Finite Volume Effects on Transverse Momentum Spectra at LHC and RHIC Using a Blast-Wave Model with Planck Transformed Temperatures
We investigate finite volume effects on the transverse momentum spectra of charged pions produced in the most central heavy-ion collisions at RHIC and LHC energies. A cylindrically symmetric finite volume Boltzmann-Gibbs blast-wave model is employed that fully incorporates the finite longitudinal extent of the fire cylinder at kinetic freeze-out. The model applies Planck transformations to convert the local rest frame temperature and chemical potential of each fluid element into laboratory frame values, ensuring full Lorentz covariance. This approach is compared with the conventional infinite volume blast-wave model, in which the thermodynamic parameters remain defined in the local rest frame while the particle momenta are expressed in the laboratory frame. Both models are fitted to the experimental transverse momentum distributions of charged pions measured by the HADES, STAR, PHENIX, and ALICE collaborations over the center-of-mass energy range $\sqrt{s_{NN}} = 2.4$ GeV to $5.44$ TeV. The finite volume model with Planck transformed laboratory frame parameters yields temperature values fully consistent with relativistic thermodynamics (except for a small anomaly at $\sqrt{s_{NN}} = 193$ and $200$ GeV) and produces realistic fire cylinder volumes several times larger than the initial nuclear overlap volume. In contrast, the conventional infinite volume model yields unphysical results: infinite volume, infinite maximum half-length, and maximum longitudinal flow velocity equal to the speed of light at all energies. These findings demonstrate that a proper treatment of finite system size, together with the correct Lorentz (Planck form) transformation of the thermodynamic variables, is essential for the reliable extraction of freeze-out parameters in heavy-ion collisions.