The motion of a two-dimensional perfect fluid can be described as an area preserving rearrangement of the initial vorticity that conserve the kinetic energy. In the infinite time limit, vorticity mixing is conjectured to occur for most initial conditions. A.I. Shnirelman in ’93 introduced the concept of maximally mixed states, by requiring that any further mixing of them would necessarily change their energy, and showed they are perfect fluid equilibria. We offer a new perspective on this theory by proving that many of them can be obtained as minimizers of a variational problem. We also show that maximally mixed states, in general, need not conform to the geometry of the domain. In particular, in a straight periodic channel, we find non stationary states which can be arbitrarily close to any shear flow in L^1 of vorticity but cannot converge back to a shear flow in the long-time limit. This is a joint work with T. D. Drivas.