The accretion rate resulting in the initial mass function

Abstract The aim of this study is to provide a description of the stellar initial mass function (IMF) using the framework of non-extensive statistical mechanics. We seek to demonstrate that the Tsallis distribution, characterized by a single analytical form, can successfully reproduce the full shape of the stellar IMF, from sub-stellar objects to massive stars, without the need for multi-segment or composite models. We adopt the Tsallis distribution derived from non-extensive statistics to model the IMF and perform a detailed comparison with the Kroupa form (Mon Not R Astron Soc 322:231–246, 2001, https://doi.org/10.1046/j.1365-8711.2001.04022.x ). The model parameters, including the non-extensivity index q and the characteristic mass scale $$m_0$$ m 0 , are obtained by fitting the distribution to observational IMF data. Furthermore, we establish a dynamical interpretation of the Tsallis form through an analytical relation between the IMF and linear accretion rates. The Tsallis distribution reproduces the adopted IMF representation with high accuracy using only two parameters. The best-fit values, $$q = 1.40\pm 0.01$$ q = 1.40 ± 0.01 and $$m_0 = 0.086\pm 0.008\,M_\odot$$ m 0 = 0.086 ± 0.008 M ⊙ , yield an asymptotic power-law slope $$\alpha = 2.5$$ α = 2.5 , consistent with the Salpeter and Kroupa exponents. The model naturally reproduces the low-mass turnover near the hydrogen-burning limit and the high-mass power-law tail without requiring discontinuities. The derived analytical relation shows that, within the adopted formulation, the Tsallis form of the IMF corresponds to an accretion-rate dependence of the form $$\dot{m}=a+bm$$ m ˙ = a + b m . The non-extensivity parameter $$q$$ q may therefore provide a phenomenological characterization of fluctuations or correlations in star-forming systems. Establishing a direct connection between $$q$$ q and specific physical properties of the star-forming medium, however, requires further investigation.