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assume Boltzmann Equations collisions compared to macroscopic conductivities parallel conductivity perpendicular conventional diffusion counting of variables curl vp/p difference momentum equation diffusion relation diffusion velocities domain dropping electric field electromagnetic and frictional electromagnetic terms equation in addition explicitly extra velocity flow fluid equations form of Ohm's formal analysis frictional terms function given Grad gyro oscillations Hall effect Hall vector ignore friction independent ionization and recombination l/B for large linear homogeneous magnetic field magneto-hydrodynamic Maxwell's equations mean-free-path MH-III number density one-fluid theory order of magnitude ordinary differential equations ordinary fluid dynamics oscillating term partial differential equations particle or fluid perfect conductor plasma and gyro plasma frequency pressure terms become PrJu purely electromagnetic quasi-equilibrium is reached relative velocity relevant scalar set of ordinary single energy equation slowly varying small compared solution stress tensor supplement term Ohm's law tional two-fluid types of particles valid York University