  In the past years we have experienced an increasing interest in understanding
of the physical properties of collisionless plasmas, mostly because of the
large number of astrophysical environments, e.g. the intracluster medium (ICM),
containing magnetic fields which are strong enough to be coupled with the
ionized gas and characterized by densities sufficiently low to prevent the
pressure isotropization with respect to the magnetic line direction. Under
these conditions a new class of kinetic instabilities arises, such as firehose
and mirror ones, which were extensively studied in the literature. Their role
in the turbulence evolution and cascade process in the presence of pressure
anisotropy, however, is still unclear. In this work we present the first
statistical analysis of turbulence in collisionless plasmas using three
dimensional double isothermal magnetohydrodynamical with the
Chew-Goldberger-Low closure (CGL-MHD) numerical simulations. We study models
with different initial conditions to account for the firehose and mirror
instabilities and to obtain different turbulent regimes. We study the
probability distribution functions, spectra, structure functions and anisotropy
of density and velocity fluctuations. The results indicate that in some cases
the instabilities significantly modifies the statistical properties of
turbulence and even though preliminary and restricted to very specific
conditions, show that the physical properties of turbulence in collisionless
plasmas, as those found in the ICM, may be very different from what has been
largely believed. Implications can range from interchange of energies to cosmic
rays acceleration.
