\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \)

Quantum Kinetic Theory

4. Kinetic equation

Source term

A kinetic equation, also called transport equation, is an equation of motion for the space-time behaviour of the distribution function. The derivation and justification of such equations for various systems and circumstances is one of the tasks of statistical mechanics. However, we do not wish to discuss this in any detail. Instead, we shall rely on consistency and plausibility arguments.

Our starting point will be the conservation laws. Take, for example, the energy conservation law (3.7). After substitution of (2.6) and (3.8), this equation may be written as

1

$$ \be {\sum\limits_k \int {d{\omega _k}}} \ep_k \left(\ptt f_k + \vecv_k \cdot \nabla f_k\right)=0 {\rm{ }}{\rm{.}} \la{3.1} \ee $$

This implies that the kinetic equation must be of the form

2

$$ \be \ptt f_k(x,p) + \vecv_k \cdot \nabla f_k (x,p)= C_k(x,p){\rm{ }}{\rm{,}} \la{3.2} \ee $$

with a source term such that

3

$$ \be {\sum\limits_k \int {d{\omega _k}}} \ep_k C_k =0 {\rm{ }}{\rm{.}} \la{3.3} \ee $$

The physical significance of $C_k$ may not be clear at this point, but it seems intuitively obvious that $C_k$ will involve the interaction between the particles. The constraint (4.3) expresses then the fact that in any particle interaction energy is strictly conserved.

By a similar reasoning we arrive at the conclusion that we also must have

4

$$ \be {\sum\limits_k \int {d{\omega _k}}} \vecp C_k =0 {\rm{ }}{\rm{,}} \la{3.4} \ee $$

5

$$ \be {\sum\limits_k \int {d{\omega _k}}} q_{{\rm A} k} C_k =0 {\rm{ }}{\rm{.}} \la{3.5} \ee $$
  • These relations reflect the conservation laws of momentum and charge on the microscopic level.

Collision invariants