Freeze-Out Problem in Hydrokinetic Approach to $A+A$ Collisions

A new method for evaluating spectra and correlations in the hydrodynamic approach is proposed. It is based on an analysis of the Boltzmann equations (BE) in terms of probabilities for constituent particles to escape from the interacting system. The conditions of applicability of the Cooper-Frye freeze-out prescription are considered within the method. The results are illustrated with a nonrelativistic exact solution of BE for an expanding spherical fireball as well as with approximate solutions for ellipsoidally expanding ones.

Figure 1

The space-time $(t,r)$ dependence of the emission function averaged over momenta for an expanding spherically symmetric fireball containing 400 particles with mass 1 GeV and with cross section $\sigma =40\mathrm{m}\mathrm{b}$, initially at rest and localized with Gaussian radius parameter $R=7\mathrm{f}\mathrm{m}$ and temperature ${\mathrm{T}}_0=0.130\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 2

The transverse (bottom, at $p_L=0$) and longitudinal (top, at $p_T=0$) spectra of particles, escaped until $t=8\mathrm{f}\mathrm{m}/c$ from an expanding ellipsoidally symmetric fireball of the same particles as in Fig. 1, initially at rest and localized with Gaussian radius parameters $X_1=X_2=7\mathrm{f}\mathrm{m}$, $X_3=0.7\mathrm{f}\mathrm{m}$, and temperature ${\mathrm{T}}_0=0.300\mathrm{G}\mathrm{e}\mathrm{V}$. Dashed lines correspond to spectra calculated according to CFp, applied to the l.eq. distribution function at $T(t)=0.063\mathrm{G}\mathrm{e}\mathrm{V}$.