We present, in a coherent rigorous way, a generalization of the relativistic kinetic theory of particles that includes the monoparametric (parameter q) family of thermal equilibrium distributions of Tsallis thermostatistics. The q-depending entropy function leads to a local distribution function with a well-defined generalized temperature (not depending on q) but with other quantities such as the energy density, pressure, particle number density, etc., depending on q. In general, this extension forms the basis for relativistic thermodynamics of equilibrium and nonequilibrium transport processes that includes nonextensivity. Astrophysics and cosmology are two of the areas where Tsallis’ theory may find important applications. This is due to the nonextensive thermodynamical behaviour associated with the long range of gravitational interaction. Thus, a relativistic nonextensive kinetic theory may be appropriate for the treatment of a number of real systems, like the system of galaxies, hot gases of particles and plasmas. Also, it may be used to study a number of interesting problems in cosmology, like, for instance, the distribution of bubbles sizes, and consequently the level of inhomogeneity generated, in a scalar-tensor theory of gravity, as well as to apply it to another constituents, if we could extend the known history of the universe back to quark/hadron transition and electroweak phase transition.
