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

This paper generalizes the first-order out-of-equilibrium correction to the relativistic Boltzmann distribution function using the projection method and Chapman-Enskog expansion to derive constitutive equations that explicitly incorporate frame and representation freedom, thereby coupling dissipative fluxes to all state variable derivatives and weak external electromagnetic fields.

The Gist

Imagine a bustling city square filled with millions of people (the gas molecules) moving in all directions. Sometimes they bump into each other (collisions), and sometimes they are pushed by a gentle wind or a magnetic field (external forces). Physicists want to predict how this crowd moves as a whole—how the density changes, how fast the "average" person is moving, and how the temperature shifts.

This paper is about building a better rulebook for predicting that crowd's behavior when things are slightly chaotic (out of equilibrium), specifically when the rules of Einstein's relativity apply (where nothing moves faster than light).

Here is the breakdown of their work using simple analogies:

1. The Old Problem: A Broken Compass

For decades, scientists used a method called the Chapman-Enskog expansion to predict how gases behave. Think of this method as a recipe for baking a cake. It works great for normal cakes (non-relativistic gases). However, when scientists tried to use this same recipe for "relativistic cakes" (gases moving near light speed), the result was a disaster. The old recipes predicted that the cake would spontaneously explode or behave in ways that broke the laws of physics (instability).

Because of this, scientists stopped using this method for relativistic fluids for a long time, fearing it was fundamentally broken.

2. The New Approach: The "Projection" Method

The authors of this paper decided to try the recipe again, but with a very specific, rigorous technique called the projection method.

Imagine you are trying to describe the movement of the crowd. You have two main ways to define "where the crowd is":

• The Particle Frame: You define the crowd's center based on where the people are.
• The Energy Frame: You define the crowd's center based on where the energy (heat/movement) is.

In the past, scientists argued that you had to pick one of these definitions and stick to it. If you picked the wrong one, your math broke.

3. The Big Discovery: Two "Knobs" to Turn

The main breakthrough in this paper is showing that you don't have to pick just one definition. The authors found that there are two independent "knobs" you can turn to fix the math and make it work for any situation.

Knob 1: The "Frame" (Who is the Observer?)

This is about where you decide to stand to measure the crowd.

• The paper shows that you can choose to measure the crowd from the perspective of the particles, the energy, or any mix in between.
• The Analogy: Imagine watching a parade. You can stand on the sidewalk (Particle Frame), or you can ride on a float with the marching band (Energy Frame). The paper proves that the math works perfectly whether you stand on the sidewalk or ride the float, as long as you adjust your calculations correctly. This resolves the old fear that the math was "unstable."

Knob 2: The "Representation" (How do we write the rules?)

This is a more subtle freedom. Even after you pick where to stand, you still have a choice in how you write down the rules for the crowd's behavior.

• The authors show that you can add certain "correction terms" to your equations. These terms don't change the final physical reality (the crowd still moves the same way), but they change the mathematical description of the forces.
• The Analogy: Think of writing a story. You can describe a car crash as "The car hit the wall" or "The wall hit the car." The event is the same, but the sentence structure is different. The authors found a way to structure the "sentence" of the fluid's laws so that it remains stable and causal (nothing happens before it causes it), regardless of which "sentence structure" you prefer.

4. The Result: A Universal Rulebook

By turning these two knobs, the authors derived a general set of equations (constitutive equations).

• These equations connect the "forces" (like temperature changes or pressure gradients) to the "fluxes" (like heat flow or viscosity).
• Crucially, these new equations are stable. They don't explode. They are causal (effects happen after causes). And they are hyperbolic (information travels at a finite speed, not instantly).

5. Why This Matters (According to the Paper)

The paper claims that by using this method, they have successfully revived the Chapman-Enskog expansion for relativistic fluids. They showed that:

1. The old fear of instability was due to being too rigid in how we chose our "frame" and "representation."
2. By allowing flexibility in these choices, we can derive theories that match the most modern, successful theories (known as BDNK theories) but are derived directly from the microscopic behavior of particles (the Boltzmann equation).
3. This provides a solid microscopic foundation for understanding how hot, fast-moving fluids (like those in neutron stars or the early universe) behave without breaking the laws of physics.

In short: The authors took a broken, old recipe for relativistic fluids, added two flexible "adjustment knobs" (Frame and Representation), and proved that with these adjustments, the recipe works perfectly, producing stable and realistic predictions for how fast-moving gases behave.

Technical Summary

1. Problem Statement

The paper addresses the derivation of first-order constitutive equations for relativistic dissipative fluids directly from kinetic theory (the relativistic Boltzmann equation). Historically, the Chapman-Enskog (CE) expansion applied to relativistic fluids was believed to lead to unstable, acausal, and ill-posed first-order theories (similar to the classical Eckart and Landau formulations). While recent developments (BDNK theories) have proposed stable first-order theories in arbitrary frames, a rigorous microscopic derivation that explicitly incorporates the freedom of frame choice and representation choice within the standard CE projection method has been lacking.

The authors aim to:

• Generalize the first-order non-equilibrium distribution function obtained via the CE expansion to include arbitrary choices of frame and representation.
• Demonstrate how these choices lead to general constitutive equations that couple dissipative fluxes to all derivatives of state variables (forces), including weak external electromagnetic fields.
• Clarify the microscopic origin of the "frame" and "representation" freedoms found in phenomenological theories.

2. Methodology

The authors utilize the Chapman-Enskog (CE) expansion combined with the projection method (following the formalism of Saint-Raymond and their previous work [14]) applied to the relativistic Boltzmann equation for a dilute gas with binary elastic collisions.

Key Steps in the Derivation:

1. Linearization: The distribution function is expanded as $f = f^{(0)} + \epsilon f^{(1)}$, where $f^{(0)}$ is the local Jüttner equilibrium distribution. The first-order correction $f^{(1)}$ satisfies a linearized integral equation involving the collision operator $L$.
2. Projection Method: The solution to the linearized equation is constructed by projecting the source term onto the orthogonal complement of the kernel of the collision operator ($\ker L = \text{span}\{1, p^\mu\}$).
3. General Solution Structure: The general solution for $f^{(1)}$ is written as the sum of a particular solution (driven by gradients of state variables) and a homogeneous solution (an arbitrary element of $\ker L$):
   $$f^{(1)} = -f^{(0)} \left[ \mathcal{P}^{\mu\nu} L^{-1}((p_\mu p_\nu)^\perp) + mA + B^\mu p_\mu \right]$$
   Here, $A$ and $B^\mu$ are undetermined coefficients representing the homogeneous part.
4. Frame Fixing (Compatibility Conditions): To fix the coefficients $A$ and $B^\mu$, the authors introduce general compatibility conditions (matching conditions) defined by arbitrary functions $g_1(\gamma), g_2(\gamma), g_3(\gamma)$:
   $$\int g_i(\gamma) f^{(1)} d\text{vol} = 0$$
   These conditions determine the specific frame (e.g., Eckart, Landau, or Trace-Fixed Particle).
5. Representation Freedom: The authors introduce parameters $\hat{\Gamma}_i$ into the particular solution. These parameters multiply terms that are of higher order in the Knudsen parameter but are retained as "on-shell" corrections. This corresponds to the representation freedom, allowing the inclusion of terms that do not alter the equilibrium state but modify the constitutive relations off-shell.

3. Key Contributions

• Unified Framework for Frames and Representations: The paper provides a rigorous microscopic derivation showing that the freedom to choose a frame (via compatibility conditions $g_i$) and the freedom to choose a representation (via parameters $\hat{\Gamma}_i$) are independent degrees of freedom in the kinetic theory.
• General Constitutive Equations: The authors derive the most general first-order constitutive equations for particle current ($J^\mu$) and energy-momentum tensor ($T^{\mu\nu}$). These equations couple dissipative fluxes (heat flow, diffusion, bulk/shear viscosity) to all possible thermodynamic forces (gradients of density, temperature, acceleration, expansion, and electromagnetic fields).
• Recovery of Known Theories: The formalism naturally recovers specific frames as special cases:
  • Eckart Frame: Fixed by setting particle diffusion to zero.
  • Landau Frame: Fixed by setting energy diffusion (heat flow) to zero.
  • Trace-Fixed Particle (TFP) Frame: A specific frame previously shown to yield hyperbolic and stable equations.
• Connection to BDNK Theories: The resulting constitutive equations are shown to have the same structural form as the recently proposed BDNK (Bemfica-Disconzi-Noronha-Kovtun) theories, providing a microscopic justification for the phenomenological parameters used in those theories.

4. Key Results

The derived constitutive equations for the dissipative quantities (particle density correction $n^{(1)}$, energy density correction $e^{(1)}$, pressure correction $p^{(1)}$, heat flux $Q^\alpha$, diffusion current $J^\alpha$, and shear stress $T^{\alpha\beta}$) take the general form:
$$\text{Flux} = \sum_i c_i \cdot \text{Force}_i$$
Where the coefficients $c_i$ depend on:

1. Microscopic Transport Coefficients: Shear viscosity ($\eta$), bulk viscosity ($\zeta$), and thermal conductivity ($\kappa$), derived from the collision kernel ($S_1, S_3, S_5$).
2. Frame Parameters ($\chi_i$): Determined by the choice of compatibility functions $g_i$. Changing the frame shifts these coefficients but preserves the fundamental tensorial structure of the forces.
3. Representation Parameters ($\hat{\Gamma}_i$): Arbitrary parameters that allow for the inclusion of specific combinations of time and space derivatives.

Specific Findings:

• Entropy Production: The authors calculate the entropy production rate and demonstrate that, regardless of the chosen frame or representation, the second-order terms in the entropy production are positive-definite, ensuring thermodynamic consistency.
• Stability and Causality: The paper argues that the inclusion of the representation freedom (non-zero $\hat{\Gamma}_i$) is crucial. It allows for the construction of transport equations that form a well-posed, hyperbolic, and causal Cauchy problem, resolving the historical instability issues of first-order relativistic hydrodynamics.
• Force-Flux Coupling: The general solution reveals that in an arbitrary frame, dissipative fluxes are coupled to a linear combination of all gradients (time and space derivatives of $n, T, u^\mu$), rather than just spatial gradients as in traditional Eckart/Landau formulations.

5. Significance

• Microscopic Foundation: This work bridges the gap between phenomenological first-order theories (like BDNK) and kinetic theory. It proves that the "freedom" to choose frames and representations in macroscopic theories is not arbitrary but arises naturally from the non-uniqueness of the solution to the linearized Boltzmann equation.
• Resolution of Instability: By explicitly showing how the representation freedom ($\hat{\Gamma}_i$) can be tuned to ensure hyperbolicity and causality, the paper provides a kinetic-theory basis for stable first-order relativistic hydrodynamics, challenging the long-held belief that first-order theories are inherently unstable.
• Generality: The derived equations are valid for charged fluids in the presence of weak electromagnetic fields and in arbitrary spacetime dimensions ($d+1$), making the results applicable to a wide range of astrophysical and high-energy physics contexts (e.g., quark-gluon plasma, neutron star mergers).

In summary, the paper establishes that the most general first-order relativistic fluid theory can be rigorously derived from the Boltzmann equation by systematically accounting for the mathematical freedoms inherent in the Chapman-Enskog expansion, leading to stable and causal transport equations.