text: The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.
  Its generalization in general relativity is
  where  is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant  phase space as opposed to fully contravariant  phase space.
  In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of quantum field theory. This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.