# Theories of Relativistic Dissipative Fluid Dynamics

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## Abstract

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## 1. Introduction and Summary

#### Notations and Conventions

## 2. Preliminaries

#### 2.1. Conserved Quantities

#### 2.2. Conservation Laws

#### 2.3. Ideal Fluid Dynamics

#### 2.4. Power Counting in Fluid Dynamics—Knudsen and Inverse Reynolds Numbers

#### 2.5. Matching Conditions

- (i)
**Ideal fluid dynamics:**As already discussed in Section 2.3, for ideal fluid dynamics, all dissipative quantities vanish, i.e., their constitutive relations are trivial,$$\delta \epsilon =\delta n=\mathrm{\Pi}={n}^{\mu}={h}^{\mu}={\pi}^{\mu \nu}=0\phantom{\rule{0.277778em}{0ex}}.$$- (ii)
**Dissipative fluid dynamics in the Landau frame:**For dissipative fluid dynamics, a possible subset of the constitutive relations reads$$\delta \epsilon =0\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{1.em}{0ex}}\delta n=0\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{1.em}{0ex}}{h}^{\mu}=0\phantom{\rule{0.277778em}{0ex}}.$$The other constitutive relations for ${n}^{\mu}$, $\mathrm{\Pi}$, and ${\pi}^{\mu \nu}$ have to be further specified; see discussion in the following sections. The first two relations (16) imply that $\epsilon \equiv {\epsilon}_{0}$, $n\equiv {n}_{0}$, i.e., the fictitious local-equilibrium reference state is chosen in such a way that its energy density and particle-number density match the energy density and the particle-number density of the actual fluid. The conditions listed in Equation (16) are also referred to as Landau matching conditions. The last relation (16) implies no energy diffusion, i.e., the fluid 4-velocity ${u}^{\mu}\equiv {u}_{L}^{\mu}$ is identical to the energy flow. This corresponds to choosing a particular reference frame for the motion of the fluid, which is usually called Landau frame [1].Projecting Equation (1b) onto ${u}_{L,\nu}$, we derive$${T}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\nu}^{\mu}{u}_{L}^{\nu}=\epsilon {u}_{L}^{\mu}\phantom{\rule{0.277778em}{0ex}},$$$${u}_{L}^{\mu}=\frac{{T}^{\mu \nu}{u}_{L,\nu}}{\sqrt{{T}^{\alpha \beta}{u}_{L,\beta}{T}_{\alpha \gamma}{u}_{L}^{\gamma}}}\phantom{\rule{0.277778em}{0ex}}.$$Note that the latter definition is an implicit equation for ${u}_{L}^{\mu}$.- (iii)
**Dissipative fluid dynamics in the Eckart frame:**Another possible subset of constitutive relations reads$$\delta \epsilon =0\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{1.em}{0ex}}\delta n=0\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{1.em}{0ex}}{n}^{\mu}=0\phantom{\rule{0.277778em}{0ex}}.$$The other constitutive relations for ${h}^{\mu}$, $\mathrm{\Pi}$, and ${\pi}^{\mu \nu}$ have to be further specified.The first two relations (19) are the same as for the Landau frame, i.e., the fictitious local-equilibrium reference state is again chosen in such a way that its energy density and particle-number density agree with the energy density and the particle-number density of the actual fluid. The conditions of Equation (19) are also referred to as Eckart matching conditions.The last relation (19) implies no particle diffusion, i.e., the fluid 4-velocity ${u}^{\mu}\equiv {u}_{E}^{\mu}$ is identical to the flow of particle number. This particular choice of reference frame is called Eckart frame [2]. Solving Equation (1a) for ${u}_{E}^{\mu}$, we derive$${u}_{E}^{\mu}\equiv \frac{{N}^{\mu}}{\sqrt{N\xb7N}}\phantom{\rule{0.277778em}{0ex}}.$$

#### 2.6. Constitutive Relations—General Considerations

- (i)
- Linear order in the product of derivatives of primary fluid-dynamical quantities and dissipative quantities, i.e., $\sim \dot{B}A$ or $\left({\nabla}^{\mu}B\right)\phantom{\rule{0.166667em}{0ex}}A$;
- (ii)
- Linear order in the derivatives of dissipative quantities, i.e., $\sim \dot{A}$ or ${\nabla}^{\mu}A$;
- (iii)
- Second order in derivatives of primary fluid-dynamical quantities, i.e., $\sim \ddot{B}$, ${\dot{B}}^{2}$, $\dot{B}{\nabla}^{\mu}B$, ${\nabla}^{\mu}\dot{B}$, $u\xb7\partial \left({\nabla}^{\mu}B\right)$, ${\nabla}^{\mu}{\nabla}^{\nu}B$, or $\left({\nabla}^{\mu}B\right)\left({\nabla}^{\nu}B\right)$,
- (iv)
- Second order in dissipative quantities, i.e., $\sim {A}^{2}$.