  We investigate the transverse momentum distribution (TMD) statistics from
three different theoretical approaches. In particular, we explore the framework
used for string models, wherein the particle production is given by the
Schwinger mechanism. The thermal distribution arises from the Gaussian
fluctuations of the string tension. The hard part of the TMD can be reproduced
by considering heavy tailed string tension fluctuations, for instance, the
Tsallis $q$-Gaussian function, giving rise to a confluent hypergeometric
function that fits the entire experimental TMD data. We also discuss the
QCD-based Hagerdon function, another family of fitting functions frequently
used to describe the spectrum. We analyze the experimental data of minimum bias
pp collisions reported by the BNL Relativistic Heavy Ion Collider (RHIC) and
the CERN Large Hadron Collider (LHC) experiments (from $\sqrt{s}=0.2$ TeV to
$\sqrt{s}=13$ TeV). We extracted the corresponding temperature by studying the
behavior of the spectra at low transverse momentum values. For the three
approaches, we compute all moments, highlighting the average, variance, and
kurtosis. Finally, we compute the Shannon entropy and the heat capacity through
the entropy derivative with respect to the temperature. We found that the
$q$-Gaussian string tension fluctuations lead to a monotonically increasing
heat capacity as a function of the center of mass energy, which is also
observed for the Hagedorn fitting function. This behavior is consistent with
the experimental observation that the temperature slowly rises with increments
of the collision energy.
