Relativistic Hydrodynamics 2026, 04: Relativistic Kinetic Theory

The relativistic Boltzmann equation introduces considerations for Lorentz invariance, but otherwise follows the same approach as the Newtonian one. It coincides with Liouville's theorem.

From the relativistic H-theorem, via application of relativistic transport fluxes to the underlying tensors, one derives the number density current, the rest-mass density current, the energy-momentum tensor, the third moment, and the entropy current.

From general relativity, the energy conditions applicable to the problems arise. Similar to the Newtonian transport equation, the generic tensor assumes binary collisions and the full contribution emerges from integration over the invariant element. The result is a relativistic conservation for the transport as a relativistic extension of the Newtonian conservation equation. As in the Newtonian case, derive continuity and energy/momentum conservation.

For fluids with quantum contributions, i.e. non-degenerate, the equilibrium equation is in the absence of external forces. It introduces a degeneracy factor for the internal spin degrees of freedom. It gives rise to rest-mass density, energy density, free and specific energy and chemical potentials, as well as a relativistic coldness through the Lorentz factor

The degenerate case runs analogue.

Relativistic perfect fluids ignore viscous effects and heat fluxes, their pressure tensor is diagonal.