2006/10/31 by Jörn Dunkel, Peter Talkner, Peter Hänggi · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Quantum Mechanics and Applications #Statistical Mechanics and Entropy #astro-ph #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1088/1367-2630/9/5/144
published as NewJ.Phys.9:144,2007 · 15 pages: extended version, references added
arxiv created 2007/03/09 · openalex publication_date 2007/05/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 and (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 analyse 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.