Relativistic fluid dynamics has regularly been used as a tool to model heavy ion collisions. The idea of using hydrodynamics to study the process of multiparticle production in high-energy hadron collisions can be traced back to work by, in particular, Landau in the early 1950s (see [12]). In the early days these phenomena were observed in cosmic rays. The idea to use hydrodynamics was resurrected as collider data became available [15] and early simulations were carried out at Los Alamos [2, 3]. More recently, modeling has primarily been focussed on reproducing data from, for example, CERN. A useful review of this active area of research can be found in [33].

In a hydrodynamics based model, a high-energy nuclear collision is viewed in the following way: In the center-of-mass frame two Lorentz contracted nuclei collide and, after a complex microscopic process, a hot dense plasma is formed. In the simplest description this matter is assumed to be in local thermal equilibrium. The initial thermalization phase is, of course, out of reach for hydrodynamics. In the model, the state of matter is simply specified by the initial conditions, e.g. in terms of distributions of fluid velocities and thermodynamical quantities. Then follows a hydrodynamical expansion, which is described by the standard conservation equations for energy/momentum, baryon number, and other conserved quantities, such as strangeness, isotopic spin, etc. (see [40] for a variational principle derivation of these equations). As the expansion proceeds, the fluid cools and becomes increasingly rarefied. This eventually leads to the decoupling of the constituent particles, which then do not interact until they reach the detector.

Fluid dynamics provides a well defined framework for studying the stages during which matter becomes highly excited and compressed and, later, expands and cools down. In the final stage when the nuclear matter is so dilute that nucleon-nucleon collisions are infrequent, hydrodynamics ceases to be valid. At this point additional assumptions are necessary to predict the number of particles, and their energies, which may be formed (to be compared to data obtained from the detector). These are often referred to as the “freeze-out” conditions. The problem is complicated by the fact that the “freeze-out” typically occurs at a different time for each fluid cell.

Even though the application of hydrodynamics in this area has led to useful results, it is clear that the theoretical foundation for this description is not a trivial matter. Basically, the criteria required for the equations of hydrodynamics to be valid are

- many degrees of freedom in the system,
- a short mean free path,
- a short mean stopping length,
- a sufficient reaction time for thermal equilibration, and
- a short de Broglie wavelength (so that quantum mechanics can be ignored).

An interesting aspect of the hydrodynamic models is that they make use of concepts largely outside traditional nuclear physics, e.g. thermodynamics, statistical mechanics, fluid dynamics, and of course elementary particle physics. This is natural since the very hot, highly excited matter has a large number of degrees of freedom. But it is also a reflection of the basic lack of knowledge. As the key dynamics is uncertain, it is comforting to resort to standard principles of physics like the conservation of momentum and energy.

Another key reason why hydrodynamic models are favored is the simplicity of the input. Apart from the initial conditions which specify the masses and velocities, one needs only an equation of state and an Ansatz for the thermal degrees of freedom. If one includes dissipation one must in addition specify the form and magnitude of the viscosity and heat conduction. The fundamental conservation laws are incorporated into the Euler equations. In return for this relatively modest amount of input, one obtains the differential cross sections of all the final particles, the composition of clusters, etc. Of course, before one can confront the experimental data, one must make additional assumptions about the freeze-out, chemistry, etc. A clear disadvantage of the hydrodynamics model is that much of the microscopic dynamics is lost.

Let us discuss some specific aspects of the hydrodynamics that has been used in this area. As we will recognize, the issues that need to be addressed for heavy-ion collisions are very similar to those faced in studies of relativistic dissipation theory and multi-fluid modeling. The one key difference is that the problem only requires special relativity, so there is no need to worry about the spacetime geometry. Of course, it is still convenient to use a fully covariant description since one is then not tied down to the use of a particular set of coordinates.

In many studies of heavy ions a particular frame of reference is chosen. As we know from our discussion of dissipation and causality (see Section 14), this is an issue that must be approached with some care. In the context of heavy-ion collisions it is common to choose as the velocity of either energy transport (the Landau–Lifshitz frame) or particle transport (the Eckart frame). It is recognized that the Eckart formulation is somewhat easier to use and that one can let be either the velocity of nucleon or baryon number transport. On the other hand, there are cases where the Landau–Lifshitz picture has been viewed as more appropriate. For instance, when ultrarelativistic nuclei collide they virtually pass through one another leaving the vacuum between them in a highly excited state causing the creation of numerous particle-antiparticle pairs. Since the net baryon number in this region vanishes, the Eckart definition of the four-velocity cannot easily be employed. This discussion is a clear reminder of the situation for viscosity in relativity, and the resolution is likely the same. A true frame-independent description will need to include several distinct fluid components.

Multi-fluid models have, in fact, often been considered for heavy-ion collisions. One can, for example, treat the target and projectile nuclei as separate fluids to admit interpenetration, thus arriving at a two-fluid model. One could also use a relativistic multi-fluid model to allow for different species, e.g. nucleons, deltas, hyperons, pions, kaons, etc. Such a model could account for the varying dynamics of the different species and their mutual diffusion and chemical reactions. The derivation of such a model would follow closely our discussion in Section 10. In the heavy-ion community, it has been common to confuse the issue somewhat by insisting on choosing a particular local rest frame at each space-time point. This is, of course, complicated since the different fluids move at different speeds relative to any given frame. For the purpose of studying heavy ion collisions in the baryon-rich regions of space, the standard option seems to be to define the “baryonic Lorentz frame”. This is the local Lorentz frame in which the motion of the center-of-baryon number (analogous to the center-of-mass) vanishes.

The main problem with the one-fluid hydrodynamics model is the requirement of thermal equilibrium. In the hydrodynamic equations of motion it is implicitly assumed that local thermal equilibrium is “imposed” via an equation of state of the matter. This means that the relaxation timescale and the mean-free path should be much smaller than both the hydrodynamical timescale and the spatial size of the system. It seems reasonable to wonder if these conditions can be met for hadronic and nuclear collisions. On the other hand, from the kinematical point of view, apart from the use of the equation of state, the equations of hydrodynamics are nothing but conservation laws of energy and momentum, together with other conserved quantities such as charge. In this sense, for any process where the dynamics of flow is an important factor, a hydrodynamic framework is a natural first step. The effects of a finite relaxation time and mean-free path might be implemented at a later stage by using an effective equation of state, incorporating viscosity and heat conductivity, or some simplified transport equations. This does, of course, lead us back to the challenging problem of designing a causal relativistic theory for dissipation (see Section 14). In the context of heavy-ion collisions no calculations have yet been performed using a fully three-dimensional, relativistic theory which includes dissipation. In fact, considering the obvious importance of entropy, it is surprising that so few calculations have been reported for either relativistic or nonrelativistic hydrodynamics (although see [59]). An interesting comparison of different dissipative formulations is provided in [82, 83].

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