Statistical mechanics of magnetohydrodynamics

A statistical mechanical formulation for the steady state of self-organized magnetohydrodynamic plasma is studied based on the empirical variational principle, delta(E - lambda H) = 0, for the steady state, where E and H denote the energy and the helicity of a magnetic field. The eigenfunctions of the curl operator are shown to span the phase space of a magnetic field in a bounded system, and the invariant measure is found. The classical ensemble theory is formulated assuming the Shannon or Renyi entropy. To avoid the divergence of the expectation values at nonzero temperature, Rose-Einstein statistics is also phenomenologically treated. It is implied that the spectra of the energy, helicity, and the helicity fluctuation obey the power law for a multiply connected domain with a nonzero cohomological field. For the toroidal system, these powers are implied to be three, three, and two, respectively. The invariant measure for the incompressible flow in a bounded domain is also given.