  We consider the statistics of density number of filaments for the propagation
of a laser beam subjected to multiple filamentation in a closed area with
reflecting boundaries. Dissipation arrests the catastrophic collapse of
filaments, causing their disintegration into almost linear waves.These waves
form a nearly gaussian random field that seeds new filaments. The evolution of
the energy distribution function of the angular spectrum and accordingly law of
the dynamics of formation of the thermodynamic characteristics of the light
field such as the effective temperature and entropy was found. It is
established that the growth rate of the thermodynamic functions and the
Hamiltonian of the system grows in proportion to the number density of
filaments. Also found that depending on the level of the average (background)
intensity of the light field can move in two steady state. The first mode is
realized for the steady state level of the background intensity does not exceed
the value, and is characterized by typical classical (linear) of a closed
thermodynamic system, a constant level of entropy and temperature, while the
number density of the filaments is equal to zero. In a strongly nonlinear
regime, steady state when the number density of filaments goes to a constant
level, depending on a cubic, but the entropy and the effective temperature rise
in a linear fashion from the distance of propagation. Patterns of development
of the number density of filaments with access to a steady state, are
nonmonotonic and significantly determined by the initial angular spectral
energy distribution of the light field. The average value of filament density
depends on the intensity of the complex way the linear growth stage is replaced
with the output of the nonlinear saturation, nonlinear growth, corresponding to
the transition to the regime of developed turbulence with growth rate in the
third degree.
