Klein–Gordon Equations, Equations of Relativistic Quantum Hydrodynamics, and Quantum Shocks in Describing Collisions of Atomic Nuclei

Equations of quantum relativistic hydrodynamics are obtained from an effective Klein–Gordon equation with allowance for dissipation. The inclusion of dissipation in the Klein–Gordon equation entails the need for introducing an additional thermal term and an equation for it. This leads to a closed system of equations that takes into account nonequilibrium processes and which makes it possible to describe the dynamics of collisions between atomic nuclei and to calculate the yields of secondaries. Solving these equations permits identifying quantum shocks and tracing the time evolution of the emerging hot spot. The calculated spectra of protons emitted in heavy-ion collisions are contrasted against available experimental data.