The generalized Maxwell–Boltzmann distribution and its application on the Jeans mass

Abstract

The classical Jeans mass criterion, a cornerstone of astrophysics, predicts the gravitational collapse of a homogeneous, self-gravitating gas cloud based on the Maxwell–Boltzmann velocity distribution. However, this framework often fails to account for the observed unstability of Bok globules, whose measured masses frequently fall below the classical threshold. In this study, we derive a generalized Jeans mass,

, from first principles by replacing the standard distribution with a Mittag-Leffler modified Maxwell–Boltzmann (MLMB) distribution. This formalism introduces a parameter,

$$ \gamma $$ γ

, which quantifies the deviation from classical statistical mechanics. We demonstrate that for specific critical values of

$$ \gamma $$ γ

, the generalized Jeans mass aligns precisely with the observed masses of unstable Bok globules, successfully resolving the classical paradox. A comparative analysis shows that our statistical mechanical generalization achieves quantitative agreement with alternative resolutions invoking dark matter backgrounds or modified gravity. This establishes the MLMB framework as a fundamental and versatile approach for modeling gravitational instability, where the parameter

$$ \gamma $$ γ

effectively captures the complex internal dynamics of molecular clouds.