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The fluctuations in the temperature and polarization of the cosmic microwavebackground are described by a hierarchy of Boltzmann equations. In its integralform, this Boltzmann hierarchy can be converted from the usual Fourier-spacebase into a position-space and causal description. We show that probabilitydensities for random flights play a key role in this description. The integralsystem can be treated as a perturbative series in the number of steps of therandom flights, and the properties of random flight probabilities imposeconstraints on the domains of dependence. We show that, as a result of thesedomains, a Fourier-Bessel decomposition can be employed in order to calculatethe random flight probability densities. We also illustrate how the H-theoremapplies to the cosmic microwave background: by using analytical formulae forthe asymptotic limits of these probability densities, we show that, as thephoton distribution approaches a state of equilibrium, both the temperatureanisotropies and the net polarization must vanish.

CMB and Random Flights: temperature and polarization in position space