Violent relaxation in one-dimensional self-gravitating system: deviation from the Vlasov limit due to finite-$N$ effect

Abstract

We investigate the effect of a finite particle number $N$ on the violent relaxation leading to the Quasi-Stationary State (QSS) in a one-dimensional self-gravitating system. From the theoretical point of view, we demonstrate that the local Poissonian fluctuations embedded in the initial state give rise to an additional term proportional to $1/N$ in the Vlasov equation. This term designates the strength of the local mean-field variations by fluctuations. Because it is of the mean-field origin, we interpret it differently from the known collision term in the way that it effects the violent relaxation stage. Its role is to deviate the distribution function from the Vlasov limit, in the collisionless manner, at a rate proportional to $1/N$ while the violent relaxation is progressing. This hypothesis is tested by inspecting the QSSs in simulations of various $N$. We observe that the core phase-space density can exceed the limiting density deduced from the Vlasov equation and its deviation degree is in accordance with the $1/N$ estimate. This indicates the deviation from the standard mean-field approximation of the violent relaxation process by that $1/N$ term. In conclusion, the finite-$N$ effect has a significant contribution to the QSS apart from that it plays a role in the collisional stage that takes place long after. The conventional collisionless Vlasov equation might not be able to describe the violent relaxation of a system of particles properly without the correction term of the local finite-$N$ fluctuations.

11 pages, 8 figures, accepted for publication in Phys. Rev. E

References in corpus (28)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• Collisionless relaxation in non-neutral plasmas
• Collisionless relaxation in gravitational systems: From violent relaxation to gravothermal collapse
• Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation
• Core-halo distribution in the Hamiltonian Mean-Field Model
• Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions
• Inconsistency in theories of violent-relaxation
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Quasi-stationary states in the self-gravitating sheet model
• Kinetic equations for systems with long-range interactions: a unified description
• Relaxation processes in one-dimensional self-gravitating many-body systems
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Statistical Mechanics of 1d Self-Gravitating Systems: The Core-Halo Distribution
• Relaxation times of unstable states in systems with long range interactions
• Relaxation in self-gravitating systems
• Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Collisional relaxation in the inhomogeneous Hamiltonian-Mean-Field model: diffusion coefficients
• Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams
• Finite $N$ corrections to Vlasov dynamics and the range of pair interactions
• Dynamics of One-dimensional Self-gravitating Systems Using Hermite-Legendre Polynomials
• The quasilinear theory in the approach of long-range systems to quasi-stationary states
• On the effectiveness of mixing in violent relaxation
• Dynamics to the universal structure of one-dimensional self-gravitating systems in the quasi-equilibrium state
• Predicting Steady States of One-dimensional Collisionless Gravitating Systems

Cited by in corpus (1)

• Chaos in violent relaxation dynamics. Disentangling micro- and macro-chaos in numerical experiments of dissipationless collapse