Hydro Primer

Posted by Steffen A. Bass on September 14th 2007
Categories: Bulk Properties,Collective Flow,Historic Milestone,Reviews,Transport Theory

This post provides a list on historically relevant work on the application of
relativistic fluid dynamics (RFD) to heavy-ion collisions. Note that the history and state of the art of hybrid hydro+micro approaches will be covered in a separate post:

The application of fluid dynamics to nuclear matter dates as far back as the liquid drop model. Post World War II the study of excited nuclear matter became an active topic of research. One of the earliest applications of fluid dynamics to that topic is the famous work by L.D. Landau:
On the multiparticle production in high-energy collisions

In the early 1970ies several groups suggested that matter in nuclear collisions would be compressed and heated through the generation of shock-waves, which could be calculated with the help of nuclear fluid dynamics. The first paper of significant impact was that of Chapline, Johnson, Teller and Weiss and the second one was by the Frankfurt group (W. Scheid, H. Mueller and W. Greiner):

  1. Highly excited nuclear matter
  2. Nuclear Shock Waves in Heavy-Ion Collisions

Note, however, that historically the application of RFD was by no means generally accepted and was treated with substantial scepticism. Here for example a comment by a very prominent theorist who at that time disputed that heavy-ion collisions would ever generate densities larger than twice nuclear matter ground-state density:
Comment on nuclear shock waves in heavy-ion collisions

The Frankfurt Group played an important role in the application of RFD to collective flow, both in terms of sidewards flow (1st reference) as well
as regarding the squeeze-out of matter perpendicular to the
reaction plane (2nd reference):

A good review of the early days of Relativistic Fluid Dynamics was written by Clare and Strottman:
Relativistic Hydrodynamics and heavy ion reactions.

RFD made its entry into the domain of ultra-relativistic heavy-ion collisions with Bjorken’s seminal paper:
Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region

A good resource to learn the nuts and bolts of RFD are the following papers:

Nearly all RFD codes utilize the Cooper-Frye Formula to calculate spectra at freeze-out – here is the original reference:
Single-particle distribution in the hydrodynamic and statistical thermodynamic models of multiparticle production

Elliptic flow at ultra-relativistic energies, the preferential emission of matter along the impact parameter axis of the overlap zone of the two colliding nuclei, was first discussed as a fluid-dynamical effect by Ollitrault:
Anisotropy as a signature of transverse collective flow
Note that this phenomenon is related to the squeeze-out of matter found at BEVALAC and SIS energies – only here the emission is in-plane, driven by the pressure gradients of the compressed matter expanding into the vacuum, whereas at lower energies the emission direction is out-of-plane, dicated by the presence of spectator matter blocking the expansion in-plane.

The main relevance of elliptic flow lies in the short timescales over which it is generated and its resulting sensitivity to the equation of state of the (QGP) medium. A selection of high-impact publications on that matter is given here:

The most recent state-of-the-art implementations of ideal RFD are fully 3+1 dimensional – there are currently two models available, one relying on an Eulerian grid (Hirano et al.) and one based on a Lagrangian (co-moving) grid (Nonaka et al):

One of the major problems in the application of ideal RFD to RHIC energies is that for the spectra to be reproduced a thermal freeze-out temperature of approx. 130 MeV needs to be used wheras data on particle ratios suggest a chemical freeze-out temperature at 160 MeV. The problem can be remedied in an ad-hoc fashion by renormalizing the spectra of kaons, protons, etc., which, however, only hides the problem w/o fixing it. Two approaches have been shown to address this problem:

  1. Partical Chemical Equilibrium approach: here one introduces a chemical potential at the critical temperature for each hadron whose yield is supposed to be frozen-out at that temperature. The PCE approach can account for the proper normalization of the spectra, however, it fails to reproduce the
    transverse momentum and mass dependence of the elliptic flow:

  2. TheHybrid Hydro+Micro approach: in this approach the hadronic phase is treated via a microscopic calculation based on a Boltzmann equation. This allows for the separate flavor-dependent freeze-out of each hadron species. The hybrid approach has been shown to reproduce spectra and elliptic flow simultatneously. We will cover this approach in a separate post.

An alternative approach to the solution of the hydrodynamic equations is provided by the smooth particle hydrodynamics method. The SPH method also alleviates some of the shortcomings of regular RFD regading the simultaneous chemical and kinetic freeze-out of all hadron species:
Smoothed particle hydrodynamics for relativistic heavy ion collisions

The current holy grail of RFD is the development of a viscous 3+1 dimensional RFD implementation. The importance of viscosity was recognized early on, see e.g. this early paper by the Frankfurt group:
Viscous fluid dynamical calculation of the reaction C-12 82-MeV/nucleon + Au-197

Very recently significant progress has been made by several groups in the formulation of a viscous fluid approach to the description of a QGP:

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