The $p_T$ spectrum of the charged particles produced in pp collisions can be described through the fragmentation of color strings by assuming the Schwinger mechanism with heavy-tailed distributions describing the string tension fluctuations. In particular, we discuss the string tension fluctuations described by the q-Gaussian distribution, and the one derived for the QCD-based Hagedorn function. As a result of the latter, the physics involved in the Hagedorn function can be understood under the string fragmentation scheme. Consequently, the system created in the collision departs from the thermal description, which is corroborated by deriving the temperature fluctuations of these approaches. We found that temperature fluctuations are a local property occurring in a small part of the system.