Entropy Production and Effective Viscosity in Heavy-Ion Collisions

Entropy production and an effective viscosity in central Au+Au collisions are estimated in a wide range of incident energies 3.3 GeV $\le \sqrt{s_{NN}}\le$ 39 GeV. The simulations are performed within a three-fluid model employing three different equations of state with and without deconfinement transition, which are equally good in reproduction of the momentum-integrated elliptic flow of charged particles in the considered energy range. It is found that more that 80\% entropy is prodused during a short early collision stage which lasts $\sim$1 fm/c at highest considered energies $\sqrt{s_{NN}}\ge$ 20 GeV. The estimated values of the viscosity-to-entropy ratio ($\eta/s$) are approximately the same in all considered scenarios. At final stages of the system expansion they range from $\sim$0.05 at highest considered energies to $\sim$0.5 lowest ones. It is found that the $\eta/s$ ratio decreases with the temperature ($T$) rise approximately as $\sim 1/T^4$ and exhibits a rather weak dependence on the net-baryon density.Comment: 10 pages, 9 figures. Version accepted for publication in the European Physical Journal

Light fragment production at CERN Super Proton Synchrotron

Recent data on the deutron and $^3$He production in central Pb+Pb collisions at the CERN Super Proton Synchrotron (SPS) energies measured by the NA49 collaboration are analyzed within the model of the three-fluid dynamics (3FD) complemented by the coalescence model for the light-fragment production. The simulations are performed with different equations of state---with and without deconfinement transition. It is found that scenarios with the deconfinement transition are preferable for reproduction rapidity distributions of deuterons and $^3$He, the corresponding results well agree with the experimental data. At the same time the calculated transverse-mass spectra of $^3$He at midrapidity do not that nice agree with the experimental data. The latter apparently indicates that coalescence coefficients should be temperature and/or momentum dependent.Comment: 7 pages, 7 figures, 1 table, version accepted for publication in Eur. Phys. J.

Estimation of the Shear Viscosity from 3FD Simulations of Au+Au Collisions at sNN=\sqrt{s_{NN}}= 3.3--39 GeV

An effective shear viscosity in central Au+Au collisions is estimated in the range of incident energies 3.3 GeV $\le \sqrt{s_{NN}}\le$ 39 GeV. The simulations are performed within a three-fluid model employing three different equations of state with and without the deconfinement transition. In order to estimate this effective viscosity, we consider the entropy produced in the 3FD simulations as if it is generated within the conventional one-fluid viscous hydrodynamics. It is found that the effective viscosity within different considered scenarios is very similar at the expansion stage of the collision: as a function of temperature ($T$) the viscosity-to-entropy ratio behaves as $\eta/s \sim 1/T^4$; as a function of net-baryon density ($n_B$), $\eta/s \sim 1/s$, i.e. it is mainly determined by the density dependence of the entropy density. The above dependencies take place along the dynamical trajectories of Au+Au collisions. At the final stages of the expansion the $\eta/s$ values are ranged from $\sim$0.05 at highest considered energies to $\sim$0.5 at the lowest ones.Comment: 4 pages, 3 figures, to be published in Eur. Phys. Journ.

Continuity in a parameter of solutions to generic boundary-value problems

We introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex H\"older space $C^{n+1,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0\leq\alpha\leq1$. The boundary conditions can contain derivatives $y^{(r)}$, with $1\leq r\leq n+1$, of the solution $y$ to the system. For parameter-dependent problems from this class, we obtain constructive criterion under which their solutions are continuous in the normed space $C^{n+1,\alpha}$ with respect to the parameter.Comment: 15 page