On the probability distributions of the force and potential energy for a system with an infinite number of random point sources

TL;DR

This paper analyzes the probability distributions of force and potential energy for a test particle interacting with infinitely many random sources, identifying different limiting behaviors based on the interaction exponent and spatial dimension.

Contribution

It introduces a framework for understanding the limiting distributions of force and energy in systems with infinite random sources, including new non-singular and singular limit classifications.

Findings

Identifies three non-singular limits: Mean Field, Thermodynamic, and Mixed.

Derives conditions for convergence of force and energy distributions.

Classifies two singular limits at specific interaction exponents.

Abstract

In this work, we study the probability distribution for the force and potential energy of a test particle interacting with $N$ point random sources in the limit $N\rightarrow\infty$. The interaction is given by a central potential $V(R)=k/R^{\delta-1}$ in a $ d$-dimensional euclidean space, where $R$ is the random relative distance between the source and the test particle, $\delta$ is the force exponent, and $k$ is the coupling parameter. In order to assure a well-defined limit for the probability distribution of the force and potential energy, we { must} renormalize the coupling parameter and/or the system size as a function of the number $N$ of sources. We show the existence of three non-singular limits, depending on the exponent $\delta$ and the spatial dimension $d$. (i) For $\delta<d$ the force and potential energy { converge} to their respective mean values. This limit is called Mean Field Limit. (ii) For $\delta>d+1$ the potential energy converges to a random variable and the force to a random vector. This limit is called Thermodynamic Limit. (iii) For $d<\delta<d+1$ the potential energy converges to its mean and the force to a random vector. This limit is called Mixed Limit Also, we show the existence of two singular limits: (iv) for $\delta=d$ the potential energy converges to its mean and the force to zero, and (v) for $\delta=d+1$ the energy converges to a finite value and the force to a random vector.