CMB and Random Flights: temperature and polarization in position space
  The fluctuations in the temperature and polarization of the cosmic microwave
background are described by a hierarchy of Boltzmann equations. In its integral
form, this Boltzmann hierarchy can be converted from the usual Fourier-space
base into a position-space and causal description. We show that probability
densities for random flights play a key role in this description. The integral
system can be treated as a perturbative series in the number of steps of the
random flights, and the properties of random flight probabilities impose
constraints on the domains of dependence. We show that, as a result of these
domains, a Fourier-Bessel decomposition can be employed in order to calculate
the random flight probability densities. We also illustrate how the H-theorem
applies to the cosmic microwave background: by using analytical formulae for
the asymptotic limits of these probability densities, we show that, as the
photon distribution approaches a state of equilibrium, both the temperature
anisotropies and the net polarization must vanish.
