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

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 $N\to \infty$. 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.

25 pages, 5 tables, Preprint Article