Quantum Kinetic Theory

This lecture note deals with the kinetic theory of quantum gases consisting of fermions, bosons or a mixture of both. The theory is concerned with non-equilibrium states and processes in particle systems whose constituents are in some way weakly coupled. That is to say, it is assumed that the interaction between the particles can be characterized by a small parameter. This can be a dimensionless coupling constant, or the ratio of the mean distance between the particles and their size or the range of the interaction. However, it is not assumed that the quantum effects arising from the Fermi or Bose statistics are small.

The value of kinetic theory is that it enables one to explicitly calculate the transport coefficients for a variety of physically interesting systems, which include dilute gasses, plasma's and electrons in metals. Despite the restriction of weak coupling and rarefaction, the theory can also be applied to dense liquids by taking quasi-particles, instead of bare ones, as the fundamental constituents of matter.

The number of applications in physics, astrophysics and, especially, in the technical sciences, is enormous, and we have to confine ourselves to the most general subjects which exhibit the basic kinetic phenomena, and the methods of treating them. Most attention will be given to neutral quantum gasses as the simplest example treated by kinetic theory. However, the presentation will be kept as general as possible to bring the general features and basic reasoning to the forefront.

• For the following we will borrow some notations from relativity theory. We will often indicate a point of space-time as \(x = x^\mu = (t, \bf{x} )\), \(\mu = 0,1,2,3 \), and particle energy-momentum by $p = p^\mu = (p_0,\vecp)$. The energy $p_0$ will always be equal to $\ep(\vecp) = \vecp^2/2m$, except where indicated differently. Gradient operator and time derivative may also be combined: $\pt_\mu = (\ptt, \nabla)$ , $\pt^\mu = (\ptt, -\nabla)$. Furthermore, we will often write $x\cdot p = x_\mu p^\mu = x^\mu p_\mu$ for $t p_0-\vecx\cdot\vecp$.
• We will use natural units $\hbar =c = k_{\rm B} = 1$ throughout.