Foundations of magnetohydrodynamics with applications to dense plasmas

A derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented [1]. Non-equilibrium thermodynamics, a macroscopic framework for describing irreversible processes, is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell’s equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. Itis then shown how these properties are connected to microscopic dynamics using the Irving-Kirkwood procedure and Green-Kubo relations. Discussions of symmetry arguments and the Onsager-Casimir relations are provided, which allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm’s law, Fourier’s law, and Newton’s law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman-Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations. Moreover, it is shown how the transport coefficients defined in kinetic theory relate to those from the Green-Kubo relations.

[1] Jarett LeVan, Scott D. Baalrud; Foundations of magnetohydrodynamics. Phys. Plasmas 1 July 2025; 32 (7): 070901.