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 ‘weakened’ versions of both the theories in which we use the whole equilibrium coarse-grained distribution function ¯ f i as a constraint instead of the total energy constraint. We use these weakened theories to construct expressions for the conditional probability Ki(τ ) that a test particle initially at the phase-space coordinate τ would endup in the ith macro-cell at equilibrium. We show that the logarithm of the ratio Rij(τ ) ≡ Ki(τ )/Kj(τ )i sdirectly proportional to the initial phase-space density f 0(τ ) for the LyndenBell theory and inversely proportional to f 0(τ ) for the Nakamura theory. We then measure Rij (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 the theories are at odds with the numerical results, both qualitatively and quantitatively. Ke yw ords: methods: numerical ‐ galaxies: haloes ‐ galaxies: kinematics and dynamics ‐ galaxies: statistics.