# Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

## TL;DR

This paper develops asymptotic orderings for the Master kinetic equation describing large systems of hard spheres, clarifying its relation to Boltzmann and Enskog equations in different density regimes.

## Contribution

It introduces physically meaningful asymptotic orderings for the Master kinetic equation, connecting it with classical kinetic equations in various regimes.

## Key findings

- Derived asymptotic regimes for dilute and dense systems.
- Established relationships between Master, Boltzmann, and Enskog equations.
- Provided a framework for approximations in large hard-sphere systems.

## Abstract

In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large $N-$body system of hard spheres,\textit{ i.e.,} formed by $N\equiv\frac{1}{\varepsilon} \gg1$ particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto \textit{et al}., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the $1-$body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of $\varepsilon$ for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of $\varepsilon.$