Generalized relativistic second order magnetohydrodynamics: A correlation function approach using Zubarev's nonequilibrium statistical operator

We use total energy-momentum conservation and the Bianchi identity (magnetic-flux conservation) to construct second-order relativistic magnetohydrodynamics in a Zubarev’s non-equilibrium statistical operator (NESO) framework. We obtain all dissipative tensors in the medium by focusing on a relativistic magnetized plasma that preserves parity and is symmetric to charge-conjugation. We also provide Kubo formulas for all transport coefficients that arise at second order. Moreover, we extend the NESO formalism to systematically take into account for nonlocal contributions.

💡 Research Summary

This paper presents a comprehensive formulation of second-order relativistic magnetohydrodynamics (RMHD) using Zubarev’s nonequilibrium statistical operator (NESO) framework. The authors construct the theory based on two fundamental conservation laws: the conservation of total energy-momentum and the conservation of magnetic flux (embodied by the Bianchi identity). This approach represents a paradigm shift from conventional RMHD, which often treats the electric field as a hydrodynamic variable and assumes infinite electrical conductivity, leading to inconsistencies when incorporating dissipative effects. Instead, this work treats only the magnetic field as a hydrodynamic degree of freedom, recognizing that magnetic flux is conserved and not subject to Debye screening, while the electric field emerges as a higher-order dissipative effect.

The core methodology involves a tensorial decomposition of the total energy-momentum tensor (T^μν) and the dual electromagnetic field-strength tensor (Ť^μν) with respect to the fluid four-velocity (u^μ) and the magnetic field direction vector (b^μ). This decomposition identifies equilibrium variables (energy density ε, parallel/perpendicular pressures p_∥ and p_⊥, magnetic flux density B) and a set of dissipative tensors (Π_∥, Π_⊥, h^μ, f^μ, π^μν_⊥, l^μ, g^μ, m^μν) that contain gradient corrections. Projecting the conservation laws onto u^ν, b^ν, and a transverse projector G^α_ν yields the equations of motion for these fundamental variables.

An entropy current analysis is performed to enforce the second law of thermodynamics locally. The divergence of the entropy current (∂μ S^μ) is expressed as a sum of products of dissipative tensors and their conjugate thermodynamic forces (e.g., expansion scalars θ∥, θ_⊥, shear tensor σ^μν_⊥, magnetic shear χ^μν, etc.). By considering parity and charge-conjugation symmetry of the plasma, linear constitutive relations are posited (e.g., π^μν_⊥ = 2η_⊥ σ^μν_⊥), defining transport coefficients like shear viscosities (η_⊥, η_∥), bulk viscosities (ζ_∥, ζ_⊥, ζ_×), and magnetic diffusion coefficients (ρ_∥, ρ_⊥).

The central theoretical achievement is the application of the NESO formalism to derive these relations from a microscopic statistical basis. The NESO constructs a nonequilibrium density operator constrained by the expectation values of conserved quantities (energy-momentum and magnetic flux). Solving the Liouville equation with a causal “switch-on” prescription provides a systematic gradient expansion for the dissipative parts of T^μν and Ť^μν. The paper revisits first-order RMHD within this framework, confirming previous results. It then proceeds to the main result: a complete set of second-order constitutive relations and the evolution equations for all dissipative quantities. These second-order terms include relaxation-type terms (analogous to Israel-Stewart theory) and nonlinear couplings, which are encoded in memory kernels and multi-point equilibrium correlation functions. Furthermore, the authors provide explicit Kubo formulas—expressions in terms of retarded equilibrium correlation functions—for every transport coefficient that appears at second order. This establishes a direct link between the macroscopic hydrodynamic description and underlying quantum field theory.

Finally, the authors note an extension of the NESO formalism to systematically account for nonlocal contributions, enhancing the theory’s capability to handle long-range correlations and memory effects. In summary, this work provides a rigorous, causal, and stable foundation for second-order relativistic magnetohydrodynamics derived from first principles, with direct applicability to strongly coupled magnetized systems in high-energy nuclear physics and astrophysics.