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→∞. The interaction is given by a central
potential V(R)=k/Rδ−1 in a d-dimensional euclidean space, where
R is the random relative distance between the source and the test particle,
δ 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
δ and the spatial dimension d. (i) For δ<d the force and
potential energy { converge} to their respective mean values. This limit is
called Mean Field Limit. (ii) For δ>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<δ<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 δ=d the
potential energy converges to its mean and the force to zero, and (v) for
δ=d+1 the energy converges to a finite value and the force to a random
vector.Comment: 25 pages, 5 tables, Preprint Articl