Superstatistics in the context of relativity

The statistical properties of space and astrophysical environments often exhibit heavy-tailed distributions, reflecting deviations from thermal equilibrium. These distributions can often be represented as a superposition of statistics, i.e., superstatistics. However, the heavy-tailed nature of these distributions can lead to significant nonphysical contributions from unrealistic superluminal particles, thereby biasing thermodynamic properties and the overall dynamics. In this work, we first quantify these unphysical effects for key thermodynamic quantities, like pressure and entropy, within the three universality classes of superstatistics, and we discuss a regularized formulation that resolves the issue without requiring a fully relativistic treatment. We then develop a relativistic extension of superstatistics by generalizing the Maxwell-Jüttner distribution to nonequilibrium settings. We derive the thermodynamic properties of the resulting distributions and analyze their asymptotic behavior in the weakly relativistic and ultrarelativistic limits. Finally, we apply our model to observational data of solar energetic protons and show that it provides an excellent fit, demonstrating its relevance for describing the statistical properties of high-energy astrophysical systems.