On the probabilistic approach to the N-body problem
  This work discusses the main analogies and differences between the
deterministic approach underlying most cosmological N-body simulations and the
probabilistic interpretation of the problem that is often considered in
mathematics and statistical mechanics. In practice, we advocate for averaging
over an ensemble of $S$ independent simulations with $N$ particles each in
order to study the evolution of the one-point probability density $\Psi$ of
finding a particle at a given location of phase space $(\mathbf{x},\mathbf{v})$
at time $t$. The proposed approach is extremely efficient from a computational
point of view, with modest CPU and memory requirements, and it provides an
alternative to traditional N-body simulations when the goal is to study the
average properties of N-body systems, at the cost of abandoning the notion of
well-defined trajectories for each individual particle. In one spatial
dimension, our results, fully consistent with those previously reported in the
literature for the standard deterministic formulation of the problem, highlight
the differences between the evolution of the one-point probability density
$\Psi(x,v,t)$ and the predictions of the collisionless Boltzmann
(Vlasov-Poisson) equation, as well as the relatively subtle dependence on the
actual finite number $N$ of particles in the system. We argue that
understanding this dependence with $N$ may actually shed more light on the
dynamics of real astrophysical systems than the limit $N\to\infty$.
