A numerical comparison of theories of violent relaxation

Using N-body simulations with a large set of massless test particles we compare the predictions of two theories of violent relaxation, the well known Lynden-Bell theory and the more recent theory by Nakamura. We derive ``weaken'' versions of both theories in which we use the whole equilibrium coarse-grained distribution function as a constraint instead of the total energy constraint. We use these weaken theories to construct expressions for the conditional probability $K_i(\tau)$ that a test particle initially at the phase-space coordinate $\tau$ would end-up in the $i$'th macro-cell at equilibrium. We show that the logarithm of the ratio $R_{ij}(\tau) \equiv K_i(\tau)/K_j(\tau)$ is directly proportional to the initial phase-space density $f_0(\tau)$ for the Lynden-Bell theory and inversely proportional to $f_0(\tau)$ for the Nakamura theory. We then measure $R_{ij}(\tau)$ using a set of N-body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both theories are at odds with the numerical results, qualitatively and quantitatively.Comment: Replaced with a revised version, which is now accepted to MNRAS. LaTeX, 12 pages, 6 figure

Relaxation of a Collisionless System and the Transition to a New Equilibrium Velocity Distribution

In this paper, we present our conclusions from the numerical study of the collapse of a destabilized collisionless stellar system. We use both direct integration of the Vlasov-Poisson equations and an N-body tree code to obtain our results, which are mutually confirmed. We find that spherical and moderately nonspherical collapse configurations evolve to new equilibrium configurations in which the velocity distribution approaches a Gaussian form, at least in the central regions. The evolution to this state has long been an open question, and in this work we are able to clarify the process responsible and to support predictions made from statistical considerations (Lynden-Bell 1967; Nakamura 2000). The simulations of merging N-body systems show a transition to a Gaussian velocity distribution that is increasingly suppressed as the initial separation of centres is increased. Possible reasons for this are discussed.Comment: 25 pages, LaTeX. Accepted for publication in Ap