Relative entropy, Haar measures and relativistic canonical velocity distributions

Abstract

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 of relative entropy. The latter characterizes probability distributions with respect to a pre-specified reference measure. To identify the canonical BGS entropy with a relative entropy is appealing for two reasons: (i) the maximum entropy principle assumes a coordinate invariant form; (ii) thermodynamic equilibrium distributions, which are obtained as solutions of the maximum entropy problem, may be characterized in terms of the transformation properties of the underlying reference measure (e.g., invariance under group transformations). As examples, we analyze two frequently considered candidates for the one-particle equilibrium velocity distribution of an ideal gas of relativistic particles. It becomes evident that the standard Jüttner distribution is related to the (additive) translation group on momentum space. Alternatively, imposing Lorentz invariance of the reference measure leads to a so-called modified Jüttner function, which differs from the standard Jüttner distribution by a prefactor, proportional to the inverse particle energy.

15 pages: extended version, references added

References in corpus (5)

• Relativistic diffusion processes and random walk models
• Covariant Equilibrium Statistical Mechanics
• One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model
• Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation
• The maximum relative entropy principle

Cited by in corpus (11)

• Relativistic Brownian Motion
• Thermal equilibrium and statistical thermometers in special relativity
• Statistical thermodynamics of a two dimensional relativistic gas
• Statistical kinetic treatment of relativistic binary collisions
• Time parameters and Lorentz transformations of relativistic stochastic processes
• A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas
• A rigorous solution to the superluminal issue in the diffusion equation
• Thermostat for a relativistic gas
• The Relativistic Maxwell-Jüttner Velocity Distribution Function
• Adiabatic Elimination in Relativistic Stochastic Mechanics
• Relativistic Lévy processes