The compressibility and other basic properties of
the nuclear equation of state, phase transitions in nuclear matter,
and nuclear interactions can be studied in relativistic heavy ion
reactions at beam energies in the range of to
. In order to search for the
existence of the quark-gluon plasma, ultra-relativistic heavy ion
collision experiments with beam energies exceeding
must be performed [56
]. Up to low
ultra-relativistic energies baryons stemming from the projectile
and the target are fully or partly stopped by each other forming a
baryon rich matter in the center of the reaction zone. This regime
is called the stopping energy region. At even larger energies the
theorectical expectation is that the (initial) baryon charge of the
target and projectile is so far apart in phase space that it cannot
be slowed down completely during the heavy ion collision. In this
so-called transparent energy regime the quanta carrying the baryon
charge will essentially keep their initial velocities, i.e., the
center of the reaction zone will be almost baryon free. However,
much energy will be deposited in this baryon free region, and the
resulting large energy density matter may form a quark-gluon
plasma.
In order for a hydrodynamic description of heavy
ion collisions to be applicable, several criteria must be
fulfilled [56]:
The first condition is satisfied reasonably well
when there are many nucleons involved in the collision and when
pion production or resonance excitations become important, i.e.,
for almost central collisions of sufficiently heavy and energetic
ions. The mean free path of a nucleon in nuclear matter scales
inversely with the nucleon-nucleon cross section, and is about at a bombarding energy of
, which is
short compared to the radii of heavy nuclei. However, the mean free
path increases with energy. The average distance it takes for a
nucleon in nuclear matter to dissipate its kinetic energy is called
the mean stopping length. At
a nucleon will
penetrate about
into a nucleus. But at larger energies
the mean stopping length may exceed the nuclear radius (there exist
effects both increasing and decreasing the mean stopping
length [56
]), i.e., the
colliding nuclei will appear partially or nearly transparent to one
another. Modifications to the hydrodynamic equations are then
necessary. The establishment of local thermal equilibrium seems to
be reasonably well satisfied in heavy ion collisions. Finally, at
bombarding energies of interest the de Broglie wavelength is about
or smaller, which is small compared to the nuclear
radius.
Hydrodynamic simulations of heavy ion collisions
are complicated by additional physical and numerical
issues [56, 63]. We will mention only a
few of these issues here.
Since ideal hydrodynamics assumes that matter is
in local thermal equilibrium at every instant, colliding fluid
elements are forced by momentum conservation to instantaneously
stop and by energy conservation to convert all their kinetic energy
into thermal energy. Thus, when immediate complete stopping is not
achieved at large beam energies, non-ideal hydrodynamics must be
considered (see, e.g., Elze et al. [82]). However, the viability
of non-ideal hydrodynamics as a causal theory is still a matter of
debate, and there are still open questions concerning the proper
relativistic generalization [56, 125]. In the
ultra-relativistic regime, where the stopping power becomes very
low, matter in the high energy density, baryon-free central region
is supposed to establish local thermal equilibrium within a
(proper) time of order , i.e., the subsequent
evolution can be described by ideal hydrodynamics.
Numerical algorithms for RHIC must scope with the
presence of (almost) vacuum in the baryon-free central region. This
can cause problems due to erroneous (i.e., numerical) acausal
transport of matter [244]. Another challenge
is posed by the phase transition to the quark-gluon plasma, which
is usually assumed to be of first order. Matter undergoing a
first-order phase transition may exhibit thermodynamically
anomalous behaviour (changes in the convexity of isentropes) which
can cause important consequences for the wave structure of the
hydrodynamic equations leading to non-uniqueness of solutions of
Riemann problems (see Section 9.1).
The performance of numerical algorithms for RHIC
(RHLLE and FCT SHASTA) in the presence of vacuum and for
thermodynamically anomalous matter were systematically explored by
Rischke et al. [244, 246
].