  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.
