Reconnections of Vortex Loops in the Superfluid Turbulent HeII. Rates of the Breakdown and Fusion processes

Kinetics of merging and breaking down vortex loops is the important part of the whole vortex tangle dynamics. Another part is the motion of individual lines, which obeys the Biot-Savart law in presence of friction force and of applied external velocity fields if any. In the present work we evaluate the coefficients of the reconnection rates $A(l_{1},l_{2},l)$ and $B(l,l_{1},l_{2})$. Quantity $A$ is a number (per unit of time and per unit of volume) of events, when two loops with lengths $l_{1}$and $l_{2}$ collide and form the single loop of length $ l=l_{1}+l_{2}$. Quantity $% B(l,l_{1},l_{2}) $ describes the rate of events, when the single loop of the length $l$ breaks down into two the daughter loops of lengths $ l_{1}$ and $l_{2}$. These quantities ave evaluated as the averaged numbers of zeroes of vector $\mathbf{S}%_{s}(\xi_{2},\xi_{1},t)$ connecting two points on the loops of $\xi_{2}$ and $ \xi_{1}$ at moment of time $t$. Statistics of the individual loops is taken from the Gaussian model of vortex tangle. PACS-number 67.40