A numerical comparison of theories of violent relaxation

Abstract

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 $K_i(τ)$ that a test particle initially at the phase-space coordinate $τ$ would end-up in the $i$'th macro-cell at equilibrium. We show that the logarithm of the ratio $R_{ij}(τ) \equiv K_i(τ)/K_j(τ)$ is directly proportional to the initial phase-space density $f_0(τ)$ for the Lynden-Bell theory and inversely proportional to $f_0(τ)$ for the Nakamura theory. We then measure $R_{ij}(τ)$ 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.