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 100 A MeV to 10 A GeV. In order to search for the existence of the quark-gluon plasma, ultra-relativistic heavy ion collision experiments with beam energies exceeding 10 A GeV must be performed . 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 :
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 0.3 fm at a bombarding energy of 200 MeV, 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 200 MeV a nucleon will penetrate about 2 fm 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 ), 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 2 fm 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. ). 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 1 fm / c, 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 . 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. [245, 247].
© Max Planck Society and the author(s)