A little-known thermomechanical relation between entropy and action, originally discovered by Boltzmann in the classical domain, was later reconsidered by de Broglie in relation to the wave-particle duality in the free propagation of single particles. In this paper we present a version adapted to the phenomenological description of the hadronization process. The substantial difference with respect to the original de Broglie scheme is represented by the universality of the temperature at which the process occurs; this in fact coincides with the Hagedorn temperature. The main results are: 1) a clear connection between the universality of the temperature and the existence of a confinement radius of the color forces; 2) a lower bound on the hadronic mass, represented by the universal temperature, in agreement with experimental data; 3) a scale invariance which allows the reproduction of the well-known hadronic mass spectrum solution of the statistical bootstrap model. The approach therefore presents a heuristic interest connected to the study of the strong interaction.

The transverse momentum distribution of J/ψ mesons produced in pp collisions at the center-of-mass energy 5 TeV, 7 TeV，and 13 TeV is described by the modified Hagedorn function which is based on the Tsallis function and Hagedorn function. The calculated results by the modified Hagedorn function are in accord with experimental data measured by the LHCb Collaboration at LHC. The related parameters are obtained and analyzed.