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We consider the statistics of density number of filaments for the propagationof a laser beam subjected to multiple filamentation in a closed area withreflecting boundaries. Dissipation arrests the catastrophic collapse offilaments, causing their disintegration into almost linear waves.These wavesform a nearly gaussian random field that seeds new filaments. The evolution ofthe energy distribution function of the angular spectrum and accordingly law ofthe dynamics of formation of the thermodynamic characteristics of the lightfield such as the effective temperature and entropy was found. It isestablished that the growth rate of the thermodynamic functions and theHamiltonian of the system grows in proportion to the number density offilaments. 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 isrealized for the steady state level of the background intensity does not exceedthe value, and is characterized by typical classical (linear) of a closedthermodynamic system, a constant level of entropy and temperature, while thenumber density of the filaments is equal to zero. In a strongly nonlinearregime, steady state when the number density of filaments goes to a constantlevel, depending on a cubic, but the entropy and the effective temperature risein a linear fashion from the distance of propagation. Patterns of developmentof the number density of filaments with access to a steady state, arenonmonotonic and significantly determined by the initial angular spectralenergy distribution of the light field. The average value of filament densitydepends on the intensity of the complex way the linear growth stage is replacedwith the output of the nonlinear saturation, nonlinear growth, corresponding tothe transition to the regime of developed turbulence with growth rate in thethird degree.

The density number of filaments in the state of the weak and optical turbulence