Non-conformal Fluids Research Articles

Causal first-order hydrodynamics from kinetic theory and holography

We show how causal relativistic Navier-Stokes equations arise from the relativistic Boltzmann equation: the causality is preserved via a judicious choice of the zero modes of the collision operator. A completely analogous procedure may be used to extract causal hydrodynamics from the fluid-gravity correspondence: again, causality of the hydrodynamic equations is preserved by a suitable choice of zero modes of the corresponding differential operators in the bulk. We give examples of zero modes which give rise to causal hydrodynamic equations for non-conformal fluids with a conserved U(1) global symmetry current.

Hydrodynamic attractor in the nonconformal Bjorken flow

Nonconformal attractor behavior is studied by solving nonconformal second-order viscous hydrodynamics with respect to boost-invariant plasmas. Numerical solutions of the relative decay rate of the enthalpy density, the inverse shear and bulk Reynolds numbers all exhibit universal patterns towards an ideal fluid description at late stages, demonstrating the existence of a nonconformal hydrodynamic attractor. However, as a generic feature, the nonconformal hydrodynamic attractor emerges only at late times, which differs drastically from the behavior observed in conformal systems. The absence of the early-time attractor in nonconformal fluids is a consequence of the early-time unstable mode, which arises from a positive-valued eigenmode associated with the free-streaming dynamics. The early-time attractor can be restored in a mixture of the shear and bulk viscous corrections, in which early-time unstable modes cancel approximately. However, to support practical simulations of hydrodynamics in high-energy heavy-ion collisions, for which the early-time nonconformal attractor is approachable in all independent modes, the quadratic coupling between the bulk pressure and the expansion rate should be enhanced by at least a factor of two through the transport coefficient ${\ensuremath{\delta}}_{\mathrm{\ensuremath{\Pi}}\mathrm{\ensuremath{\Pi}}}$.

Second order transport coefficients of nonconformal fluids from compactified Dp-branes

All the 7 dynamical second order transport coefficients of the nonconformal fluids that correspond to Dp-branes with one or more world-volume directions compactified are derived via fluid/gravity correspondence. The conditions considered in this paper include D4-brane with 1, 2 or 3 compact directions, D3-brane with 1 or 2 compact directions, as well as D2-brane with 1 direction compactified. The derived second order transport coefficients satisfy the Haack-Yarom, Romatschke and Kleinert-Probst relations.

Causality and existence of solutions of relativistic viscous fluid dynamics with gravity

A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive the most general viscous energy-momentum tensor yielding equations of motion of second order in the derivatives, which is shown to provide a novel type of generalization of the relativistic Navier-Stokes equations for which causality holds. We show how this energy-momentum tensor may be derived from conformal kinetic theory. We rigorously prove existence, uniqueness, and causality of solutions of this theory (in the full nonlinear regime) both in a Minkowski background and also when the fluid is dynamically coupled to Einstein's equations. Linearized disturbances around equilibrium in Minkowski spacetime are stable in this causal theory. A numerical study reveals the presence of an out-of-equilibrium hydrodynamic attractor for a rapidly expanding fluid. Further properties are also studied and a brief discussion of how this approach can be generalized to non-conformal fluids is presented.

( 3+1 )-dimensional anisotropic fluid dynamics with a lattice QCD equation of state

Anisotropic hydrodynamics improves upon standard dissipative fluid dynamics by treating certain large dissipative corrections nonperturbatively. Relativistic heavy-ion collisions feature two such large dissipative effects: (i) Strongly anisotropic expansion generates a large shear stress component which manifests itself in very different longitudinal and transverse pressures, especially at early times. (ii) Critical fluctuations near the quark-hadron phase transition lead to a large bulk viscous pressure on the conversion surface between hydrodynamics and a microscopic hadronic cascade description of the final collision stage. We present a new dissipative hydrodynamic formulation for nonconformal fluids where both of these effects are treated nonperturbatively. The evolution equations are derived from the Boltzmann equation in the 14-moment approximation, using an expansion around an anisotropic leading-order distribution function with two momentum-space deformation parameters, accounting for the longitudinal and transverse pressures. To obtain their evolution we impose generalized Landau matching conditions for the longitudinal and transverse pressures. We describe an approximate anisotropic equation of state that relates the anisotropy parameters with the macroscopic pressures. Residual shear stresses are smaller and are treated perturbatively, as in standard second-order dissipative fluid dynamics. The resulting optimized viscous anisotropic hydrodynamic evolution equations are derived in $3+1$ dimensions and tested in a ($0+1$)-dimensional Bjorken expansion, using a state-of-the-art lattice equation of state. Comparisons with other viscous hydrodynamical frameworks are presented.

Second-order hydrodynamics and universality in non-conformal holographic fluids

We study second-order hydrodynamic transport in strongly coupled non-conformal field theories with holographic gravity duals in asymptotically anti-de Sitter space. We first derive new Kubo formulae for five second-order transport coefficients in non-conformal fluids in $(3+1)$ dimensions. We then apply them to holographic RG flows induced by scalar operators of dimension $\Delta=3$. For general background solutions of the dual bulk geometry, we find explicit expressions for the five transport coefficients at infinite coupling and show that a specific combination, $\tilde{H}=2\eta\tau_\pi-2(\kappa-\kappa^*)-\lambda_2$, always vanishes. We prove analytically that the Haack-Yarom identity $H=2\eta\tau_\pi-4\lambda_1-\lambda_2=0$, which is known to be true for conformal holographic fluids, also holds when taking into account leading non-conformal corrections. The numerical results we obtain for two specific families of RG flows suggest that $H$ vanishes regardless of conformal symmetry. Our work provides further evidence that the Haack-Yarom identity $H=0$ may be universally satisfied by strongly coupled fluids.

Relativistic viscous fluid dynamics and non-equilibrium entropy

Fluid dynamics corresponds to the dynamics of a substance in the long wavelength limit. Writing down all terms in a gradient (long wavelength) expansion up to second order for a relativistic system at vanishing charge density, one obtains the most general (causal) equations of motion for a fluid in the presence of shear and bulk viscosity, as well as the structure of the non-equilibrium entropy current. Requiring positivity of the divergence of the non-equilibrium entropy current relates some of its coefficients to those entering the equations of motion. I comment on possible applications of these results for conformal and non-conformal fluids.

Theoretical Equation of State of Dense Nonconformal Fluids from Effective Potentials. 1. Applications to Model Systems

The approximate nonconformal (ANC) theory, successfully applied to dilute gases, is generalized to dense fluids. Monte Carlo simulations were performed for several ANC potentials, used as effective nonconformal interactions in the theory. These results were used to assess the performance of alternate equations of state (EOS). Perturbation theory illuminates the effects of nonconformality but fails to produce an EOS of ANC fluids of the needed accuracy. An accurate empirical EOS was found for supercritical fluids and used to test the applicability of the theory. EOSs were found for Lennard-Jones, two-center Lennard-Jones, linear Kihara, and square well heterodiatiomic fluids. These theoretical EOSs are in good agreement with simulation results of the same fluids. The effective potentials of the nonspherical systems are found to be state-dependent with parameters modeled by a low order virial-like series. These results give a basis for the application of the theory to real dense fluids.

SAFT-VRE: Phase Behavior of Electrolyte Solutions with the Statistical Associating Fluid Theory for Potentials of Variable Range

A modification of the statistical associating fluid theory has recently been developed to model non-conformal fluids with attractive potentials of variable range (SAFT-VR) [Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. J. Chem. Phys. 1997, 106, 4168] which gives a very good description of the phase behaviour of water and its mixtures with nonelectrolytes. In the present paper we extend the SAFT-VR approach to deal with strong-electrolyte solutions (SAFT-VRE). The water molecules are modeled as hard spheres with four attractive short-range sites to describe the hydrogen-bonding association. The electrolyte molecules are modeled with two hard spheres of different size which describe the anion and cation respectively. The mean-spherical approximation (MSA) is used for the restricted primitive model (RPM) to account for the long-range Coulombic ion−ion interactions, while the long-range water−water and ion−water attractive interactions are modeled as square-well di...

Structure of variable-width square-well fluids from the reference hypernetted chain equation

The problem of calculating the structure of square-well (SW) fluids in the context of non-conformal fluids is reviewed briefly. New and extensive results over a wide range of SW systems and thermodynamic states were obtained by means of the reference hypernetted chain (RHNC) integral equation. The results show that RHNC theory is in excellent agreement with the results of simulation, thus providing a reliable and fast procedure for obtaining the structure of fluids with discontinuous potentials. A few representative results are presented and compared with new Monte Carlo simulation data. The bridge functions obtained with the RHNC procedure are also compared with those calculated directly from the Monte Carlo radial distribution functions.

Van der waals one-fluid theory : justification and generalisation

We describe an approach to van der Waals one-fluid theory based on thermodynamic consistency and propose a method for generalising it to non-conformal fluids.