Relativistic Dissipative Magnetohydrodynamics from the Boltzmann equation for 2-particle species gas

1. Problem Statement

Relativistic Magnetohydrodynamics (RMHD) is essential for modeling fluids in strong magnetic fields, particularly in astrophysics and high-energy heavy-ion collisions (e.g., at RHIC and LHC), where magnetic fields can reach $10^{19}$–$10^{20}$ Gauss.

• Limitation of Current Models: Traditional RMHD approaches often rely on single-particle fluid approximations or standard Israel-Stewart-type theories. These theories typically assume the shear stress tensor ($\pi^{\mu\nu}$) evolves according to a single differential equation.
• The Gap: It remains unclear how the dynamics of a fluid composed of multiple charged species (specifically a locally neutral gas of opposite charges) behave under strong magnetic fields. Specifically, does the standard Israel-Stewart framework hold, or do new dynamical features emerge that require a more fundamental derivation from kinetic theory?

2. Methodology

The authors derive the equations of motion (EOM) for relativistic dissipative magnetohydrodynamics directly from the Boltzmann equation using the method of moments.

• System Setup:
  • A locally electrically neutral fluid composed of two species of massless classical particles with opposite charges ($+$ and $-$).
  • Assumptions: Vanishing dipole moment/spin (zero magnetization/polarization), vanishing electric charge chemical potential ($\mu=0$), and only elastic collisions.
• Kinetic Theory:
  • The Boltzmann equation is solved for the distribution functions $f^\pm_k$ of both species.
  • The collision term is calculated using the 14-moment approximation combined with the massless particle assumption.
  • Cross-sections are simplified: $\sigma_{++} = \sigma_{--} \equiv \sigma_T$ and a constant cross-section is assumed for derivation.
• Derivation of Equations:
  • The authors calculate the time derivatives of the total shear stress tensor ($\pi^{\mu\nu} = \pi^{\mu\nu}_+ + \pi^{\mu\nu}_-$) and the relative shear stress tensor ($\delta\pi^{\mu\nu} = \pi^{\mu\nu}_+ - \pi^{\mu\nu}_-$).
  • Decomposition: Crucially, the shear stress tensor is decomposed with respect to the magnetic field direction ($b^\mu$) using a complete, normalized, orthogonal basis $(u^\mu, b^\mu, \ell^\mu_\pm)$. This splits the tensor into:
    1. Longitudinal component ($\pi_\parallel$): Parallel to the magnetic field.
    2. Semi-transverse component ($\pi_\perp$): Mixed terms.
    3. Transverse component ($\pi_{\perp\perp}$): Orthogonal to both flow and magnetic field.
• Application to Bjorken Flow:
  • The derived equations are applied to the Bjorken flow scenario (a standard model for heavy-ion collisions) with the magnetic field oriented transversely.
  • The system is reduced to a set of coupled second-order differential equations for the energy density ($\epsilon$) and the normalized shear stress components ($\pi_\parallel/\epsilon$, $\pi_{\perp\perp}/\epsilon$, and the relative component $\delta\hat{\pi}_{\perp\perp}/\epsilon$).

3. Key Contributions

• Derivation of Multi-Component EOM: The paper demonstrates that in a two-species charged gas, the shear stress tensor does not follow a single evolution equation. Instead, it splits into distinct components (longitudinal, semi-transverse, and transverse), each governed by its own unique dynamical equation.
• Coupling via Magnetic Field: The equations reveal a direct coupling between the total shear stress and the relative shear stress via the magnetic field strength ($B$). Specifically, terms proportional to $|q|B/T$ appear, linking the evolution of the total tensor to the relative tensor.
• Beyond Israel-Stewart: The authors show that the standard Israel-Stewart formalism, which typically treats shear stress as a single relaxation mode, is insufficient to describe the system under strong magnetic fields.

4. Results

The authors numerically solved the derived equations for the Bjorken flow scenario.

• Oscillatory Dynamics: For moderately strong to high magnetic fields, the transverse component of the shear stress tensor ($\pi_{\perp\perp}$) exhibits oscillatory behavior.
• Dependence on $\eta/s$: These oscillations are particularly prominent for higher values of the specific shear viscosity to entropy density ratio ($\eta/s$).
• Longitudinal vs. Transverse:
  • The longitudinal component shows a slight hint of oscillation under strong fields but is less affected than the transverse component.
  • The semi-transverse component decouples from the others in this specific flow configuration.
• Failure of Standard Theory: The observed oscillations cannot be captured by conventional Israel-Stewart-like theories, which predict monotonic relaxation to equilibrium.

5. Significance

• Theoretical Implications: This work challenges the universality of the Israel-Stewart framework in magnetized plasmas. It suggests that for systems with strong magnetic fields (like the Quark-Gluon Plasma created in LHC/RHIC collisions), a more nuanced, multi-component hydrodynamic description is necessary.
• Physical Insight: The oscillatory behavior indicates that the magnetic field induces a restoring force or a specific mode of propagation in the dissipative sector of the fluid that is absent in non-magnetized or weakly magnetized fluids.
• Future Applications: The findings provide a more accurate theoretical foundation for interpreting experimental data from heavy-ion collisions where strong transient magnetic fields are generated. It calls for the development of new hydrodynamic models that incorporate these multi-component dynamics to accurately simulate the evolution of the early universe or astrophysical objects with intense magnetic fields.