# On the Boltzmann-Grad limit of the Master Kinetic Equation

## TL;DR

This paper investigates the Boltzmann-Grad limit of the Master kinetic equation for hard-sphere systems, focusing on the non-commutative properties of the limit operator and its implications for deriving the Boltzmann equation.

## Contribution

It provides a detailed analysis of the Boltzmann-Grad limit operator's non-commutative nature within the Master kinetic equation framework.

## Key findings

- The form of the limit equation depends on the prescription of the limit operator.
- The non-uniqueness of the limit equation affects the derivation of the Boltzmann equation.
- Implications for the global prescription of the Boltzmann equation are discussed.

## Abstract

In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. The statistical description is couched in terms of the Master kinetic equation, i.e., the kinetic equation which realizes the axiomatic "\textit{ab initio}" approach to the classical statistical mechanics of finite hard-sphere systems recently developed (Tessarotto \textit{et al.}, 2013-2017). The issue addressed here concerns the prescription of the BG-limit operator and specifically the non-commutative property of $\mathcal{L}_{BG}$ with the free-streaming operator which enters the same kinetic equation.\ It is shown\ that the form of the resulting limit equation remains in principle non-unique, its precise realization depending critically on the way the action of the same operator is prescribed. Implications for the global prescription of the Boltzmann equation are pointed out.