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

TL;DR

The paper presents a unified relativistic thermodynamics framework by tying thermodynamic potentials to relativistic dynamics. It distinguishes a fundamental Hamiltonian-based potential, using momentum as an independent state variable, from a conjugate negative-Lagrangian potential, using velocity as the state variable, and derives three classes of transformations: Non-Planck (from the Hamiltonian with $\mathbf{P}$), Planck (from the conjugate Lagrangian with $\mathbf{v}$), and Ott (from the total energy as a function of $\mathbf{v}$, which is not a thermodynamic potential). It shows Planck transformations reproduce consistent temperature and chemical-potential behavior across ensembles, while Ott transformations introduce inconsistencies, particularly in moving quark-gluon plasma and finite-volume blast-wave models. The work validates Planck-type transformations in ultrarelativistic QGP via Stefan–Boltzmann limit and implements Boltzmann-Gibbs and Tsallis blast-wave models for finite-volume firecylinders, demonstrating Planck-based results align global and local distributions, whereas Ott-based results do not. These findings have practical implications for modeling relativistic heavy-ion collisions and interpreting thermodynamic quantities in moving media. $$T_0, p_0, \mu_0$$ in the rest frame and their Lorentz-transformed counterparts are central to connecting equilibrium thermodynamics with relativistic dynamics across ensembles, with the rest energy $E_0$ acting as the crucial bridge between the two theories.

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 $α$. 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.

Figures (4)

• Figure 1: (Color online) The temperature $T$ and pressure $p$ in the laboratory reference frame $K$ are expressed as functions of the system's momentum $\mathbf{P}$, where the system is at rest in the moving reference frame $K_{0}$. This pertains to an ultrarelativistic ideal gas of quarks and gluons with $N_{f} = 2+1$ flavors in a system of mass $E_{0}=1$ GeV in a volume $V_{0} = 1$ fm$^{3}$ at $\mu_{0} = 0$. The temperatures $T$ (Non-Planck), $T$ (Planck), $T_{*}$, and $T_{0}$ were computed using Eqs. (\ref{['x20']}), (\ref{['p8']}), (\ref{['p14']}), and (\ref{['57']}), respectively. Similarly, the pressures $p$ (Non-Planck), $p$ (Planck), $p_{*}$, and $p_{0}$ were determined using Eqs. (\ref{['x21']}), (\ref{['p9']}), (\ref{['p15']}), and (\ref{['58']}), respectively.
• Figure 2: (Color online) The equation of state for an ultrarelativistic ideal gas of quarks and gluons with $N_{f} = 2+1$ flavors in a system of mass $E_{0}=1$ GeV with $P = 2$ GeV at $\mu_{0} = 0$. For additional details, refer to the caption of Figure \ref{['fig1']}.
• Figure 3: (Color online) Comparison of the Maxwell-Boltzmann global equilibrium transverse momentum distribution (solid line) for $\pi^{+}$ pions with the Maxwell-Boltzmann local equilibrium transverse momentum distributions from the blast-wave (BW) model, using Planck transformations (long dashed line) and Ott transformations (dotted line). The distributions are evaluated at a temperature of $T=100$ MeV and a chemical potential of $\mu=10$ MeV in the laboratory reference frame $K$ at a rapidity of $y=0$. Calculations are based on a firecylinder at freeze-out with a radius of $R=2$ fm and a maximum spacetime rapidity of $\eta_{\text{max}}=3$. For the BW model, a transverse flow velocity of $\bar{v}_{T\mathrm{f}}=0.6$ is used. Transverse momentum distribution (\ref{['t4']}) with rest-frame parameters (dashed line), defined in the rest frame of a fluid element, are evaluated at $T_0 = 100$ MeV and $\mu_0 = 10$ MeV.
• Figure 4: (Color online) Comparison of the Maxwell-Boltzmann global equilibrium transverse momentum distribution of the Tsallis-3 statistics in the zeroth-order approximation (solid line) for $\pi^{+}$ pions with the Maxwell-Boltzmann local equilibrium transverse momentum distributions of the Tsallis blast-wave (TBW) model with rest-frame parameters (dashed line) and using Planck transformations (long dashed line) and Ott transformations (dotted line), at a non-extensivity parameter of $q=1.05$. Further details are provided in the caption of Figure \ref{['fig3']}.