  Rayleigh-B\'enard convection, i.e. the flow of a fluid between two parallel
plates that is driven by a temperature gradient, is an idealised setup to study
thermal convection. Of special interest are the statistics of the turbulent
temperature field, which we are investigating and comparing for three different
geometries, namely convection with periodic horizontal boundary conditions in
three and two dimensions as well as convection in a cylindrical vessel, in
order to work out similarities and differences. To this end, we derive an exact
evolution equation for the temperature probability density function (PDF).
Unclosed terms are expressed as conditional averages of velocities and heat
diffusion, which are estimated from direct numerical simulations. This
framework lets us identify the average behaviour of a fluid particle by
revealing the mean evolution of fluid of different temperatures in different
parts of the convection cell. We connect the statistics to the dynamics of
Rayleigh-B\'enard convection, giving deeper insights into the temperature
statistics and transport mechanisms. We find that the average behaviour is
described by closed cycles in phase space that reconstruct the typical
Rayleigh-B\'enard cycle of fluid heating up at the bottom, rising up to the top
plate, cooling down and falling down again. The detailed behaviour shows subtle
differences between the three cases.
