Israel–Stewart, BRSSS, and DNMR Transient Hydrodynamics

Müller–Israel–Stewart, BRSSS, and DNMR are related but nonidentical constructions. MIS promotes dissipative stresses to transient variables constrained by an entropy-current or moment ansatz; BRSSS classifies conformal Landau-frame constitutive data through second order; DNMR derives transient equations and coefficient relations from a specified kinetic moment closure with simultaneous Knudsen and inverse-Reynolds counting.

Required background. Conventional Relativistic Navier–Stokes Instability and Acausality motivates transient completion. Moment Hierarchies and Closure Schemes supplies the kinetic closure logic.

Helpful background. Strong Hyperbolicity, Stability, and Causal Propagation separates the conditions that a transient model must satisfy.

A minimal transient shear channel

Promote the shear stress $\pi^{\mu\nu}$ to an independent field. In the simplest local rest-frame channel,

$$\tau_\pi\partial_t\pi^{xy}+\pi^{xy} = -\eta\,\partial_x v_y,$$

while momentum conservation gives

$$w\partial_t v_y+\partial_x\pi^{xy}=0.$$

Eliminating $\pi^{xy}$ yields

$$\tau_\pi\partial_t^2v_y +\partial_t v_y -\frac{\eta}{w}\partial_x^2v_y=0.$$

The dispersion relation is

$$\tau_\pi\omega^2+i\omega-\frac{\eta}{w}k^2=0.$$

At small $k$,

$$\omega_{\mathrm h} = -i\frac{\eta}{w}k^2+O(k^4), \qquad \omega_{\mathrm{nh}} = -\frac{i}{\tau_\pi} +i\frac{\eta}{w}k^2+O(k^4).$$

The hydrodynamic shear pole is preserved, and a nonhydrodynamic relaxation pole appears. The high-$k$ characteristic speed in this isolated channel is

$$v_{\mathrm{sh}}^2=\frac{\eta}{w\tau_\pi}.$$

Thus $\eta>0$ and $\tau_\pi>0$ give damping, while causal shear characteristics require $v_{\mathrm{sh}}\le1$. The coupled sound, bulk, heat, and charge sectors impose additional inequalities; this one-channel result is not a complete causality certificate.

Müller–Israel–Stewart

The MIS family introduces independent dissipative quantities such as bulk pressure $\Pi$, heat or diffusion current $q^\mu$, and shear stress $\pi^{\mu\nu}$. Let $\Delta^\mu{}_{\nu}=\delta^\mu{}_{\nu}-u^\mu u_\nu$ be the mixed transverse projector and $\Delta^{\mu\nu}{}_{\alpha\beta}$ its symmetric transverse-traceless counterpart. Schematically,

$$\tau_\Pi D\Pi+\Pi=-\zeta\theta+\cdots,$$ $$\tau_q\Delta^\mu{}_\nu Dq^\nu+q^\mu = \text{first-order thermal force}+\cdots,$$ $$\tau_\pi\Delta^{\mu\nu}{}_{\alpha\beta}D\pi^{\alpha\beta} +\pi^{\mu\nu} = \eta\sigma^{\mu\nu}+\cdots.$$

The ellipses are not optional decoration. They contain expansion, acceleration, vorticity, cross-coupling, and nonlinear terms whose form depends on the MIS variant and matching assumptions. Israel and Stewart derive a transient thermodynamic/kinetic framework with additional variables and relaxation times Israel and Stewart 1979, §§III–V, pp. 352–372.

“MIS” is therefore a family label. Causality and stability require coefficient and state inequalities; nonlinear well-posedness is established only for particular sectors and formulations. One cannot cite the presence of $\tau_\pi$ as a proof.

BRSSS

BRSSS classifies the stress tensor of a conformal fluid in Landau frame through second derivatives. Besides $\eta$, the flat-space second-order data include a relaxation-type coefficient $\tau_\pi$ and nonlinear conformal tensors with coefficients commonly denoted $\lambda_1,\lambda_2,\lambda_3$; curvature response adds $\kappa$. Baier et al. give the complete conformal basis and identify its Kubo and microscopic meaning Baier et al. 2008, §§3–4, pp. 6–17, Open PDF.

As a constitutive gradient expansion, BRSSS does not by itself define an exact hyperbolic evolution equation. A common numerical implementation rewrites part of the constitutive relation as a relaxation equation for $\pi^{\mu\nu}$. That completion agrees with BRSSS through the declared order but fixes higher-order terms that the second-order basis does not determine.

Consequently:

• BRSSS is conformal; generic bulk dynamics is outside its original scope.
• A BRSSS coefficient is not automatically an MIS entropy-current coefficient with the same name.
• A “BRSSS evolution” must state the particular relaxation completion.

DNMR

DNMR begins with a relativistic Boltzmann equation and expands the distribution around local equilibrium in irreducible moments. The closure keeps selected dissipative moments as independent variables and counts both

$$\mathrm{Kn}\sim\frac{\ell_{\mathrm{micro}}}{L}, \qquad \mathrm{Re}^{-1}\sim \frac{\text{dissipative stress}}{\text{equilibrium pressure}}.$$

Terms of the same combined order can be lost in a simpler 14-moment or entropy-current truncation. The resulting relaxation times, cross-couplings, and nonlinear coefficients depend on the collision kernel, particle content, moment basis, and truncation. Denicol et al. derive this systematic transient closure and its coefficient relations Denicol et al. 2012, §§II–VI, pp. 2–18, Open PDF.

DNMR is thus a kinetic derivation of a class of transient equations, not a synonym for every second-order hydrodynamic model. Its microscopic origin does not by itself prove nonlinear causality after moment truncation.

Framework dictionary

Feature	MIS	BRSSS	DNMR
Primary purpose	Transient irreversible thermodynamics	Complete conformal constitutive basis through second order	Controlled kinetic moment closure
Extra variables	Dissipative stresses/fluxes	None in the constitutive classification; often added in a completion	Selected kinetic moments as transient variables
Counting	Near-equilibrium/entropy-current or moment assumptions	Derivative expansion and conformal symmetry	Knudsen plus inverse Reynolds
Coefficients	Phenomenological or kinetic, variant dependent	Independent conformal transport data	Related by specified collision model and closure
Bulk viscosity	Allowed	Zero in conformal theory	Allowed for nonconformal kinetic systems
Hyperbolicity	Conditional	Not defined by the gradient basis alone	Conditional after closure
Common misuse	Treating one minimal equation as all MIS	Calling any relaxation model BRSSS	Treating kinetic origin as an automatic theorem

The broader status comparison is in the relativistic hydrodynamic consistency reference.

Bjorken-flow comparison

For longitudinal expansion, define the shear correction $\Phi$ by

$$p_L=p-\Phi, \qquad p_T=p+\frac12\Phi.$$

Energy conservation is

$$\frac{\mathrm d\epsilon}{\mathrm d\tau} = -\frac{\epsilon+p-\Phi}{\tau}.$$

A generic transient equation has the structure

$$\tau_\pi\frac{\mathrm d\Phi}{\mathrm d\tau} +\Phi = \frac{4\eta}{3\tau} -\delta_{\pi\pi}\frac{\Phi}{\tau} -\frac{\lambda_1}{2\eta^2}\Phi^2 +\cdots.$$

Minimal MIS may omit or choose the last terms; a BRSSS completion fixes the conformal second-order combination through its coefficients; DNMR supplies $\delta_{\pi\pi}$ and other terms from the chosen kinetic closure. Matching only $\eta$ and $\tau_\pi$ does not make the three evolutions identical.

Use the second and third columns to keep three constructions apart. Inspect which data are independent relaxation variables, which are conformal gradient coefficients, and which arise only after choosing a kinetic or transient closure.

Israel–Stewart promotes dissipative stresses to relaxing variables; BRSSS classifies second-order conformal constitutive data; DNMR derives a particular moment closure and power counting from kinetic theory. A numerical transient implementation of BRSSS adds a completion not fixed by the gradient series alone. The diagram is schematic, and adjacency does not make these frameworks equivalent.

In text: compare the retained variables, counting scheme, relaxation spectrum, nonlinear second-order terms, and coefficient origin. Matching $\eta$ and $\tau_\pi$ fixes neither the full transient equations nor their characteristic and stability properties.

Exercise

For the minimal shear telegrapher system, determine the conditions for damping and subluminal shear characteristics.

Solution

The nonhydrodynamic root at $k=0$ is $-i/\tau_\pi$, so damping requires $\tau_\pi>0$. The hydrodynamic root is $-i(\eta/w)k^2$, so with $w>0$ it requires $\eta\ge0$. At large $k$ the real part approaches

$$\operatorname{Re}\omega_\pm \sim \pm\sqrt{\frac{\eta}{w\tau_\pi}}\,k.$$

Hence the isolated shear channel is causal only if

$$0\le\frac{\eta}{w\tau_\pi}\le1.$$

Sound and other channels can impose stronger conditions, so this is necessary for the toy sector, not sufficient for a full theory.

Where this leads

BDNK First-Order Causal Hydrodynamics obtains causal characteristics using general-frame constitutive terms rather than independent stress variables. Hydrodynamic Attractors and Asymptotic Gradient Expansions uses transient nonhydrodynamic modes to explain divergent gradient series.

References

• Baier, Rudolf, Paul Romatschke, Dam T. Son, Andrei O. Starinets, and Mikhail A. Stephanov. 2008. “Relativistic Viscous Hydrodynamics, Conformal Invariance, and Holography.” Journal of High Energy Physics 2008 (4): 100. DOI. Open PDF.

• Denicol, Gabriel S., Harri Niemi, Etele Molnár, and Dirk H. Rischke. 2012. “Derivation of Transient Relativistic Fluid Dynamics from the Boltzmann Equation.” Physical Review D 85: 114047. DOI. Open PDF.

• Israel, Werner, and John M. Stewart. 1979. “Transient Relativistic Thermodynamics and Kinetic Theory.” Annals of Physics 118: 341–372. DOI.