Flux-Conservative Relativistic Viscous Hydrodynamics

• The paper introduces a flux-conservative formulation of relativistic viscous hydrodynamics that embeds second-order dissipation terms to ensure causality and stability.
• It outlines the transition from first-order Navier–Stokes to second-order MIS-type methods, enabling robust numerical simulations in heavy-ion collisions.
• The framework combines energy–momentum conservation with relaxation-type corrections to accurately model the quark–gluon plasma dynamics.

The flux-conservative formulation of relativistic viscous hydrodynamics represents a framework in which the dynamical equations governing the evolution of dissipative relativistic fluids are written in explicit divergence (or conservation) form, rendering them particularly amenable to both theoretical analysis and high-resolution numerical simulation. The structure of these equations is fundamentally dictated by the local conservation of energy–momentum, and, in modern formulations, is extended to ensure causality and stability through the inclusion of appropriate second-order (in gradients) terms. This approach has direct relevance for describing both weakly and strongly coupled systems, with notable applications in modeling the quark–gluon plasma in ultrarelativistic heavy-ion collisions.

1. Fundamentals of the Flux-Conservative Structure

The central principle is that the time evolution of a relativistic viscous fluid is regulated by conservation laws, most notably the local conservation of the energy–momentum tensor: $\partial_\mu T^{\mu\nu} = 0$ where the energy–momentum tensor is decomposed into ideal and dissipative parts: $T^{\mu\nu} = T^{\mu\nu}_0 + \Pi^{\mu\nu}$ with

$$T^{\mu\nu}_0 = \varepsilon u^\mu u^\nu - p\Delta^{\mu\nu}$$

$\Delta^{\mu\nu} = g^{\mu\nu} - u^\mu u^\nu$

Here, $\varepsilon$ is the local energy density, $p$ the pressure, $u^\mu$ the fluid four-velocity ($u^2=1$), and $g^{\mu\nu}$ the metric tensor.

The flux-conservative formulation leverages the divergence form of the conservation law and splits it into evolution equations for the hydrodynamic fields (energy density, fluid velocity, and dissipative stresses), typically via projections along $u^\mu$ (time-like) and $\Delta^\mu_\alpha$ (space-like): $$D \equiv u^\mu \partial_\mu,\quad \nabla^\alpha \equiv \Delta^{\alpha\mu} \partial_\mu$$ yielding coupled equations for $\varepsilon$ and $u^\mu$ that explicitly include viscous contributions: $$D\varepsilon + (\varepsilon+p)\nabla_\mu u^\mu - \Pi^{\mu\nu}\nabla_{\langle\mu} u_{\nu\rangle} = 0$$

$$(\varepsilon+p)Du^\alpha - \nabla^\alpha p + \Delta^\alpha_\nu\partial_\mu \Pi^{\mu\nu} = 0$$

Here, $\langle\cdots\rangle$ denotes symmetric, traceless, and transverse projections as appropriate for shear stresses.

2. First-Order (Navier–Stokes) vs Second-Order (MIS-type) Dissipation

In the relativistic context, naive first-order (Navier–Stokes) viscous terms,

$$\Pi^{\mu\nu} = \eta \nabla^{\langle\mu} u^{\nu\rangle} + \zeta \Delta^{\mu\nu} (\nabla_\alpha u^\alpha)$$

with $\eta$ and $\zeta$ the shear and bulk viscosity, respectively, lead to parabolic (diffusive) PDEs. Such systems are acausal—allowing arbitrarily fast signal propagation in the ultraviolet regime (as seen from the group velocity for linearized shear perturbations, $d\omega/dk \to \infty$ as $k\to\infty$) and are numerically unstable without artificial regularization.

To ensure causality and stability, a second-order (MIS-type) gradient expansion is employed. Supplementary dynamical equations for the shear-stress tensor components (and, if present, for bulk and heat current) are introduced, typically in relaxation-type form: $$\pi^{\mu\nu} + \tau_\pi \left[ \Delta^\mu_\alpha \Delta^\nu_\beta D \pi^{\alpha\beta} + \frac{4}{3} \pi^{\mu\nu} \nabla_\alpha u^\alpha \right] = \eta \nabla^{\langle\mu} u^{\nu\rangle} + \cdots$$ Additional second-order transport coefficients (e.g., $\tau_\pi$, $\kappa$, $\lambda_1, \lambda_2, \lambda_3$) can be systematically enumerated, especially in conformal theories, and may include curvature and non-linear terms: $$\pi^{\mu\nu} = \eta \nabla^{\langle\mu} u^{\nu\rangle} - \tau_\pi \left[ \Delta^\mu_\alpha \Delta^\nu_\beta D \pi^{\alpha\beta} + \frac{4}{3} \pi^{\mu\nu} \nabla_\alpha u^\alpha \right] + \cdots$$ This upgrade yields a set of hyperbolic, first-order-in-time PDEs suitable for stable, causal numerical evolution.

3. Numerical Implementation: Conservative Algorithms and Stability

In practice, the entire system—including the conservation of energy–momentum and the evolution equations for dissipative quantities—is recast in the generic balance-law form,

$\partial_t U + \nabla \cdot F(U) = S(U)$

where $U$ is the vector of evolved fields (energy, momentum, and dissipative stresses), $F(U)$ denotes the fluxes associated with each field, and $S(U)$ are (possibly geometrically induced) source terms.

This form is ideally suited for finite-volume and shock-capturing algorithms, which guarantee discrete conservation and high resolution across sharp gradients (such as those encountered in heavy-ion collision simulations). Crucially, the inclusion of genuine physical viscosity damps high-wavenumber numerical noise and obviates the need for artificial (numerical) viscosity typically invoked in ideal hydrodynamics.

The equations are often implemented in coordinate systems adapted to the physical symmetry (e.g., Milne coordinates for boost-invariant expansion), and the conservative form ensures that the discretization respects the relevant global (and local) conservation laws.

4. Microscopic Input: Transport Coefficients from Kinetic Theory and AdS/CFT

The regime of applicability and the specific values of the second-order transport coefficients depend critically on the underlying microscopic dynamics.

• In weakly coupled systems with quasiparticle descriptions, kinetic theory provides estimates for coefficients such as $\tau_\pi$ and $\eta/s$ (cf. Arnold–Moore–Yaffe),

$$\tau_\pi \approx \text{const}/T \cdot (\eta/s),\quad \lambda_2 \approx -2\eta\tau_\pi$$

• For strongly coupled theories, the AdS/CFT correspondence yields precise values for conformal fluids, as in $\mathcal{N}=4$ super-Yang–Mills theory,

$$c_s^2 = 1/3, \quad \eta/s = 1/(4\pi), \quad \tau_\pi = (2 - \ln2)/(2\pi T),\quad \text{etc.}$$

Insertion of these values into the MIS-type (or flux-conservative) framework yields hydrodynamic equations that maintain causality, as all mode group velocities are bounded. For example, the maximum transverse group velocity is given by

$$v_T^{\text{max}} = \sqrt{\eta_0/( \tau_\pi(\varepsilon_0 + p_0)) }$$

5. Phenomenological Applications: Heavy-Ion Collisions

The flux-conservative viscous hydrodynamic equations underpin state-of-the-art simulations of ultrarelativistic heavy-ion collisions, including the time evolution of the quark–gluon plasma. In the canonical boost-invariant (Bjorken) scenario the equations reduce to a coupled set of ODEs, for instance,

$$\partial_\tau \varepsilon + \frac{\varepsilon + p}{\tau} - \frac{\pi^\xi_\xi}{\tau} = 0$$

$$\partial_\tau \pi^\xi_\xi = -\frac{\pi^\xi_\xi}{\tau_\pi} + \frac{4\eta}{3\tau\tau_\pi} - \frac{4\pi^\xi_\xi}{3\tau} - \frac{\lambda_1}{2\tau_\pi \eta^2} (\pi^\xi_\xi)^2$$

and are solved numerically using conservative finite-difference schemes.

Direct comparisons with RHIC data (multiplicities, elliptic flow $v_2$, etc.) demonstrate the predictive power of this approach. Simulations reproduce, e.g., the 20% reduction in $v_2$ at $p_T \sim 1.5$ GeV for $\eta/s = 0.08$, closely matching experimental measurements and supporting the near-minimal viscosity of the QGP.

6. Theoretical Robustness and Regimes of Applicability

The flux-conservative formulation can be derived independenty from:

• An extended entropy current and the generalized second law of thermodynamics (ensuring nonnegative entropy production and yielding relaxation-type equations for dissipative quantities),
• The systematic gradient expansion in the underlying (quantum) field theory,
• Matching moments from kinetic theory (Boltzmann equation).

Each derivation clarifies the coefficients’ regime of validity and constrains the physical applicability of the theory. For instance, in strongly coupled plasmas described by AdS/CFT, the transport coefficients extracted are directly inserted.

Second-order (MIS-type) flux-conservative formulations are causal and stable, while first-order (pure Navier–Stokes) forms are ruled out for relativistic regimes due to instabilities and acausal behaviors.

7. Summary Table: Key Formulas and Features

Aspect	Flux-Conservative Formulation	Significance
Conservation law	$\partial_\mu T^{\mu\nu} = 0$	Fundamental evolution basis
Energy-momentum tensor	$$T^{\mu\nu} = \varepsilon u^\mu u^\nu - p \Delta^{\mu\nu} + \pi^{\mu\nu}$$	Including both ideal and viscous terms
Constitutive equation	$$\pi^{\mu\nu} + \tau_\pi [\cdots] = \eta\nabla^{\langle\mu} u^{\nu\rangle} + \cdots$$	Second-order, relaxation-type evolution
Discretization	$\partial_t U + \nabla\cdot F(U) = S(U)$	Suitable for finite-volume/shock-capturing schemes
Microscopic coefficients	$\eta/s$, $\tau_\pi$, $\lambda_{1,2,3}$ from kinetic theory/AdS/CFT	Input for causal, stable evolution
Numerical application	Bjorken flow: $$\partial_\tau \varepsilon + (\varepsilon+p)/\tau - \pi^\xi_\xi/\tau = 0$$	Benchmark for QGP time evolution

In conclusion, the flux-conservative formulation provides a theoretically robust, numerically stable, and phenomenologically validated framework for relativistic viscous fluids. It is central to contemporary analyses of both fundamental theory (across weak and strong coupling) and quantitative interpretation of high-energy experimental data, most notably in the dynamics of heavy-ion collisions (0902.3663).