On the probabilistic approach to the N-body problem

Abstract

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 $N\to \infty$.

arXiv comment15 pages, 6 figures. Accepted for publication in MNRAS

References in corpus (27)

• A Universal Density Profile from Hierarchical Clustering
• The Inner Structure of LambdaCDM Halos I: A Numerical Convergence Study
• Statistical mechanics and dynamics of solvable models with long-range interactions
• Galaxies in N-body simulations: overcoming the overmerging problem
• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• NBODY6++GPU: Ready for the gravitational million-body problem
• A Very Fast and Momentum-Conserving Tree Code
• Towards optimal softening in 3D N-body codes: I. Minimizing the force error
• On the physical origin of dark matter density profiles
• A new approach to simulating collisionless dark matter fluids
• An adaptively refined phase-space element method for cosmological simulations and collisionless dynamics
• Kinetic theory of spatially inhomogeneous stellar systems without collective effects
• Statistical mechanics of collisionless orbits. I. Origin of central cusps in dark-matter halos
• Discreteness effects in cosmological N-body simulations
• The dynamical structure of dark matter haloes
• ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation
• Quasi-stationary states in the self-gravitating sheet model
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Direct Integration of the Collisionless Boltzmann Equation in Six-dimensional Phase Space: Self-gravitating Systems
• Statistical Mechanics of 1d Self-Gravitating Systems: The Core-Halo Distribution
• Secondary infall and dark matter haloes
• Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory
• Gravitational Collapse in One Dimension
• Do galaxies form a spectroscopic sequence?
• Vlasov versus N-body: the Hénon sphere
• Dark matter halo's and self similarity
• Integer Lattice Dynamics for Vlasov-Poisson

Cited by in corpus (1)

• Phase-space structure analysis of self-gravitating collisionless spherical systems