Microscopic Foundations of Relativistic Fluid Dynamics

Preface
	References
Contents
About the Authors
1 Relativistic Fluid Dynamics
	1.1 Thermodynamics
	1.2 Relativistic Ideal Fluid Dynamics
		1.2.1 Conserved Currents in an Ideal Fluid
		1.2.2 Equations of Motion
		1.2.3 Covariant Thermodynamics and Entropy Production
	1.3 Relativistic Dissipative Fluid Dynamics
		1.3.1 Matching Conditions
		1.3.2 Tensor Decomposition of τµν
		1.3.3 Definition of the Local Rest Frame and Equations of Motion
		1.3.4 Relativistic Navier–Stokes Theory
		1.3.5 Gradient Expansion and Navier–Stokes Theory
	1.4 Causal Fluid Dynamics
		1.4.1 Diffusion Equation and Acausality in Heat Conduction
		1.4.2 Transient Theory of Fluid Dynamics
	1.5 Transient Thermodynamics and Israel–Stewart Theory
	1.6 Non-hydrodynamic Modes and the Origin of the Relaxation Time
2 Linear Stability and Causality
	2.1 Fluid-Dynamical Equations Linearized Around Global Equilibrium
	2.2 Linearized Fluid-Dynamical Equations in Fourier Space
		2.2.1 Tensor Decomposition in Fourier Space
		2.2.2 Longitudinal and Transverse Components
	2.3 Ideal Fluid Dynamics
	2.4 Relativistic Navier–Stokes Theory
		2.4.1 Transverse Modes
		2.4.2 Longitudinal Modes
		2.4.3 Causality and Stability of Navier–Stokes Theory
	2.5 Transient Theory of Fluid Dynamics
		2.5.1 Transverse Modes in the Rest Frame
		2.5.2 Longitudinal Modes in the Rest Frame
		2.5.3 Stability for a Moving Background
		2.5.4 Causality of Wave Propagation
	2.6 Summary
3 Analytical Solutions and Transient Dynamics
	3.1 Fluid Dynamics in a General Metric System
	3.2 Bjorken Flow
		3.2.1 Coordinates and Kinematic Properties
		3.2.2 Fluid-Dynamical Equations
		3.2.3 Ideal-Fluid Limit
		3.2.4 Relativistic Navier–Stokes Theory
		3.2.5 Transient Theory of Fluid Dynamics
		3.2.6 Gradient Expansion of Transient Fluid Dynamics
	3.3 Gubser Flow
		3.3.1 Coordinates and Kinematical Properties
		3.3.2 Fluid-Dynamical Equations of Motion
		3.3.3 Ideal-Fluid Limit
		3.3.4 Relativistic Navier–Stokes Theory
		3.3.5 Transient Theory of Fluid Dynamics
		3.3.6 Gradient Expansion of Transient Fluid Dynamics
		3.3.7 Divergence of the Gradient Expansion
		3.3.8 Domain of Applicability of the Gradient Expansion
		3.3.9 Slow-Roll Expansion
		3.3.10 Divergence of the Slow-Roll Series
		3.3.11 Attractor Solution
	3.4 Summary
4 Microscopic Origin of Transport Coefficients: Linear-Response Theory
	4.1 Preliminaries
	4.2 Equivalence Between Gradient Expansion and Taylor Series
	4.3 The Role of the Analytical Structure of tildeGR(ω)
		4.3.1 tildeGR(ω)  with One Pole
		4.3.2 tildeGR(ω) with Two Poles
		4.3.3 tildeGR(Q) with N Poles
	4.4 Applications
		4.4.1 The Linearized Boltzmann Equation
		4.4.2 Linear-Response Theory and Metric Perturbations
	4.5 Discussion
	4.6 Summary
5 Fluid Dynamics from Kinetic Theory: Traditional Approaches
	5.1 Matching Fluid-Dynamical with Kinetic Degrees of Freedom
		5.1.1 Macroscopic Conservation Laws
		5.1.2 Fluid-Dynamical Variables and Matching Conditions
	5.2 Chapman–Enskog Theory
		5.2.1 Solving the Chapman–Enskog Expansion:  Zeroth- and First-Order Solutions
		5.2.2 Minimal Truncation Scheme
	5.3 Israel–Stewart Theory
		5.3.1 14-Moment Approximation
		5.3.2 Matching Procedure
		5.3.3 Moment Equations
		5.3.4 Calculation of the Collision Integrals
		5.3.5 Hydrodynamic Equations of Motion
	5.4 Summary
6 Method of Moments: Equilibrium Reference State
	6.1 Moment Expansion
	6.2 Equations of Motion for the Irreducible Moments
	6.3 Generalized Collision Term
		6.3.1 Computation of the Linear Collision Term
		6.3.2 Computation of the Nonlinear Collision Term
	6.4 Summary
	6.5 Appendix 1: Irreducible Projection Operators
	6.6 Appendix 2: Thermodynamic Integrals and Properties
	6.7 Appendix 3: Orthogonality of the Irreducible Tensors
	6.8 Appendix 4: Orthogonal Polynomials
7 Method of Moments: Convergence Properties
	7.1 Boltzmann Equation and Fluid-Dynamical Variables
	7.2 Bjorken Flow
		7.2.1 Method of Moments
		7.2.2 Moment Equations
		7.2.3 Results
	7.3 Gubser Flow
		7.3.1 Method of Moments
	7.4 Summary
8 Fluid Dynamics from the Method of Moments
	8.1 Power Counting
	8.2 Resummed Transient Relativistic Fluid Dynamics
	8.3 Resummed Transient Relativistic Fluid Dynamics: 14 Dynamical Variables
	8.4 Transport Coefficients
		8.4.1 14-Moment Approximation
		8.4.2 23-Moment Approximation and Beyond
	8.5 Discussion: Navier–Stokes Limit and Causality
	8.6 Resummed Transient Relativistic Fluid Dynamics: 23 Dynamical Variables
	8.7 Comparisons with Microscopic Theory
		8.7.1 Stationary Shock Solutions
	8.8 Summary
	8.9 Appendix 1: Transport Coefficients in Eq. (8.44)
	8.10 Appendix 2: Calculation of the Collision Integrals
		8.10.1 Particle-Diffusion Current
		8.10.2 Shear-Stress Tensor
	8.11 Appendix 3: Calculation of γ1(2)
		8.11.1 14-Moment Approximation
		8.11.2 23-Moment Approximation
	8.12 Appendix 4: Transport Coefficients in Sect. 8.6
9 Method of Moments: Anisotropic Reference State
	9.1 Fluid-Dynamical Variables
	9.2 Anisotropic State
	9.3 Expansion Around the Anisotropic Distribution Function
	9.4 Equations of Motion for the Irreducible Moments
	9.5 Collision Integrals
	9.6 Summary
	9.7 Appendix 1: Irreducible Projection Operators
	9.8 Appendix 2: Generalized Thermodynamic Integrals
	9.9 Appendix 3: Generalized Thermodynamic Integrals  in the Equilibrium Limit
	9.10 Appendix 4: Orthogonality of the Irreducible Tensors
	9.11 Appendix 5: Orthogonal Polynomials