Theoretical Evidence for Planck Transformation of Temperature: Boltzmann-Gibbs and Tsallis Blast-Wave Models

Abstract: In physics, thermodynamic quantities are governed by two mutually exclusive relativistic transformations: the Planck and Ott transformations. This study derives both transformations from the fundamental principles of relativistic mechanics and equilibrium thermodynamics, treating the fundamental thermodynamic potential as a Lagrangian or Hamiltonian function for a moving system. We show that the Planck transformations align with the core principles of both Hamiltonian and Lagrangian mechanics, whereas the Ott transformations deviate from the foundational principles of Hamiltonian mechanics. These findings are supported by analysis of an ultrarelativistic ideal gas of quarks and gluons in the Stefan-Boltzmann limit. Additionally, we develop Boltzmann-Gibbs and Tsallis blast-wave models for finite-volume systems. By comparing local equilibrium transverse momentum distributions of hadrons -- derived from these models using the Planck and Ott transformations -- with global equilibrium distributions from Boltzmann-Gibbs and Tsallis formalisms, using identical parameters in the laboratory reference frame, we find that only the Planck transformations yield accurate results, while the Ott transformations produce significant discrepancies.