## Abstract

The non-equilibrium equation of state is found in the approximation of the functional on the local density, and its application to the description of the emission of protons and pions in heavy ion collisions is considered. The non-equilibrium equation of state is studied in the context of the hydrodynamic approach. The compression stage, the expansion stage, and the freeze-out stage of the hot spot formed during the collisions of heavy ions are considered. The energy spectra of protons and subthreshold pions produced in collisions of heavy ions are calculated with inclusion of the nuclear viscosity effects and compared with experimental data for various combinations of colliding nuclei with energies of several tens of MeV per nucleon.

### Keywords

- local density functional
- hydrodynamics
- non-equilibrium equation of state
- heavy ions
- hot spot
- nuclear viscosity
- protons
- subthreshold pions

## 1. Introduction

The main object of studying heavy ion collisions is to study the equation of state (EOS) of nuclear matter. Along with molecular dynamics and the Vlasov dynamic equation, nuclear hydrodynamics is an effective method for describing the interaction of heavy ions with medium and intermediate energies (see, e.g., [1]). Typically, the equilibrium EOS is used [1]; it involves the local thermodynamic equilibrium in the system. A hybrid model was proposed for use at high energies in [2, 3]. It includes a fast non-equilibrium stage and the subsequent description of the dynamics of a nucleus-nucleus collision based on equilibrium relativistic hydrodynamics of an ideal fluid. We showed in our works [4, 5, 6, 7, 8, 9, 10, 11] that local thermodynamic equilibrium is not immediately established in the process of collisions of heavy ions, since the non-equilibrium component of the distribution function, which leads to the formation of a collisionless shock wave, is important at the compression stage.

The kinetic equation for finding the distribution function of nucleons is used in this paper. It is solved in conjunction with the equations of hydrodynamics, which are essentially local conservation laws of mass, momentum, and energy. As a result, the non-equilibrium equation of state is found in the approximation of the functional on the local density. Since the emitted secondary particles (nucleons, fragments, and pions) contain the basic information about the EOS, it is necessary to know the differential cross sections for the emission of these particles. The energy spectra of protons and subthreshold pions with allowance for nuclear viscosity are analyzed in this paper as a follow-up to our works [11, 12, 13] devoted to the energy spectra of protons and fragments in which viscosity was neglected.

By subthreshold production, we mean the generation of

The pion production threshold during the collision of heavy ions decreases owing to collective effects and the internal motion of nucleons. These effects are naturally taken into account using the hydrodynamic approach, which explicitly includes the many-particle nature of colliding heavy ions. In the case of low energies, the hydrodynamics should be modified to take into account the non-equilibrium EOS, which describes the transition from the initial non-equilibrium state to the state of local thermodynamic equilibrium.

Such an approach to describing the temporal evolution of the resulting hot spot includes a compression stage and an expansion stage taking into account the nuclear viscosity that we found. The calculated energy spectra of protons and pions produced in nuclear collisions (both identical and different in mass) at an energy of 92 MeV per nucleon in the case of protons and 94 MeV per nucleon in the case of subthreshold pions are in agreement with the available experimental data [1, 14], respectively.

## 2. Non-equilibrium equation of state in a local density approximation

If the energies of colliding heavy ions are less than 300 MeV per nucleon (pion production threshold in free nucleon-nucleon collisions), we use the kinetic equation to find the nucleon distribution function

where ^{−3,} binding energy *K* = 210 MeV; and

Equation (1) with allowance for the hydrodynamic equations obtained from (1) by taking the corresponding moments with a weight of 1,

where the distribution function

So, the initial moments

The kinetic terms are.

which corresponds to diagonal tensor of pressure *I* and *T* and

To find density ρ, velocity field *q*, and temperatures *T* and

## 3. Hydrodynamic stage

We simplify the description of the time evolution of colliding nuclei distinguishing the compression stage, the expansion stage, and the freeze-out stage of the resulting hot spot. We reduce the interaction between two nuclei to the interaction between their overlapping regions. This can be interpreted as a hot spot formation process. In this case, we take into account the conservation laws. Shock waves with changing front diverging in opposite directions are formed at the stage of compression during the interaction between overlapping regions of colliding nuclei [5, 6, 7, 8, 9].

In the process of compression, when the shock wave reaches the boundaries of the hot spot, the density reaches its maximum value. The dependence of the maximum compression ratio

The relaxation factor at the energy region of

A compressed and heated hot spot (a ball with radius

The heat flux for a local equilibrium distribution function is

The velocity field is found from Eq. (3) in the approximation of a homogeneous but time-dependent density of hot spot

where

However, the deviation of the distribution function

where

where *l* ≈ 3 fm. In our case, the temperature is ^{−1} s^{−1}. It coincides in the order of magnitude with the gas estimate

## 4. Double differential cross sections of the emission of protons and pions: comparison with the experimental data

Protons and pions are emitted when the nuclear system reaches a critical density. The cross section of the emission of protons (pions) is found from the condition that the number of particles

where

Here *K* = 210 MeV, i.e., with the same which was taken for the best description of the experiment in the calculations that we performed in [8, 9] at energies of 250 and 400 MeV per nucleon for colliding Ne and U nuclei.

We present the proton spectra in the ^{40}Ar + ^{40}Ca *1*), 50° (*2*), 70° (*3*), and 90° (*4*) for the energy of projectile nuclei of ^{40}Ar of 92 MeV per nucleon (Figure 2). In Figure 2, the solid curves correspond to our calculation, the histograms correspond to the calculations performed by the method of solving the VUU equation [1], and the dots are the experimental data from [1].

As can be seen, our calculation (this is not the Monte Carlo method and not histograms) is in good agreement with the experimental data. This is especially true for small angles of emission of protons (30°, 50°, and 70°). Our approach has an advantage over the more detailed method of solving the VUU equation [1], since the solid curves (but not histograms) are the result of the calculation. Note here that simple cascade models, as mentioned in [1], cannot describe these experimental data at all.

We compared our data with the available experimental data on the emission of pions. Figure 3 illustrates the comparison of our calculated (solid lines) and experimental [14] (dots) double differential cross sections for the reactions of ^{16}O ions collide with ^{27}Al nuclei (curve *1*), ^{58}Ni nuclei (curve *2*), and ^{197}Au nuclei (curve *3*) at energies of ^{16}O ions of

Figure 4 illustrates the comparison of the calculations (solid lines) with the experimental data [14] (dots) for the reaction ^{16}O + ^{27}Al → ^{16}O ions of 94 MeV per nucleon at pion emission angles of 70° (curve *1*), 90° (curve *2*), and 120° (curve *3*). The calculation is in agreement with the experimental data if its parameters are constant. In all the illustrations under consideration, the agreement of calculation with the experiment was achieved without introducing fitting parameters and is more successful than our previous works [11, 19, 20].

## 5. Conclusions

Thus, the idea of using the hydrodynamic approach with a non-equilibrium equation of state in describing collisions of heavy ions is further developed in this work. The non-equilibrium equation of state is found in the approximation of the 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, 12, 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 heavy ions with intermediate energies better than the equation of state corresponding to traditional hydrodynamics, which initially implies the local thermodynamic equilibrium.

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 solution of the Vlasov-Uling-Uhlenbeck kinetic equation.

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

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

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.