  Families of $N$ interacting curves are considered, with long range, mean
field type, interaction. A family of curves defines a 1-current, concentrated
on the curves, analog of the empirical measure of interacting point particles.
This current is proved to converge, as $N$ goes to infinity, to a mean field
current, solution of a nonlinear, vector valued, partial differential equation.
In the limit, each curve interacts with the mean field current and two
different curves have an independence property if they are independent at time
zero. This set-up is inspired from vortex filaments in turbulent fluids,
although for technical reasons we have to restrict to smooth interaction,
instead of the singular Biot-Savart kernel. All these results are based on a
careful analysis of a nonlinear flow equation for 1-currents, its relation with
the vector valued PDE and the continuous dependence on the initial conditions.
