The density number of filaments in the state of the weak and optical   turbulence
  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. 
A.D. Bulygin
physics.optics