Relative entropy, Haar measures and relativistic canonical velocity distributions

The thermodynamic maximum principle for the Boltzmann-Gibbs-Shannon (BGS)entropy is reconsidered by combining elements from group and measure theory.Our analysis starts by noting that the BGS entropy is a special case ofrelative entropy. The latter characterizes probability distributions withrespect to a pre-specified reference measure. To identify the canonical BGSentropy with a relative entropy is appealing for two reasons: (i) the maximumentropy principle assumes a coordinate invariant form; (ii) thermodynamicequilibrium distributions, which are obtained as solutions of the maximumentropy problem, may be characterized in terms of the transformation propertiesof the underlying reference measure (e.g., invariance under grouptransformations). As examples, we analyze two frequently considered candidatesfor the one-particle equilibrium velocity distribution of an ideal gas ofrelativistic particles. It becomes evident that the standard J\"uttnerdistribution is related to the (additive) translation group on momentum space.Alternatively, imposing Lorentz invariance of the reference measure leads to aso-called modified J\"uttner function, which differs from the standardJ\"uttner distribution by a prefactor, proportional to the inverse particleenergy.Comment: 15 pages: extended version, references adde