**Author details**

functional on the local density. The differential cross sections of the emission of protons and the production of subthreshold pions in heavy ion collisions are uniformly described with the same fixed parameters of the equation of state and in the same approach as in the previous papers [11–13], which describe the differential cross sections for the formation of protons and light fragments. It is shown that the non-equilibrium equation of state included in the hydrodynamic equations allows us to describe the experimental energy spectra of protons produced in collisions of

corresponding to traditional hydrodynamics, which initially implies the local ther-

In comparison with previous works, the inclusion of the effects of nuclear viscosity, which we found in the relaxation approximation for the kinetic equation, is new. This did not add new parameters in describing the temporal evolution of nuclear collisions. The relaxation time τ, which determines the nuclear viscosity coefficient η, turned out to be close to the value found on the basis of the behavior of nuclear Fermi liquid [15] and is not a fitting parameter. When describing the emission of protons and fragments, the inclusion of the viscosity of the medium is

The highlighting of proton (pion) emission after the temporal evolution of the resulting hot spot and the contribution to the particle emission cross sections during the fusion of "spectators" (non-overlapping regions of colliding nuclei) were significant in calculating the cross sections. This made it possible to describe the differential cross sections of the emission of protons (pions) for collisions of nuclei in various combinations. Highlighting this feature of our approach can be useful in comparison with other ways of pion production in heavy ion collisions, for example [21, 22], based on the solution of the Vlasov-Uling-Uhlenbek equation. These works include a range of higher energies of colliding heavy ions (more than 300 MeV per nucleon) and the production of pions by means of Δ-isobar production. We

included this channel at low subthreshold energies, not limited to the production of thermal pions. However, this channel appears on the higher energy tails of the

Studies of the formation of protons, fragments, and subthreshold production of pions may be of interest for the development of a scientific program planned with radioactive beams in Dubna using the COMBAS facility [24], which is designed to

study nuclear collisions in the energy range of 20–100 MeV per nucleon.

This simplified hydrodynamic approach including a description of the stages of compression, expansion, and freeze-out of a substance during heavy ion collisions turned out to be no worse than a more detailed approach based on the Monte Carlo

heavy ions with intermediate energies better than the equation of state

solution of the Vlasov-Uling-Uhlenbeck kinetic equation.

not so significant, and the pions are very sensitive to the viscosity.

modynamic equilibrium.

*Density Functional Theory Calculations*

energy spectra of pions [23].

**102**

A.T. D'yachenko<sup>1</sup> \* and I.A. Mitropolsky<sup>2</sup>

1 Emperor Alexander I St. Petersburg State Transport University, St. Petersburg, Russia

2 Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, Gatchina, Russia

\*Address all correspondence to: dyachenko\_a@mail.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
