Empirical formulae to describe some physical properties of small groups
  of protogalaxies with multiplicity
  By means of identical cubic elements, we generate a partition of a volume in
which a particle-based cosmological simulation is carried out. In each cubic
element, we determine the gas particles with a normalized density greater than
an arbitrarily chosen density threshold. By using a proximity parameter, we
calculate the neighboring cubic elements and generate a list of neighbors. By
imposing dynamic conditions on the gas particles, we identify gas clumps and
their neighbors, so that we calculate and fit some properties of the groups so
identified, including the mass, size and velocity dispersion, in terms of their
multiplicity (here defined simply as the number of member galaxies). Finally,
we report the value of the ratio of kinetic energy to gravitational energy of
such dense gas clumps, which will be useful as initial conditions in
simulations of gravitational collapse of gas clouds and clusters of gas clouds.
