The Initial Mass Function as the Equilibrium State of a Variational Process: why the IMF cannot be sampled stochastically

TL;DR

The paper addresses why the stellar initial mass function (sIMF) should be viewed as the equilibrium state of a deterministic fragmentation process rather than a stochastic sampling distribution. It applies the Maximum Entropy principle to gas fragmentation in molecular clumps, deriving a MaxEnt IMF that naturally takes the form of a power law with slope $\alpha$, and shows this leads to a unique ordered mass sequence identical to the optimal sampling prescription. The key contribution is a first-principles foundation for deterministic star formation: given the fragmentation physics and a fixed embedded-cluster mass $M_{\rm ecl}$, the MaxEnt solution selects a single mass sequence $\{m_i\}$ via $m_i = Q(i/N)$, with no Poisson noise in high-mass bins and a natural cap $m_{\rm max*} \approx 150\,M_\odot$. This framework links fragmentation physics to the observed sIMF features and suggests a principled interpretation for extragalactic IMF inferences, while offering avenues to incorporate metallicity, turbulence, and cosmic-ray effects in future work.

Abstract

The stellar initial mass function (sIMF) is often treated as a stochastic probability distribution, yet such an interpretation implies Poisson noise that is inconsistent with growing observational evidence. In particular, the observed relation between the mass of the most massive star formed in an embedded cluster and the cluster's total stellar mass supports a deterministic sampling process, known as optimal sampling. However, the physical origin of optimal sampling has not been formally established in the literature. In this work, we show that the stellar mass distribution implied by optimal sampling emerges from applying the Maximum Entropy principle to the fragmentation of star-forming clumps, whose structure is set by density-dependent cooling in the optically thin regime. Here, the maximum entropy leads to unbiased distributions. By applying calculus of variations to minimize the entropy functional obtained assuming fragmentation, we recover the power-law form of the sIMF, and we show that any distribution deviating from the sIMF violates the Maximum Entropy principle. This work provides a first-principles foundation for the deterministic nature of star formation. Thus, the sIMF is the distribution resulting from a maximally unbiased system.

Figures (2)

• Figure 1: A comparison between 3 relevant masses as a function of embedded cluster mass, $M_{\rm ecl}$: the most massive star constrained by optimal sampling ($m_{\rm max}$, blue dashed line), the average stellar mass in the embedded cluster ($m_{\rm avg}$, red dot-dashed line) and the geometric mean mass ($m_{\rm geo}$, solid green line) computed according to Eq. \ref{['eq:geomass_cont']}. The blue curve is the $m_{\rm max}-M_{\rm ecl}$ relation. Up until $M_{\rm ecl}\lesssim 10^4 \, M_{\odot}$ where $m_{\rm max}\lesssim 150 \, M_{\odot}$, the average and geometric masses increase. On the left panel, for $M_{\rm ecl}\gtrsim 10^4 \, M_{\odot}$ the average and geometric masses remain constant. On the right panel, they increase because of the variability of the sIMF. We assumed the specific variability as given in marks+12yan+21, and computed with pyIGIMF$\,$Gjergo+2025 for solar metallicity. In this case, the sIMF starts to become top-heavy and consequently, both average and geometric mass increase.
• Figure :