# Distribution Probability of Force for a Physical System of N Random Particles

## TL;DR

This paper investigates the distribution of forces in a large system of randomly distributed particles, extending classical limit theorems to N-body problems with various force laws and analyzing their asymptotic behaviors.

## Contribution

It introduces a mathematical framework connecting the generalized CLT with N-body force calculations, exploring different limiting regimes and force behaviors.

## Key findings

- Derived the force distribution for large N under a 1/r^δ force.
- Analyzed the limit force behavior in Vlasov and Fluctuation regimes.
- Explored the impact of Lévy exponent α on force limits.

## Abstract

The present paper attempts to address a discussion on mathematical grounds of a model to associate the generalized version of the CLT and the $N$-body problem related to the calculation of the force on a single star or particle due to the $N-1$ stars or particles whenever they are randomly distributed in the space and $N\rightarrow\infty$. We calculate the resultant force on a test particle immersed in a $N$-particle system under a $1/r^\delta$ force ($\delta>0$) and discuss the limit force under different approaches referred to as the Vlasov limit and Fluctuation Limit. Also one shows the behaviour of the limit force in different domains for the L\'evy exponent ($\alpha$).