  This work reports on a study of the spatially coarse-grained velocity
dispersion in cosmological N-body simulations (OCDM and Lambda-CDM models) as a
function of time (redshifts z=0-4) and of the coarsening length (0.6-20 Mpc/h).
The main result is the discovery of a polytropic relationship I_1 ~ rho^{2-eta}
between the velocity-dispersion kinetic energy density of the coarsening cells,
I_1, and their mass density, rho. The exponent eta, dependent on time and
coarsening scale, is a compact measure of the deviations from the naive virial
prediction eta_virial=0. This relationship supports the ``polytropic
assumption'' which has been employed in theoretical models for the growth of
cosmological structure by gravitational instability.
