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 K-i(tau) that a test particle initially at the phase-space coordinate T would endup in the ith macro-cell at equilibrium. We show that the logarithm of the ratio R-ij(tau) equivalent to 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 the theories are at odds with the numerical results, both qualitatively and quantitatively.