Multicomponent Van der Waals model of a nuclear fireball in the freeze-out stage

A two-component van der Waals gas model is proposed to describe the hadronic stages of the evolution of a nuclear fireball in the cooling stage. At the first stage of hadronization, when mesons dominate, a two-component meson model ($\pi^0$ and $\pi^+$ -mesons) with an effective two-particle interaction potential of a rectangular well is proposed. At the late-stage hadronization, when almost all mesons have decayed, a two-component nucleon model of protons and neutrons is proposed with the corresponding effective rectangular nucleon potential. The saddle point method has been applied for analytical calculations of the partition function. This made it possible to uniformly obtain analytical expressions for both the pressure and density, taking into account the finite dimensions of the system, and the analytical expressions for chemical potentials. It is assumed that the proposed models and derived formulas can be used to analyze experimental data connected to the quantitative characteristics of the particle yields of different types in the final state from the hadronic stages of the evolution of a nuclear fireball, as well as to determine the critical parameters of the system in high-energy nucleus-nucleus collisions.


Introduction
Experimental observations of elliptical flow in non-central collisions of heavy nuclei at high energies provide substantial evidence that a state of quark-gluon plasma appears and thermalization occurs.This phenomenon is associated with the fact that particles collide with each other multiple times.For this state, one can introduce the concept of temperature, viscosity, density, and other thermodynamic quantities that characterize the substance.In these terms, one can describe and study the phenomena that occur during the cooling of a hadron gas formed after a phase transition from the state of a quark-gluon plasma.It is believed that at a critical temperature (T > 150 MeV, the so-called Hagedorn temperature), hadrons "melt," and a phase transition of the hadron gas (hadron matter) into the quark-gluon phase occurs.Therefore, a reverse transition from the quark-gluon phase to the hadron phase is also possible.Therefore, in recent decades, statistical models of hadron gas have been actively used to describe the data of the Large Hadron Collider (LHC), the Relativistic Heavy Ion Collider (RHIC), and even earlier to describe the data of the Alternating Gradient Synchrotron (AGS) and the Super Proton Synchrotron (SPS) on the particle yields in relativistic nuclear-nuclear (A + A) collisions at high energies [1], [2].
The van der Waals (vdW) model, which takes into account hadron-hadron interactions at short distances, is especially useful in this description [3] - [10].This is due to the fact that considering the effect of repulsion (excluded volumes) prevents an undesirably high density at high temperatures, a problem that arises in ideal gas models [11].Additionally, collisions of heavy high-energy ions in the LHC produce a large number of different particle types.The count of these particles is not fixed.
Therefore, the formalism of the Grand Canonical Ensemble (GCE) is an adequate mathematical framework for these phenomena.In this case, thermodynamic quantities do not depend on the number of particles but on the chemical potentials.For many years, researchers have proposed and applied different versions of the vdW models.These models have been primarily used to describe experimental data on the number of particles at high energies, where tens or even hundreds of hadrons of different types can be generated.Naturally, this generation process is limited only by the energy of collisions.Among these models, the model proposed in [11] should be noted.In this model, phenomenological parameters of the radii of the hard core, R ii and R ij , are introduced.These parameters significantly change the number of yielded particles with different types N i (where i is the particle type) and are mainly confirmed by experimental data.To describe more subtle effects in the dependence of the hadronic gas pressure on density, various authors (e.g., [12], [13]) proposed the development of this model [11].Here, the effects of attraction between hadrons at a large distance have been taken into account, leading to the appearance of a corresponding contribution to the pressure as P attr ∼ −an 2 (n is the density).For a multicomponent gas, the parameter a, corresponding to attraction, transforms into parameters a ij , and the repulsive parameter b transforms into parameters b ij .At the same time, the parameters of the effective potential corresponding to attraction and repulsion depend on the effective radii of repulsion R 0 i and attraction R i , as follows: is the depth of the effective potential well [12].
However, even this vdW model cannot be properly developed when considering a finite nuclear system.So, in the case of nuclear collisions, a nuclear fireball with dimensions < r >∼ 7 − 10 fm is observed.In a fairly general case, this problem (without considering the effects of reflection from the system wall) has been solved for a two-component system.In this case, the GCE formalism leads to the use of a double sum, which, in turn, can be transformed into a multidimensional integral.
This integral can be integrated using the saddle point method in the vicinity of a saddle point with coordinates N * 1 , N * 2 [12].Undoubtedly, it would be beneficial to apply this model to the analysis of experimental data obtained from collisions of heavy nuclei at CERN.One version of such a model was concisely presented in [14].It was believed that the collision energies were not high enough, and one could limit oneself to only two particle types: protons and neutrons.It was assumed that characteristic temperatures did not exceed the thresholds at which new particles could form.Therefore, the model itself should transparently possess a non-relativistic limit, while adhering to the conservation law of the total number of nucleons without generating new particles (kinetic freeze-out, cessation of elastic collisions).
The successive stages of the evolution of a nuclear fireball are schematically shown in Figure 1.
Moving from left to right: the initial stage with two touching ultrarelativistic nuclei; the state of a hot and superdense nuclear system; gluon and quark-pair creation; quark-gluon phase, representing deconfined nuclear matter expanding hydrodynamically; hadronization and chemical freeze-out (inelastic collisions cease); kinetic freeze-out (elastic collisions cease).However, at the penultimate stage, after transitioning to the hadronic gas phase, the temperature of the nuclear fireball is approximately T > 135 MeV in units where k B = 1 (this corresponds to the stage of hadronization and chemical freeze-out, characterized by inelastic collisions, as depicted in Figure 1).
A more detailed and comprehensive description of the mathematical framework of the model [11] - [14] is presented in this article.Some finer effects are estimated, including additional corrections for pressure, density, and root-mean-square (RMS) fluctuations.For situations where temperatures As the investigation of the hadron fireball and, consequently, the quark-gluon plasma, is expected to be connected with a deeper comprehension of the early universe's evolution, this also underscores the significance of the presented study.

I. ONE-COMPONENT VDW GAS
According to various estimates, the duration of the nuclear fireball's existence (t > 10 −22 s, see Figure 1) exceeds the characteristic time of nuclear interaction t ′ ∼ 10 −23 − 10 −24 s.This duration of the fireball's existence is compared with the relaxation time τ ∼ 10 −21 − 10 −22 s for sufficiently small local volumes (subsystems) into which the fireball can be divided.
Therefore, it can be assumed that at each moment in time exceeding the relaxation time, a local statistical equilibrium has had time to establish in the subsystem.In other words, such a local focal area is quasi-stationary, allowing the application of methods from statistical physics.Since all thermodynamic potentials, along with entropy and volume, are positive (extensive) quantities, the corresponding potentials (values) of the entire system (fireball) can be determined as the sum of the corresponding thermodynamic potentials of quasi-closed subsystems [17].Accordingly, at each moment in time, a standard representation of the distribution function of a diluted quasi-ideal van der Waals gas in the canonical ensemble (CE) for such quasi-closed subsystems can be provided.In the approximation of pair interactions and under the condition B(T )N/V ≪ 1, this quantity takes the form [17]: where, respectively, N and m are the number and mass of particles, V and T are the volume and temperature of the gas.Formula (1) uses the notation [11]: where K 2 (z) is the modified Bessel function, and the second virial coefficient in (1) has the form: and includes pairwise interaction of particles, U = U ij , (i ̸ = j).
In relativistic limit m ≫ T one can easy obtain, given the asymptotes of the Bessel function: The pressure in the system is easy to find from the partition function: Note that if the Stirling formula is used in the partition function for the factorial: then the final pressure formula (4) will not change.
*MODEL.In accordance with the above, all computations for subsystems will be conducted utilizing methods from statistical physics.This encompasses not only local statistical equilibrium but also the fulfillment of a condition of statistical (thermodynamic) constraint: (N → N A ), where In such a scenario, the final formulas can be applied to the nuclear fireball due to the mentioned additivity of thermodynamic potentials and volume.Given that the number of particles generated within the fireball reaches around 3-5 thousand during high-energy nuclear interactions, this assumption is reasonably grounded during the initial stages of its evolution.
Of course, at later stages of evolution, this assumption becomes somewhat dubious, as the number of nucleons N within the non-relativistic threshold is constrained by the law of baryon number conservation and equals N ∼ 200 (heavy element nuclei collide with mass number A ∼ 200).
However, on later stages, the duration of the fireball's existence increases, resulting in an extended relaxation time.Considering these factors, as well as the fact that we can always confine ourselves to the initial stage (see Section 3), it can be considered that the approximation of this model is reasonably justified.It is well-known that practical applications of the van der Waals equation often go beyond the conditions under which the virial approximation was derived, as supported by experience.Therefore, despite the fact that computations in the model are performed using the saddle point method under the condition B(T ) < 0, the final formulas extend to a region where the second virial coefficient B(T ) is not necessarily negative.
From the partition function Z(V, T, N ) one can also get: free energy and the derivative of the chemical potential which in the statistical limit has the form: Then, we obtain the Grand partition function (GPF) Z(V, T, µ) from the partition function Z(V, T, N ) taking into account the above physical considerations (see, e.g.[18], [19]): At high temperatures (which, for example, are realized during collisions of heavy ions in the GCE, and △N/T → dN ′ ) one can turn from the sum to the integral using the Euler-Maclaurin formula.
In this case, the first integral term remains and the logarithm of the statistical sum is introduced into the exponent.Let's denote this indicator by Φ(N ′ ) : Further integration is performed by the saddle point method [20], since at high temperatures the integrand has a strongly pronounced maximum.We obtain the following expression for finding the maximum point ( N * ) for the integrand from the extremum condition imposed on the saddle point: where µ * is the chemical potential at the saddle point.
As a result, we obtain: where the second derivative of the exponent Φ at the saddle point is defined as follows: The pressure in the GCE is defined as follows in terms of the temperature and the logarithm of the GPF (see, e.g., [18]): It's easy to show that pressure (12), taking into account (11) and ( 5), can be rewritten as follows: where the saddle point, ξ = N * (V, T, µ * )/V , is defined according to (10) and ( 5) as ξ = ϕ(m, T ) exp (µ * (ξ)/T ).The parameter ξ can be eliminated from Eq. ( 13) using the definition of density, which in the thermodynamic limit turns into the well-known formula [17]: In the thermodynamic limit (N → N A , V → ∞ ) the chemical potential of the saddle point µ * from Eq. (10) when is determined by the well-known thermodynamic equation Eq. ( 5).
Both equations (Eq.( 13) and Eq. ( 14)) in parametric form (the saddle point ξ acts as a parameter) determine the relationship between pressure P , temperature T , and density n .We obtain the state equation in GCE by excluding explicitly this parameter from the system of Eq. ( 13) and Eq. ( 14): Of course, the resulting state equation is implicitly a parametric equation, since the saddle point ξ (and, hence, n ) determines the chemical potential µ according to Eq. ( 5) and Eq. ( 10), as: It's crucial that the resulting formula considers the impact of the finite volume of the system, denoted as V s , on pressure.However, the exact nature of this contribution remains unclear to the author.There's a possibility that this might be an unphysical outcome, which could potentially be mitigated by accounting for subsequent terms in the expansion through the saddle-point method.Nevertheless, until a comprehensive analysis is conducted and a quantitative assessment is performed, we will treat this contribution as genuine.It's important to note that this contribution becomes negligible in the thermodynamic limit, where the distinction between CE and GCE disappears.
If we disregard the correction obtained from the volume of dP and assume that B(T )n << 1, then by making the following substitution in the right-hand side of Eq. ( 15): taking into account Eq. ( 16), it will become the following: Thus, the equation of state with interaction can be obtained by making the substitution µ− > µ int = µ − B(T )n in the equation of state of the ideal gas [11], [23].These equations are density functionals, which, according to (5), at a fixed chemical potential, are found from the solution of a transcendental equation n = ϕ(T, m) exp(µ/T − 2B(T )n).Assuming B(T )n << 1, this formula can be replaced with (14) where, according to (10), ξ is expressed in terms of µ: The RMS fluctuations of pressure and density calculated by known formulas (see, e.g., [17] [22]) give estimates of the found corrections to the corresponding quantities:

II. TWO-COMPONENT VDW GAS
Let's examine the procedure for incorporating excluded volume and attraction in the van der Waals model for a two-component hadron gas consisting of two types of particles labeled as "i" and "j", with N 1 and N 2 being the quantities of particles of the first and second types.In this scenario, the partition function takes the following form [11]: where m (1) , N 1 (m (2) , N 2 ) are, respectively, the masses and number of particles of the 1st (2nd) sorts, and the two-particle potential has the following form: After a trivial integration over momenta, this expression takes the following form: where the notation introduced is the same as in the first section.
This expression for the pair-interactions approximation (U (123) ≪ U (12) ) and a weakly ideal gas (2N B/V ≪ 1 ) can be rewritten as follows (see [11], [12] ): Here the notation Bij = 2 B ii +B jj has been introduced.The two-particle partition function Z(V, T, µ 1 , µ 2 ) in GCE is expressed in terms of the two-particle partition function Z(V, T, N 1 , N 2 ) in CE [17], [12], as Here, as in the one-dimensional case, when T ≫ U and N i → N A , the sum over the number of particles approximately becomes an integral, since △N/T → dN ′ : where the coordinates of the saddle point N * i (i = 1, 2) are found from the extremum conditions: Substituting the value of the partition function into the definition of pressure in the GCE [18], we obtain the following expression [12]: where Using such a mathematical apparatus, one can organically introduce the law of conservation of chemical potentials.The latter are related to the condition imposed on the integrand when finding the saddle point.In the thermodynamic limit the chemical potential determined by the extremum condition coincides with the definition of the chemical potential itself: ] is the definition of free energy (10).
We get from the definition of density The virial expansion ( 26) can be rewritten, taking into account Eq. ( 28), as a two-component vdW equation in the approximation b ij N i /V ≪ 1 and a ij /T b ij ≪ 1) : where dP , according to Eq. ( 26), takes into account the finite size of the fireball.
When formula (29) was derived, the expression Bij ≈ bij − ãij /T was used (see, e.g., [12]), and for each type of particles the corresponding parameters of attraction and repulsion were introduced: γ is a phenomenological parameter reflecting the complexity of the problem.Introducing quantities ãij constrained by the condition ãij + ãji = 2γa ij .

III. TWO-COMPONENT ASYMMETRIC VDW MODEL WITH NON-CONSERVATION OF PARTICLE NUMBER
As experiments related to the formation of quark-gluon plasma focus on heavy nucleus collisions (A+A) with very high energies exceeding 1 GeV per nucleon, it is assumed that at the initial stages of freeze-out, mesons of different types dominate (chemical freeze-out).Therefore, to describe nuclear interactions during this freeze-out stage beyond the threshold for producing new particles (T > 135 MeV), a generalization of the van der Waals model is proposed for a medium-sized meson fireball [16]: Here, r 0 = 1.1 − 1.2 f m, < a >, < b > represent the mean semiaxes of the ellipsoid, and < A > denotes the mass number of nuclei remaining in the fireball after the collision.The model assumes that the fireball is primarily composed of mesons, considering that the number of nucleons is much smaller than the number of mesons (N pn ∼ 200 − 300 << N π,ρ,ω ∼ 3000 − 5000).Contributions from other particles are neglected in the model.Thus, the following natural assumptions are summarized in the model: 1.) The average energies of nucleon-nucleon interactions do not exceed the threshold for producing heavy mesons.Therefore, the model is limited to two types of particles ("0" corresponds to π 0 -mesons, "+" corresponds to π + -mesons).
2.) Since reactions producing π + -mesons are more likely than reactions producing π 0 -mesons, it is assumed that n 0 = kn + = n, where k < 1, n 0 represents the density of π 0 -mesons, and n + represents the density of π + -mesons.For instance, this corresponds to a higher probability of π +meson production in reactions like p+d → d+n+π + compared to π 0 -meson production in reactions like p + d → d + p + π 0 (also, the lifetime of π + -mesons is longer than that of π 0 -mesons).

3.) An effective phenomenological potential of meson interaction
where (i, j) = +, 0. "(0+)" denotes the interaction of π 0 -mesons with π + -mesons, "(++)" denotes the interaction of π + -mesons, and "(00)" denotes the interaction of π 0 -mesons.For a gas composed b ij .This transformation occurs concurrently with the dependence of the effective potential's attraction and repulsion parameters on the effective radii associated with repulsion (R 0 i ) and attraction (R i ).Specifically, the relationship can be expressed as follows: represents the depth of the effective potential well.
As the parameters of the scalar component of the effective phenomenological rectangular well potential are chosen in such a way as to approximately yield the same pressure and density values as the effective mesonic potential (see Figure 2, where, for instance, the interaction of π 0 -mesons with π + -meson corresponds to U (+,0) ), the effective mesonic potential (a) can be substituted with a similar effective phenomenological rectangular well potential (b).
4.) It is assumed that the hard-core radius of the π 0 -meson is much smaller than the hard-core radius of the π + -meson: R 0 0 ≪ R 0 + .The hard-core radius of the π + -meson is considered to be known.
Average pressure and density fluctuations are easily found within the framework of the proposed model, similarly to formulas Eq. ( 20) and Eq. ( 21): The following results are obtained (Fig. 3, Fig. 4).Such data have been used (Fig. 3): T = 147 MeV, the effective radius of the π + -meson, R 0 + = 0.46 f m, and π 0 -meson, R 0 0 = 0.01 fm, the average value of the volume of the meson fireball is taken as the value < V f >∼ 600 f m 3 , k = 0.5, the parameter of the potential depth, u (+,0) 0 ∼ 80 − 100 MeV.One can clearly see (Fig. 4) an increase in the correction dP/ < P > at low densities, which is typical in the final stages of the freeze-out.Functional dependences for pressure, obtained by Eq. 29, and the ratio of dP to RMS pressure fluctuations are shown in Figs. 5 and 6.
It can be seen that the correction dP makes a nonzero contribution to the total pressure also in this case.On the other hand, it is negligibly small almost everywhere in comparison with the contribution from fluctuations.The correction makes a contribution comparable to fluctuations only in the region near zero density that is nonphysical for a nuclear fireball.But it can be neglected in this region, as can be seen from Fig. 5.

V. MULTICOMPONENT VDW GAS
It is possible to extend the above analysis to the vdW gas with multiple components, consisting of any number of different particles.By integrating over the momentum of the particles and making some modifications similar to those done in the first example, one can obtain an expression for the It is possible to calculate the pressure in the Grand Canonical Ensemble for the vdW gas with multiple components by integrating over the namber of the particles and making appropriate modifications to the formula derived for a single-component gas.
The particle densities n p = ∂P (T, µ 1 , ..., µ K )/∂µ p along with the pressure are obtained as the solutions of the system of related equations depending on the parameter of the saddle points ξ p (p = 1, ..., K).
The HG model in the grand canonical ensemble formulation does not have fixed numbers for In this approach, the system properties are determined by the pressure function (35).The chemical potentials µ i (where i = 1, ..., K) are defined as a combination of the baryonic µ B , strange µ S , and electric µ Q chemical potentials, with coefficients of expansion (γ B ) i , (γ S ) i , and (γ Q ) i respectively.
Interestingly, despite the crudeness of such a one-component approximation for the real multicomponent vdW gas of the hadron fireball, as shown in Fig. 7, a good qualitative and quantitative agreement with the results of calculations by other authors is obtained for the chemical potential (see, for example, [25], [30] and [26]- [29]).

VI. SUMMARY
The impact of considering the excluded volume and attraction is analyzed in the case of a twocomponent gas: (i) π 0 -and π + -mesons (model from Section 3); (ii) protons and neutrons (model from Section 4).The calculations were performed in the Canonical and Grand Canonical ensembles using the saddle point method for the two-component system.Particles interact with hard-core potentials at short distances and relatively high potentials at long distances (effective attraction radii).For such effective interparticle interactions, an equation of state with corrections that account for the finite dimensions of the nuclear fireball, as well as the root-mean-square fluctuations of pressure and density, has been derived.The pressure correction vanishes in the thermodynamic limit, in accordance with statistical physics, where there is no distinction between different statistical ensembles.The developed approach of integrating a large statistical sum using the saddle point method allows for obtaining both the equation of state and expressions for chemical potentials uniquely, and it can be easily extended to the case of a multi-component system (Section 5).
Interestingly, despite the simplicity of the single-component approximation, the obtained behavior of the baryon chemical potential qualitatively, and sometimes quantitatively, reproduces the corresponding calculations of other authors made under different QCD approximations.
Thus, it is anticipated that the developed model can be useful in analyzing experimental data related to the study of various stages of nuclear fireball evolution, which occurs, in particular, in experiments investigating the quark-gluon plasma.
The research was conducted within the framework of the initiative scientific topic 0122U200549

Figure 1 :
Figure 1: Successive stages of nuclear fireball evolution (The figure is taken from [15])

Figure 2 :
Figure 2: Scalar part of the effective phenomenological meson-meson potential

Figure 5 :
Figure 5: Dependence of nucleon pressure P 33 on nucleon density, n p = kn n = n , in the two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm)

Figure 6 :
Figure 6: The ratio of correction from the size of the nucleon fireball to pressure dP to the value of the RMS pressure fluctuation < P > depending on the density of nucleons, n p = kn n = n

Figure 7 :
Figure 7: The result of our calculations [31] using formula (5) for the meson and nucleon stages of the evolution of the hadron fireball

Formulas for pressure and
density obtained through the saddle point method can be employed to analyze experimental data regarding the relative abundance of particles of different types and critical parameters in high-energy nuclear-nuclear collisions.As an example of such application for the chemical freeze-out stage (model from Section 3), a generalization of the presented van der Waals model to the case of an asymmetric two-component model (π 0 -and π + -mesons) with effective phenomenological hard-core and attractive parameters has been proposed.The ratio of the pressure correction to the root-mean-square value of pressure fluctuation is assessed for the case of an asymmetric two-component meson fireball model.An increase in the correction at low density values corresponding to the final freezing stages has been identified.It has been found that the contribution to pressure, considering different radii and the finiteness of the nuclear fireball, in comparison to root-mean-square fluctuations, becomes noticeable in the case of the meson model with particle non-conservation (model from Section 3, corresponding to the chemical freeze-out stage).However, this correction can be neglected in the final stages of freeze-out when nucleons begin to dominate (model from Section 4, corresponding to the kinetic freeze-out stage).