

The zeroth order moment equation describes the conservation of particle number, also called conservation of mass.

We derive the corresponding conservation equation by integrating the Wigner equation over all velocities $k \in \mathbb{R}$

$$\begin{gather*}\begin{split}& \frac{\partial}{\partial t} \int dk f(x,k) = & ... ...m{eq}}(x,k) \frac{n(x)}{n_{\mathrm{eq}}(x)} \Bigr) . \end{split}\end{gather*}$$ (8.6)

By definition the Wigner potential $V_{\mathrm{w}}(x,k)$ is an odd function in $k$. Hence the contribution from the integral

$$\int dk \int dk' V_{\mathrm{w}}(x, k - k') f(x, k') = \int dk' f(x, k', t) \int dk V_{\mathrm{w}}(x, k - k')$$ (8.7)

vanishes. This property expresses conservation of particle number from potential scattering. In a similar way the term stemming from scattering also vanishes as we have

$$\int dk f(x,k) = 2 \pi n(x) = n(x) \frac{\int dk f_{\mathrm{eq}}(x,k)}{n_{\mathrm{eq}}(x)} .$$

With the introduction of the (particle) current density $j(x,t)$:

$$j(x,t) = \frac{\hbar}{2 \pi m^{*}} \int dk k f(x, k, t)$$ (8.8)

we finally get the continuity equation (in vector notation):

$$\frac{\partial n}{\partial t} = - \nabla \cdot {\mathbf{j}}$$

The aim is to find a discretization of the full Wigner equation which conserves mass.