Evolution equations for the vorticity distribution function in the two-dimensional case

For an ensemble of rectilinear vortex filaments, a closed evolution equation for the vorticity distribution function is derived from the Liouville equation using the Prigozhin-Balescu approach. The equation includes a convective Helmholtz operator and a quasi-linear elliptic operator of the second order with nonlocal coefficients. The local sign of the dissipative coefficient matrix is determined by the instantaneous vorticity distribution. It is shown that generally over the entire flow region, the evolution involves a monotonic increase in the information entropy and the vorticity distribution tends to become stationary.