Study on relativistic transformations for thermodynamic quantities: Boltzmann-Gibbs and Tsallis blast-wave models

Abstract

This study derives the relativistic transformations of thermodynamic quantities from the Lorentz transformations applied to the four-momentum components of a thermodynamic system, which is stationary in the inertial reference frame $K_0$ and moves at constant velocity relative to the laboratory frame $K$. Thermodynamic variables are introduced into the formalism via the zeroth component of the four-momentum in $K_0$, representing the system's internal energy. By treating the three-momentum as an independent state variable, thermodynamic quantities are defined by differentiating the zeroth component of the four-momentum (the Hamiltonian) in the reference frame $K$ with respect to the independent state variables, yielding the fundamental thermodynamic potential. This approach results in the Non-Planck transformations, which differ from the Planck transformations by a factor of $\alpha$. In contrast, by adopting the three-velocity as an independent state variable, thermodynamic quantities are obtained by differentiating the negative Lagrangian, derived from the zeroth component of the four-momentum via Legendre transformations, with respect to the independent state variables, producing the conjugate fundamental thermodynamic potential. This yields the Planck transformations. Conversely, the Ott transformations are derived from the zeroth component of the four-momentum by treating velocity as an independent state variable. This approach conflicts with the principles of mechanics, resulting in an energy that does not qualify as a thermodynamic potential. To validate these findings, we analyze an ultrarelativistic ideal gas of quarks and gluons within the Stefan-Boltzmann limit. Furthermore, we develop consistent Boltzmann-Gibbs and Tsallis blast-wave models for finite-volume freeze-out firecylinders in heavy ion collisions, incorporating Planck and Ott transformations.