  We present the results of an attempt to adapt the distribution function
formalism to characterize large-scale structures like clusters of galaxies that
form in a cosmological N-body simulation. While galaxy clusters are systems
that are not strictly in equilibrium, we show that their evolution can
nevertheless be studied using a physically motivated extension of the language
of equilibrium stellar dynamics. Restricting our analysis to the virialized
region, a prescription to limit the accessible phase-space is presented, which
permits the construction of both the isotropic and the anisotropic distribution
functions $f(E)$ and $f(E,L)$. The method is applied to models extracted from a
catalogue of simulated clusters. Clusters evolved in open and flat background
cosmologies are followed during the course of their evolution, and are found to
transit through a sequence of what we define as `quasi-equilibrium' states. An
interesting feature is that the computed $f(E)$ is well fit by an exponential
form. We conclude that the dynamical evolution of a cluster, undergoing
relaxation punctuated by interactions and violent mergers with consequent
energy-exchange, can be studied both in a qualitative and quantitative fashion
by following the time evolution of $f(E)$.

Astrophysics