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 $Ψ$ 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 $Ψ(x,v,t)$ 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 $N\to\infty$.