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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 $Ntoinfty$.
As an entry for the 2012 Gordon-Bell performance prize, we report performance results of astrophysical N-body simulations of one trillion particles performed on the full system of K computer. This is the first gravitational trillion-body simulation i
This paper contains a rigorous mathematical example of direct derivation of the system of Euler hydrodynamic equations from Hamiltonian equations for N point particle system as N tends to infinity. Direct means that the following standard tools are n
We present a new symplectic integrator designed for collisional gravitational $N$-body problems which makes use of Kepler solvers. The integrator is also reversible and conserves 9 integrals of motion of the $N$-body problem to machine precision. The
We present a pedagogical discussion of Similarity Renormalization Group (SRG) methods, in particular the In-Medium SRG (IMSRG) approach for solving the nuclear many-body problem. These methods use continuous unitary transformations to evolve the nucl
The general consensus in the N-body community is that statistical results of an ensemble of collisional N-body simulations are accurate, even though individual simulations are not. A way to test this hypothesis is to make a direct comparison of an en