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The evolution a network of vortex loops due to the fusion and breakdown inthe turbulent superfluid helium is studied. We perform investigation on thebase of the "rate equation" for the distribution function $n(l)$ of number ofloops in space of their length $l$. There are two mechanisms for change ofquantity $n(l)$. Firstly, the function changes due to deterministic process ofmutual friction, when the length grows or decreases depending on orientation.Secondly, the change of $n(l)$ occurs due to random events when the loopcrosses itself breaking down into two daughter or two loops collide merginginto one larger loop. Accordingly the "rate equation" includes the "collision"term collecting random processes of fusion and breakdown and the deterministicterm. Assuming, further, that processes of random colliding are fastest we arein position to study more slow processes related to deterministic term. In thisway we study the evolution of full length of vortex loops per unit volume-socalled vortex line density ${\cal L}(t)$. It is shown this evolution to obeythe famous Vinen equation. In conclusion we discuss properties of the Vinenequation from the point of view of the developed approach.Comment: Presentation at QFS2006, submitted to JLT

Evolution of a Network of Vortex Loops in the Turbulent Superfluid Helium; Derivation of the Vinen Equation