abstract: The Onsager-Joyce-Montgomery theory concerns equilibrium statistical mechanics for 2-D turbulent vortex dynamics. In an effort to rigorously establish this theory, we encounter a special Hamilton-Jacobi equation in space of probability measures. To be sufficiently focused, after a brief outline of the overall program, I focus on the uniqueness result (comparison principle) of the equation using techniques from mass transportation and optimal control. The talk is based on joint works with past and present collaborators.