Derivation of transient relativistic fluid dynamics from the Boltzmann equation (1202.4551v1)

Abstract: In this work we present a general derivation of relativistic fluid dynamics from the Boltzmann equation using the method of moments. The main difference between our approach and the traditional 14-moment approximation is that we will not close the fluid-dynamical equations of motion by truncating the expansion of the distribution function. Instead, we keep all terms in the moment expansion. The reduction of the degrees of freedom is done by identifying the microscopic time scales of the Boltzmann equation and considering only the slowest ones. In addition, the equations of motion for the dissipative quantities are truncated according to a systematic power-counting scheme in Knudsen and inverse Reynolds number. We conclude that the equations of motion can be closed in terms of only 14 dynamical variables, as long as we only keep terms of second order in Knudsen and/or inverse Reynolds number. We show that, even though the equations of motion are closed in terms of these 14 fields, the transport coefficients carry information about all the moments of the distribution function. In this way, we can show that the particle-diffusion and shear-viscosity coefficients agree with the values given by the Chapman-Enskog expansion.

• The paper derives a rigorous formulation of transient relativistic fluid dynamics by retaining the full moment expansion of the Boltzmann equation.
• It employs a systematic power-counting scheme in Knudsen and inverse Reynolds numbers to overcome the limitations of the Israel-Stewart approximation.
• The approach validates transport coefficients against Chapman-Enskog theory and demonstrates convergence with up to 41 moments in ultrarelativistic gas simulations.

Derivation of Transient Relativistic Fluid Dynamics from the Boltzmann Equation

The paper "Derivation of transient relativistic fluid dynamics from the Boltzmann equation" by Denicol et al. offers a systematic approach to deriving relativistic fluid dynamics using the method of moments without the constraints of traditional approximations. This work is significant as it attempts to bridge the gap between kinetic theory and macroscopic fluid dynamics in a rigorous manner while addressing known shortcomings in existing models.

The research primarily focuses on overcoming the limitations of the Israel-Stewart 14-moment approximation by extending the derivation of equations from the Boltzmann equation without truncating the series expansion prematurely. The primary goal is to incorporate the entire moment expansion and develop equations of motion based on a power-counting scheme in Knudsen and inverse Reynolds numbers. This retains the full spectrum of terms necessary for a more accurate depiction of relativistic fluid behavior.

Notably, the authors demonstrate that fluid-dynamical equations can be closed with just 14 dynamical variables while still accounting for second-order effects in Knudsen number (Kn) and inverse Reynolds numbers. This contrasts with the Israel-Stewart theory, which inherently misses significant microscopic details by neglecting first-order terms in Knudsen number.

Key Features and Claims

1. Systematic Power-Counting Scheme: By employing a systematic power-counting in both Knudsen and inverse Reynolds numbers, the authors derive causal and stable fluid equations. This method allows for a clearer understanding of how different scales and forces contribute to the fluid's behavior over time.
2. Inclusion of All Moments: Unlike previous theories that truncate the moment series to 14 moments, this research maintains all terms in the moment expansion. It systematically determines which terms contribute to physical fluid dynamics relevant to relativistic contexts, ensuring that transient dynamics are adequately captured.
3. Agreement with Chapman-Enskog Theory: The consistency between calculated transport coefficients, including shear viscosity and particle diffusion coefficients, with those from Chapman-Enskog theory, illustrates the validity of the derived equations. This agreement supports the approach taken to derive these transport coefficients without neglecting higher-order terms.
4. Extension to Ultraralativistic Gases: The authors notably advance their method with specific applications to ultrarelativistic classical gases, demonstrating both the practicality and robustness of their theoretical framework through numerical evaluations.
5. Numerical Convergence: Utilizing up to 41 moments in calculations, the research evidences convergence towards more accurate transport coefficients, supporting the robustness and reliability of the approach for practical computations in complex fluid dynamic scenarios.

Implications and Future Directions

This paper's findings have important implications for scenarios where relativistic fluid dynamics are crucial, such as in heavy-ion collisions and astrophysical phenomena. The methodology presented provides a more rigorous foundation for future developments in the modeling of relativistic fluids, allowing for better predictions and understanding of physical experiments and observations.

The proposed framework could pave the way for more detailed and accurate simulations in high-energy physics and cosmology. As computational methods advance, incorporating the complete moment expansion could enhance the fidelity of simulations, providing key insights into the transient behaviors and dynamical stability of such systems.

Future work could further explore the integration of these findings with quantum hydrodynamics or validate the predictions against real-world experimental data from high-energy physics experiments. Additionally, the extension of this methodology to non-relativistic domains could reveal new insights into classical fluid dynamics by bridging the gap between kinetic and continuum descriptions there as well.

In summary, Denicol et al.'s work is a careful exposition of how thorough consideration of microscopic scales in the Boltzmann equation can fundamentally improve the theoretical underpinnings of relativistic fluid dynamics, offering a pathway toward more comprehensive physical models.