First-order general constitutive equations for relativistic fluids using the projection method in the Chapman-Enskog expansion of the Boltzmann equation

The first-order out of equilibrium correction to the distribution function, obtained by implementing the projection method for the perturbed relativistic Boltzmann equation using the Chapman-Enskog method, is generalized in order to explicitly include the freedom of choice for thermodynamic frame and representation. It is shown how this procedure leads to general constitutive equations that couple the thermodynamic fluxes to all forces including a weak external electromagnetic field. Special cases of the resulting force-flux relations have been shown to lead to physically sound theories for relativistic fluids.

💡 Research Summary

This paper presents a rigorous microscopic derivation of general first-order constitutive equations for relativistic dissipative fluids, starting from the relativistic Boltzmann equation. The authors achieve this by employing a formal implementation of the Chapman-Enskog (CE) expansion known as the “projection method.”

The work is motivated by the historical criticism that traditional relativistic CE expansions lead to unstable first-order theories (like those of Eckart and Landau). Recent developments in stable, causal first-order hydrodynamic theories have renewed interest in establishing their microscopic foundations. The authors’ approach generalizes Saint-Raymond’s non-relativistic projection method to relativity. This method involves decomposing the function space into the kernel of the linearized collision operator L (spanned by collision invariants 1 and p^μ) and its orthogonal complement. Projecting the first-order CE equation onto this orthogonal complement guarantees the existence of a solution, which can then be inverted using L^(-1).

The key result is the general first-order solution for the non-equilibrium distribution function f^(1). This solution consists of three distinct parts: 1) A particular solution involving the inverse collision operator L^(-1) acting on the projected thermodynamic forces. 2) Terms containing free parameters (denoted Γ^0, Γ^1, Γ^2) that can be added without altering the order of the approximation, utilizing the zeroth-order balance equations. This reflects the “freedom of representation.” 3) An arbitrary element of the kernel of L, parameterized by coefficients A and B^μ. This constitutes the homogeneous solution, and fixing these coefficients corresponds to choosing a “thermodynamic frame” via compatibility conditions.

The paper meticulously distinguishes between these two freedoms. The “frame” freedom (homogeneous solution) is related to the choice of variables used to describe the non-equilibrium state (i.e., how one defines n, u^μ, T out of equilibrium). The “representation” freedom (the Γ parameters) is crucial for obtaining the complete set of force-flux couplings present in modern phenomenological theories and is identified as the element that allows the resulting hydrodynamic equations to be hyperbolic, causal, and stable.

The authors show how specific choices for the compatibility conditions (integrals involving arbitrary functions g1(γ), g2(γ), g3(γ)) yield different frames, such as the particle frame or the energy frame. They demonstrate that their general formalism, starting from the particular solution in a specific frame (the Trace-Fixed-Particle frame), can be generalized through these two freedoms to reproduce the full scope of first-order constitutive relations, including couplings to weak external electromagnetic fields. Thus, the paper successfully bridges a rigorous kinetic theory calculation with the general form of first-order relativistic hydrodynamic theories currently under investigation.