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

Quantum Kinetic Theory

2. Distribution function

Non-equilibrium

On the macroscopic level, the state of a many-particle system is described by the conserved charge densities $N_{\rm A}(x)$, the energy density $E(x)$, and the momentum density $\vecG(x)$. Furthermore, one may define a space- and time-dependent entropy density $S(x)$. Since entropy is not conserved, $S(x)$ satisfies a balance equation of the form

1

$$ \be {\ptt}S(x) + \nabla \cdot {J_{S}}(x) = \sigma (x) {\rm{ }}{\rm{,}} \la{1.1} \ee $$

where $J_{S}$ is the entropy flow, and $\sg$ the local entropy production (per unit volume and unit time) which is never negative to be in accord with the second law of thermodynamics:

2

$$ \be \sigma (x) \ge 0 {\rm{ }}{\rm{.}} \la{1.2} \ee $$
Exercise 1
  • Show by integrating over all space that (2.1) implies that the total entropy never decreases.

A non-equilibrium state of a system is characterized by a non-zero entropy production. However, at this stage we do not yet know how the entropy production is related to the irreversible processes which may occur in the system. Therefore, we shall use for now a more heuristic criterion for non-equilibrium behaviour: we shall declare a system to be outside equilibrium if the distribution function

3

$$ \be f_k (x,p) = {n_k}(p) + \delta {f_k}(x,p) \la{1.3} \ee $$

differs by a non-zero amount $\dl f_k$ from the equilibrium Fermi or Bose distribution function $n_k (p)$.

Since, in general, the properties of a non-equilibrium system are non-stationary and non-uniform, the distribution function is space and time dependent. To keep the discussion general we assume that there are a number $k = 1,2,...,$ of different particle species in the system. However, we shall confine ourselves to systems only slightly out of equilibrium which means that $\dl f_k$ is small, and that quadratic terms may be neglected.

Macroscopic variables

Let us now write down the particle density of component $k$ :

4

$$\be N_k (x) = {\int {d\omega }_k}\,{f_k}(x,p){\rm{ }}{\rm{,}} \la{1.4} \ee $$

where we introduced the notation

5

$$ \be d{\omega _k}: = {g_k}\frac{{{d^3}p}}{{{{(2\pi )}^3}}}{\rm{ }}{\rm{.}} \la{1.5} \ee $$

with $g_k$ the spin—weight factor, i.e. the number of independent spin states of the particles. It is customary to regard $f_k$ as giving the distribution of the particles in $\mu$-space:

$\triangleright\,\, f_k(x,p) \Dl^3 p \Dl^3 x =$ the number of particles of species $k$ which at time $t$ are situated in the small volume $\Dl^3 x$ at the point $\vecx$ with momenta lying in the range $(\vecp - \half \Dl \vecp , \vecp + \half \Dl \vecp)$.

However, it must be stressed that this definition is not really necessary. It is sufficient to assume that the density can be expressed in terms of an ancillary quantity $f_k (x,p)$ which in equilibrium reduces to the Fermi or Bose distribution function $n_k (p)$. In terms of this ancillary quantity the energy density and entropy density of species $k$ take the forms:

6

$$ \be {E_k}\left( x \right) = {\rm{ }}\int {d{\omega _k}} \,{\varepsilon _k}\, {f_k}\left( x,p \right){\rm{ }}{\rm{,}} \la{1.6} \ee $$

7

$$ \be {S_k}\left( x \right)= {\rm{ }}- \int {d{\omega _k}}\, [{f_k}\log{f_k}-\eta (1 + \eta {f_k})\log\left( {1 + \eta {f_k}} \right)]{\rm{ }}{\rm{,}} \la{1.7} \ee $$
  • These expressions are the direct generalizations of the energy and entropy densities of a quantumgas in equilibrium by means of the substitution $n_k (p)\rightarrow f_k (x,p)$; $\et = +1$ for bosons, and $\et= —1$ for fermions.
  • Because of the absence of any interaction terms, the expressions are only valid for weakly coupled or dilute particle systems. Nevertheless, this theory of non—equilibrium "ideal" gases is very important as a model and guideline for the construction of more complicated theories.