diff --git "a/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt" "b/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3dAyT4oBgHgl3EQf1vn1/content/tmp_files/load_file.txt" @@ -0,0 +1,525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf,len=524 +page_content='UDK 539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='17 Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Krivenko-Emetov* Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, Ukraine National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine Corresponding author: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='kryvenko-emetov@kpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='ua;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' krivemet@ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='net MULTICOMPONENT VAN DER WAALS MODEL OF A NUCLEAR FIREBALL IN THE FREEZE-OUT STAGE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the first stage of hadronization, when mesons dominate, a two-component meson model ( 0 \uf070 - and \uf02b \uf070 -mesons) with an effective two-particle interaction potential of a rectangular well is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The saddle point method has been applied for analytical calculations of the partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Keywords: fireball,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' freeze-out,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' van der Waals equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' effective nuclear capability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Grand Canonical Ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' pressure fluctuation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' quark-gluon plasma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' experimental data Introduction Experimental observations of an elliptical flow in non-central collisions of heavy nuclei at high energies provide much evidence that in these collisions of nuclei a state of quark-gluon plasma appears and thermalization occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' which is associated with the fact that particles collide with each other more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For this state, one can introduce the concept of temperature, viscosity, density, and other thermodynamic quantities that characterize the substance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is believed that at a critical temperature ( 150 \uf03e T MeV, the so-called Hagedorn temperature), hadrons "melt" and a phase transition of the hadron gas (hadron matter) into the quark-gluon phase occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 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 Alternating Gradient Synchrotron (AGS) and Super proton synchrotron (SPS), on the particle yields in a relativistic nuclear-nuclear ( A A \uf02b ) collision at high energies [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The van der Waals (vdW) model, taking into account hadron-hadron interactions at short distances, is especially useful in this description [3-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This is due to the fact that taking into account the effect of repulsion (off volumes) leads to the prevention of an undesirably high density at high temperatures, which appears in ideal gas models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In addition, collisions of heavy high-energy ions in the LHC produce a large number of different types of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The number of these particles is not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the formalism of the Grand Canonical Ensemble (GCE) is one of the adequate mathematical formalisms for these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the thermodynamic quantities do not depend on the number of particles, but on the chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For many years, researchers have proposed and applied different versions of vdW models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' These models have been mainly used to describe experimental data on the number of particles at high energies, when tens or even hundreds of hadrons of different types can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Naturally, this generation process is limited only by the energy of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Among these models, the model proposed in [11] should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this model, the phenomenological parameters of the radii of the hard-core ii R and ij R are introduced, which significantly changes the number of the yield particles with different types i N (i is the particle sort) and is mainly confirmed by experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In order to describe more subtle effects in the dependence of the hadronic gas pressure on density, various authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' [12, 13]) proposed the development of this model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Here, the effects of attraction between hadrons at a large distance have been taken into account, which leads to the appearance of a corresponding contribution to the pressure as 2 an Pattr \uf02d ~ ( n is the density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For a multicomponent gas, the parameter a , corresponding to attraction, transforms into parameters ij a , and the repulsive parameter b transforms into parameters ij b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the same time, the parameters of the effective potential corresponding to attraction and repulsion depend on the effective radii of repulsion 0 iR and attraction iR as follows: \uf028 \uf029 ij ij ij ij b c u a \uf02d 0 ~ , \uf028 \uf029 3 0 0 3 2 j i ij R R b \uf02b \uf070 \uf03d , \uf028 \uf0293 3 2 j i ij R R c \uf02b \uf070 \uf03d , ij u0 is the depth of the effective potential well [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, even this vdW model cannot be developed properly when considering a finite nuclear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' So, in the case of nuclear collisions, a nuclear fireball with dimensions 10 7 \uf0b8 \uf03e \uf03c ~ r Fm is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In a fairly general case, this problem (without taking into account the effects of reflection from the system wall) has been solved for a two-component system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the GCE formalism leads to the use of a double sum, which, in turn, can be transformed into a multidimensional integral, which can be integrated by the saddle point method in the vicinity of a saddle point with coordinates \uf02a \uf02a 2 1 N N , [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, it would be good to apply this model to the analysis of experimental data obtained for collisions of heavy nuclei at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' One of the variants of such a model was presented in [14] in a concise form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It was believed that the collision energies were not high enough, and one could limit oneself to only two varieties: protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It was assumed that the characteristic temperatures do not exceed the temperatures at which new particles can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The temperatures of the nuclear fireball are of the order of 140 135 \uf0b8 ~ T MeV in units 1 \uf03d B k (freeze- out phase, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1) at the stage after the transition to the hadronic gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the model itself should have a transparent nonrelativistic limit, taking into account the law of conservation of the total number of nucleons without the generation of new particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The successive stages of the evolution of a nuclear fireball are schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' From left to right: two touching ultrarelativistic nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' the state of a hot and superdense nuclear system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' quark-gluon phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' hadronization and chemical freeze-out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' kinetic freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The successive stages of the evolution of a nuclear fireball A more detailed and consistent description of the mathematical apparatus of the model [14] is proposed in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Some more subtle effects are estimated (additional corrections for pressure, density, and root-mean-square (RMS) fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A new two-component meson model [16] has been proposed for the case of temperatures above the production threshold ( 135 \uf03e T MeV ), when the number of mesons is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" 1 One-component vdW gas According to various estimates, the lifetime of a nuclear fireball is much longer than the characteristic nuclear interaction time 23 22 10 10 \uf02d \uf02d \uf0b8 ~ 't c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is of the order of the relaxation time 23 21 10 10 \uf02d \uf02d \uf0b8 \uf074 ~ c for sufficiently small local fireball volumes (subsystem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='01 1 10 1100 t (fm/c)Therefore, we will assume that at each moment of time exceeding the relaxation time, a local statistical equilibrium has time to be established in the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' That is, such a local fireball region is quasi-stationary, and therefore the methods of statistical physics can be applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since all thermodynamic potentials, as well as entropy and volume, are additive (extensive) quantities, therefore, the corresponding potentials (values) of the entire system (fireball) can be defined as the sum of the corresponding thermodynamic potentials of quasi-closed subsystems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Then, at each moment of time, one can give a standard representation of the partition function of a rarefied quasi-ideal van der Waals gas in the canonical ensemble (CE) for such quasi-closed subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In the approximation of pair interaction and condition \uf028 \uf029 1 \uf03c\uf03c V N T B , this quantity has the form [17]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029N N N T B V m T N N T V Z \uf02d \uf06a \uf03d , !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' , , 1 , (1) where, respectively, N and m are the number and mass of particles, V and T are the volume and temperature of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Formula (1) uses the notation [11]: \uf028 \uf029 \uf028 \uf029 T m K T m dp T p m p m T 2 2 2 0 2 2 2 2 2 exp 2 1 \uf070 \uf03d \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02b \uf02d \uf070 \uf03d \uf06a \uf0f2 \uf0a5 , , (2) where \uf028 \uf029z K2 is the modified Bessel function, and the second virial coefficient in (1) has the form: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 dV T U T B \uf0f2 \uf0a5 \uf02d \uf02d \uf03d 0 exp 1 2 1 (3) and includes pairwise interaction of particles, \uf0e5 \uf03c \uf03d j i ij U U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In relativistic limit T m \uf03e\uf03e one can easy obtain, given the asymptotes of the Bessel function: \uf028 \uf029 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6\uf02d \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf070 \uf06a T m mT m T exp 2 2 3 ~ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This formula further leads to the effect of exponential suppression of the particle yields with large mass, which is important in the study of quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure in the system is easy to find from the partition function: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029N T B V TN N T V Z V T N T V \uf02d \uf03d \uf0b6 \uf0b6 \uf03d , , , , ln P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (4) Note that if the Stirling formula is used in the partition function for the factorial: \uf028 \uf029N e N N N \uf070 \uf0bb 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' , then the final pressure formula (4) will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' THE MODEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' According to the above, all calculations for subsystems will be carried out by methods of statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This implies, in addition to the local statistical equilibrium, the fulfillment of the condition of the statistical (thermodynamic) boundary: A N N \uf0ae , where A N is the Avogadro constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the final formulas can be applied to the nuclear fireball due to the indicated additivity of thermodynamic potentials and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since the number of generated particles in a fireball is about 4-6 thousand during high-energy nucleus-nucleus interactions, this assumption is more or less justified at the first stages of its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, at later stages of evolution, since the number of nucleons at the nonrelativistic boundary is limited by the baryon number conservation law and equal to ( heavy element nuclei collide with mass number ), this assumption is, in general, somewhat doubtful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Of course, at later stages of evolution, this assumption is, in general, somewhat doubtful, since the number of nucleons n the nonrelativistic limit is confined by the baryon number conservation law and is equal to 300 200 \uf0b8 \uf03d N (the nuclei of heavy elements collide with the mass number 200 ~ A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, the practical application of the van der Waals equation quite often goes beyond the conditions under which the virial approximation has been obtained as experience shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Considering this fact, as well as the fact that we can always restrict ourselves to the first stage (see Section 3), we believe that our approximation is sufficiently justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Although calculations by the saddle point method are made when \uf028 \uf029 0 \uf03c T B , however, for the reasons stated above, the final formulas are extended to region where the second virial coefficient \uf028 \uf029 T B is not necessarily negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' From the partition function \uf028 \uf029 N T V Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' one can also get: free energy \uf028 \uf029 \uf03d N T V F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf05b \uf05d N T V Z T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ln \uf02d \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' chemical potential \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf02b \uf06a \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf03d \uf06d V N T B T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='m V N T N N T V F 2 ln ln ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (5) and the derivative of the chemical potential which in the statistical limit has the form: \uf028 \uf029 \uf028 \uf029\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 V T T B V T T B N T N V V P N 2 2 lim A N N 2 \uf0ae \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf02b \uf03d \uf0b6 \uf0b6 \uf02d \uf03d \uf0b6 \uf06d \uf0b6 \uf0ae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (6) Then, we obtain the Grand partition function (GPF) \uf028 \uf029 \uf06d , ,T V Z from the partition function \uf028 \uf029 N T V Z , , taking into account the above physical considerations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [18, 19]): \uf028 \uf029 \uf028 \uf029 N T V Z T N T V N , , , , \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf06d \uf03d \uf06d \uf0e5exp Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" (7) At high temperatures (which, for example, are realized during collisions of heavy ions in the GCE, and ' dN T N \uf0ae \uf044 ) one can turn from the sum to the integral using the Euler-Maclaurin formula." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the first integral term remains and the logarithm of the statistical sum is introduced into the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" Let's denote this indicator by \uf028 \uf029' N \uf046 : \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0f2 \uf0f2 \uf0a5 \uf0a5 \uf046 \uf03d \uf02b \uf06d \uf03d \uf06d 0 0 exp ln exp ' ' ' , ' ' , , N dN T T N V Z N dN T T V Z ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (8) Further integration is performed by the saddle point method [16], since at high temperatures the integrand has a strongly pronounced maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We obtain the following expression for finding the maximum point ( \uf02a N ) for the integrand from the extremum condition imposed on the saddle point: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf02a \uf02a \uf03d \uf02a \uf03d \uf02a \uf02a \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf03d \uf06d N N N N N N N T V Z N T N , , ln , (9) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 m T V N T N , \uf06a \uf02d \uf0bb \uf06d \uf02a \uf02a \uf02a ln ln , (10) where \uf02a \uf06d is the chemical potential at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' we obtain: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf02a \uf02a \uf02a \uf02a \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf02d \uf06d \uf070 \uf06a \uf0b4 \uf0b6 \uf046 \uf0b6 \uf070 \uf03d \uf02a \uf02a \uf02a N V N T B T V e N N m T N N T V N N N N N exp 2 2 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (11) where the second derivative of the exponent \uf046 at the saddle point is defined as follows: \uf028 \uf029 \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02b \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02b \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf046 \uf0b6 \uf02a \uf02a \uf02a \uf02a \uf03d \uf03d \uf03d \uf02a \uf03d N N N N N N N N N N T V Z N T N T N N 2 2 2 2 2 2 ln 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 0 2 2 1 1 1 2 2 \uf03c \uf03d \uf02b \uf02b \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02b \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf03d \uf02a \uf02a \uf03d \uf03d \uf02a \uf02a \uf02a V T B V T B N N N T N T N N N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' because \uf02a \uf03d \uf02a \uf02d \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 \uf02a N N T N N N 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf02a \uf02a \uf03d \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf03d \uf0f7\uf0f8 \uf0f6 \uf0e7\uf0e8 \uf0e6 \uf0b6 \uf06d \uf0b6 N N N N N N T V Z N T 2 2ln 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure in the GCE is defined as follows in terms of the temperature and the logarithm of the GPF (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [18]): \uf028 \uf029 \uf028 \uf029 V T V T T P \uf06d \uf03d \uf06d , , , Z ln .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" (12) It's easy to show that pressure (12)," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' taking into account (5) and (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' can be rewritten as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf078 \uf078 \uf02d \uf078 \uf02d \uf078 \uf0bb \uf0fa \uf0fa \uf0fb \uf0f9 \uf0ea \uf0ea \uf0eb \uf0e9 \uf078 \uf0b6 \uf046 \uf0b6 \uf02d \uf078 \uf02d \uf078 \uf0bb \uf06d \uf02a \uf03d \uf02a \uf02a V T B T B T V N T B T T P N N 2 ln 1 2 ln 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (13) where the saddle point \uf028 \uf029 V T V N \uf02a \uf02a \uf06d \uf03d \uf078 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' is defined according to (5) and (10) as \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 T N T m N \uf02a \uf06d \uf06a \uf0bb \uf078 exp ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The parameter \uf078 can be eliminated from equation (13) using the definition of density, which in the thermodynamic limit turns into the well-known formula [1]: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf05b \uf05d \uf078 \uf02d \uf078 \uf0ae \uf02d \uf078 \uf02d \uf078 \uf03d \uf06d \uf0b6 \uf06d \uf0b6 \uf03d T B V T B T P n 2 1 2 1 2 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (14) In the thermodynamic limit ( A N N \uf0ae , \uf0a5 \uf0ae V ) the chemical potential of the saddle point \uf02a \uf06d from (10) when \uf028 \uf029 \uf028 \uf029 V N T B N N 2 1\uf02d \uf03d \uf02a turns into the chemical potential \uf06d ( \uf06d \uf0ae \uf06d\uf02a ), which is determined by the well-known thermodynamic equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Both equations (13) and (14) in parametric form (the saddle point \uf078 acts as a parameter) determine the relationship between pressure P , temperature T , and density n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We obtain the state equation in GCE by excluding explicitly this parameter from the system of equations (13) and (14): \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d dP n T B Tn n T P \uf02b \uf02b \uf0bb \uf06d 1 , , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (15) Of course, the resulting state equation is implicitly a parametric equation, since the saddle point \uf078 (and, hence, and n ) determines the chemical potential \uf06d according to (5) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is important that the resulting formula takes into account the contribution to the pressure of the finite volume of the system, s V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This contribution naturally vanishes in the thermodynamic limit, where there is no difference between CE and GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Only the last of the three terms remains for large but finite volumes, as in the nuclear fireball model discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4: \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf05b \uf05d \uf028 \uf029 \uf05b \uf05d s V V V n T B T n T B n T B V T dP s 2 ln ln 1 2 \uf02d \uf0ae \uf02d \uf02b \uf03d \uf0ae lim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (16) RMS fluctuations of pressure and density calculated by known formulas (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [20]) give estimates of the found corrections to the corresponding quantities: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d n T B V n T n P V Tn P s 2 1 2 2 \uf02b \uf0b6 \uf0b6 \uf03e\uf03d \uf044 \uf03c ~ , (17) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d n T B V N V N T n N N 2 1 1 2 2 2 \uf02d \uf0b6 \uf06d \uf0b6 \uf03e\uf03d \uf044 \uf03c \uf02a \uf03d \uf02a ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (18) 2 Two-component vdW gas Let us consider the procedure for taking into account the excluded volume and attraction in the vdW model for the case of a two-component hadron gas of two types of particles “i ” and “ j ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 N and 2 N are the number of particles of the first and second sorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In this case, the partition function has the form: \uf028 \uf029 \uf03d 2 1 N N T V Z , , , \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 T U r d r d T m T m N N N k k N l l N N 12 1 2 3 1 1 3 2 1 2 1 exp !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 2 1 2 1 \uf02d \uf0b4 \uf0b4 \uf0b4 \uf06a \uf06a \uf03d \uf0f2\uf0d5 \uf0f2\uf0d5 \uf03d \uf03d , , , (19) This expression for the pair-interactions approximation ( \uf028 \uf029 \uf028 \uf029 12 123 U U \uf03c\uf03c ) and a weakly ideal gas ( 1 2 \uf03c\uf03c V NB ) can be rewritten as follows [11]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0b4 \uf06a \uf06a 2 1 2 1 2 1 2 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1 N N T m T m N N N N T V Z , , ~ , , , \uf028 \uf029 \uf028 \uf029 2 1 1 12 2 22 2 21 1 11 N N N B N B V N B N B V ~ ~ \uf02d \uf02d \uf0b4 \uf02d \uf02d \uf0b4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (20) Here we have introduced the notation: jj ii ij ii ij B B B B B \uf02b \uf03d 2 ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" The two-particle partition function \uf028 \uf029 2 1 \uf06d \uf06d , , ,T V Z in GCE is expressed in terms of the two-particle partition function \uf028 \uf029 2 1 N N T V Z , , , in CE [12, 17], as: \uf028 \uf029 \uf028 \uf029 \uf03d \uf03d \uf06d \uf06d \uf06d \uf02b \uf06d \uf0a5 \uf0a5 \uf0f2 \uf0f2 2 1 2 0 1 0 2 2 1 2 2 1 1 e d d ' , ' , , ' ' , , , ' ' N N T V Z N N T T V N N Z \uf028 \uf029 \uf028 \uf029 2 1 2 0 1 0 2 exp d d ' , ' ' ' N N N N T \uf046 \uf03d \uf0f2 \uf0f2 \uf0a5 \uf0a5 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (21) Integration of (21) by the saddle point method [21] leads us to the following result: \uf028 \uf029 \uf028 \uf029 \uf02a \uf02a \uf02a \uf02a \uf02a \uf02a \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf06d \uf02b \uf06d \uf0a2 \uf0a2 \uf046\uf0a2\uf0a2 \uf070 2 1 2 2 1 1 2 1 exp 2 N N T V Z T N N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ~ Z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' where the coordinates of the saddle point \uf02a i N ( 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d i ) are found from the extremum conditions: \uf028 \uf029 0 \uf03d \uf0a2 \uf0b6 \uf0a2 \uf0a2 \uf046 \uf0b6 i j i N N N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 22 21 12 11 2 1 c c c c N N det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d \uf0a2 \uf0a2 \uf046\uf0a2\uf0a2 \uf02a \uf02a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf02a\uf0a2 \uf03d \uf0a2 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf0a2 \uf0b6 \uf0a2 \uf0b6 \uf0a2 \uf0a2 \uf046 \uf0b6 \uf03d N N j i j i ij N N N N c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Substituting the value of the partition function into the definition of pressure in the GCE [18], we obtain the following expression [12]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf0fa\uf0fb \uf0f9 \uf0ea\uf0eb \uf0e9 \uf02d \uf078 \uf078 \uf02b \uf02d \uf078 \uf02d \uf078 \uf02d \uf078 \uf02b \uf078 \uf06d \uf06d \uf03d \uf06d \uf06d V C B B B B T V T V T T P 2 ln ln 2 1 21 12 22 2 2 11 2 1 2 1 2 1 2 1 ~ ~ ~ , , , , , Z , (22) where 21 2 12 1 22 2 11 1 B B B B C ~ ~ \uf078 \uf078 \uf02d \uf078 \uf078 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Using such a mathematical apparatus, one can organically introduce the law of conservation of chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The latter are related to the condition imposed on the integrand when finding the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In the thermodynamic limit the chemical potential determined by the extremum condition coincides with the definition of the chemical potential itself: \uf028 \uf029 i j i i i N N N T V F \uf0b6 \uf0b6 \uf03d \uf06d \uf0ae \uf06d\uf02a , , , , where \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d 2 1 2 1 ln N N T V Z T N N T V F , , , , , , \uf02d \uf03d is the definition of free energy (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We get from the definition of density \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d ji ij j ii i i i j i i B B B T P n ~ ~ ~ , , \uf02b \uf078 \uf02b \uf078 \uf02d \uf078 \uf06d \uf0b6 \uf06d \uf06d \uf0b6 \uf03d 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (23) The virial expansion (22) can be rewritten,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' taking into account (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' as a two-component vdW equation in the approximation 1 \uf03c\uf03c V N b i ij and \uf028 \uf029 1 \uf03c\uf03c ij ij Tb a ): \uf028 \uf029 \uf03d \uf06d \uf06d 2 1 2 1 n n T P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 dP n a n a n n a n a n n b n b Tn n b n b Tn \uf02b \uf02b \uf02d \uf02b \uf02d \uf02d \uf02d \uf02b \uf02d \uf02d \uf03d 1 12 2 22 2 2 21 1 11 1 1 12 2 22 2 2 21 1 11 1 1 1 ~ ~ ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (24) where dP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' according to (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' takes into account the finite size of the fireball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' When formula (24) was derived, the expression T a b B ij ij ij ~ ~ ~ \uf02d \uf0bb was used (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', [12]), and for each type of particles the corresponding parameters of attraction and repulsion were introduced [11]: \uf028 \uf029 jj ii ii ij ij a a a a a \uf02b \uf0b4 \uf067 \uf0bb 2 ~ , \uf028 \uf029 jj ii ij ii ij b b b b b \uf02b \uf03d 2 ~ , \uf067 is a phenomenological parameter reflecting the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3 The asymmetric two-component freeze-out model with non-conservation of the number of particles The considering nucleus-nucleus collisions \uf028 \uf029 A A\uf02b have very high energies, more than 1 GeV per nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' At the same time, mesons of different sorts dominate in the initial freeze-out stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, to describe the nucleus-nucleus interactions at this stage of the freeze-out above the production threshold of new particles ( 135 \uf03e T MeV), we propose a generalization of the vdW model to a medium-sized nuclear fireball [16]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf03e\uf03e \uf03c\uf03c \uf070 \uf03e \uf03c \uf03e \uf03c \uf070 \uf03e \uf02b \uf03c \uf03e \uf03c A r b a V V V f f f 3 0 2 4 3 4 3 2 ~ ~ ~ max min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Here 2 1 1 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf0b8 \uf03d r Fm, \uf03e \uf03c a , \uf03e \uf03c b are the mean semiaxes of the ellipsoid, and \uf03e\uf03e \uf03c\uf03c A is the mass number of nuclei left in the fireball after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In our considerations we assume that the fireball consists, mainly, of mesons, given that the number of nucleons is much less than the number of mesons ( \uf03c\uf03c \uf0b8 300 200 ~ pn N 5000 4000 \uf0b8 \uf070\uf072\uf077 ~ N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We neglect the contribution of other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we introduce the following additional natural assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The average internucleon energies do not exceed the production threshold of the heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we restrict ourselves to two sorts of particles ("0" is the 0 \uf070 -meson, "+" is the \uf02b \uf070 -meson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Since \uf02b \uf070 -meson production reactions are twice as likely as 0 \uf070 -meson production reactions, we assume that n kn n \uf03d \uf03d \uf02b 0 , where, 1 \uf03c k , 0 n is the 0 \uf070 -meson density, and \uf02b n is the \uf02b \uf070 -meson density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This corresponds to a more probable production of the \uf02b \uf070 -mesons in reactions \uf02b \uf070 \uf02b \uf02b \uf0ae \uf02b n d d p , 0 \uf070 \uf02b \uf02b \uf0ae \uf02b p d d p than production of the 0 \uf070 -mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We introduce the effective potential of the meson interaction \uf028 \uf029j i U , where \uf028 \uf029 \uf07b \uf07d 0, , \uf02b \uf03d j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' That is, "(0+)" is the interaction of 0 \uf070 -mesons with \uf02b \uf070 -mesons, "(++)" is the interaction of \uf02b \uf070 -mesons \uf028 \uf029 \uf028 \uf029 \uf0ef \uf0ee \uf0ef \uf0ed \uf0ec \uf0b3 \uf02b \uf02b \uf03c \uf0a3 \uf02b \uf02b \uf03c \uf0a5 \uf03d r R R if R R r R R if u R R r if U j i j i j i j i j i j i 0 0 0 0 0 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (25) Since the effective rectangular potential a well leads to approximately the same values of pressure and density as the real potential (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2, where \uf028 \uf029 U U \uf03d \uf02b0 , \uf028 \uf029 \uf065 \uf03d \uf02b0 0 u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the real meson-meson potential (a) can be replaced by a similar effective rectangular potential (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Meson-meson potential 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept that the 0 \uf070 -meson hard-core radius is much smaller than the \uf02b \uf070 -meson hard-core radius: 0 0 0 \uf02b \uf03c\uf03c R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The radius of the hard-core of the \uf02b \uf070 -meson is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Average pressure and density fluctuations are easily found within the framework of the proposed model, similarly to formulas (18) and (19): \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d Tn B V n T P f \uf02b \uf03e \uf03c \uf03e \uf044 \uf03c 1 ~ , (26) \uf028 \uf029 \uf028 \uf029 \uf05b \uf05d Tn B V n V n f f \uf02d \uf03e \uf03c \uf03e \uf03c \uf03e \uf044 \uf03c 1 1 ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (27) The following results are obtained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Such data have been used (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3): 3 142, \uf03d T MeV, the effective radius of the \uf02b \uf070 -meson, 45 0 0 , \uf03d \uf02b R Fm, and 0 \uf070 -meson, 01 0 0 0 , \uf03d R Fm, the average value of the volume of the meson fireball is taken as the value 600 ~ \uf03e \uf03c f V Fm 3 , 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf03d k , the parameter of the potential depth, \uf028 \uf029 100 80 0 0 \uf0b8 \uf02b ~ , u MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' One can U来 U*clearly see (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4) an increase in the correction \uf03e \uf03c P dP at low densities, which is typical in the final stages of the freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Dependence of the meson pressure P (24) on the meson density n kn n \uf03d \uf03d \uf02b 0 for the two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Ratio of the correction to pressure dP from the size of the meson fireball to the value of the RMS pressure fluctuation \uf03e \uf03c P (26) as a function of the meson density n kn n \uf03d \uf03d \uf02b 0 4 Two-component model of a nucleon fireball at the final stage of the freeze-out The average lifetime of mesons dominating in the initial stages of the freeze-out is relatively short ( 16 8 10 10 \uf02d \uf02d \uf0b8 \uf074 ~ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=" That's why they decay pretty quickly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Accordingly, baryons, namely protons and neutrons, begin to dominate at the final stage of freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' In addition, as shown above, the effects of the finite volume size become noticeable at sufficiently low density values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' This formally corresponds to just such final stages of the fireball evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, despite a certain doubt about the existence of a fireball at such late stages, when the boundary between the gas and the aggregate of individual nucleons gradually disappears, to describe the nucleus-nucleus P(T=142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='3, n), MeV/Fm-3 10 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30 10 20 30 40 50 FdP/

2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30interactions at the last stage of the freeze-out, which is below the production threshold of new particles ( 135 \uf03c T MeV), in [14] the following generalization of the vdW model to the nucleon fireball was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept the following simplifications by analogy with the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The average energies of internucleon collisions do not exceed the production threshold of other hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, we restrict ourselves to two varieties (“ p ” is the proton, “ n ” is the neutron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We take the relation between the density of protons and neutrons in the form n n n n p \uf03d \uf02b following from the law of conservation of the baryon number, A N Z \uf03d \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We assume that the nucleon composition of colliding nuclei is known as such n p kn n \uf03d , where 1 \uf03c k , since heavy nuclei have an excess of neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The effective potential of the proton-neutron, proton-proton and neutron-neutron interactions, which leads to approximately the same values of pressure and density as the real potential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3), can be represented by analogy to (25) as \uf028 \uf029j i U , where \uf028 \uf029 n p j i , , \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The hard-core radius of the proton is assumed to be known, 5 0 0 , \uf03d p R Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We accept that the radius of the neutron is much less than the radius of the proton: 0 0 p n R R \uf03c\uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' We get from equation (27): \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 dP n a n n n k Tn T P \uf02b \uf02d \uf064 \uf02b \uf062 \uf02d \uf061 \uf02d \uf02b \uf03d \uf06d \uf06d \uf02a \uf02a \uf02a \uf02a \uf02a \uf02a 2 2 2 1 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (28) where \uf028 \uf029 k n n \uf02b \uf03d \uf02a 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k is a dimensionless quantity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k b k b b k b 22 2 21 12 11 \uf02b \uf02b \uf02b \uf03d \uf061 ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 22 21 12 11 b k b b k b \uf02b \uf02b \uf02b \uf03d \uf062 ~ ~ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' k b b b b b b k b kb 21 12 21 22 12 11 2 22 11 ~ ~ ~ ~ \uf02b \uf02b \uf02b \uf03d \uf064 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' \uf028 \uf029 22 21 12 2 11 a k a a k a a \uf02b \uf02b \uf02b \uf03d \uf02a ~ ~ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It follows from the condition 0 1 0 2 R R \uf03c\uf03c that 11 22 b b \uf03c\uf03c , \uf062 \uf040 \uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' By analogy with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (18) and (19), we find the corresponding average fluctuations of pressure and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Functional dependences for pressure, obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (28), and the ratio of dP to RMS pressure fluctuations are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Dependence of nucleon pressure P (28) on nucleon density, n kn n n p \uf03d \uf03d , in the two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The ratio of correction from the size of the nucleon fireball to pressure dP to the value of the RMS pressure fluctuation \uf03e \uf03c P depending on the density of nucleons, n kn n n p \uf03d \uf03d It can be seen that the correction dP makes a nonzero contribution to the total pressure also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' On the other hand, it is negligibly small almost everywhere in comparison with the contribution from fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The correction makes a contribution comparable to fluctuations only in the region near zero density that is nonphysical for a nuclear fireball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' But it can be neglected in this region, as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Summary The effect of taking into account the excluded volume and attraction is analyzed in the case of a two-component gas: (i) 0 \uf070 - and \uf02b \uf070 -mesons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (ii) protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The calculations have been performed in the Canonical and Grand Canonical ensembles by the saddle point method for a two- component system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The particles interact with the potentials of the hard-core at short distances and with relatively high potentials at large distances (effective attraction radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' For effective P(T=142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' n), MeV/Fm-3 10 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 N15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30 10 20 30 40 50 FdP/

1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='8 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='7 n, Fm-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='30interparticle interactions of this type, an equation of state has been obtained with corrections that take into account the finite dimensions of the nuclear fireball, as well as RMS fluctuations of pressure and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The pressure correction disappears in the thermodynamic limit, when, according to statistical physics, there is no difference between various statistical ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The formulas for pressure and density obtained by the saddle point method can be used to analyze experimental data concerning the relative number of the yield particleshe of various sorts and critical parameters in high-energy nuclear-nucleus collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A generalization of the presented vdW model to the asymmetric case of a two-component model ( 0 \uf070 - and \uf02b \uf070 -mesons) with realistic parameters of the hard-core and attraction has been proposed as an example of such a use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The ratio of the pressure correction to the RMS value of pressure fluctuation is estimated for the case of an asymmetric two-component meson fireball model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' An increase in the correction has been found at low density values corresponding to the final stages of freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' It is found that the contribution to pressure and relative fluctuations, taking into account different radii and the finiteness of the nuclear fireball, is noticeable in the case of the meson model with nonconservation of the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' However, this correction can be neglected for the final stages of the freeze-out, when nucleons begin to dominate (the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Therefore, the developed model is applicable in the analysis of experimental data on the study of the initial meson phase of a nuclear fireball (the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 3), which occurs, in particular, in experiments on the study of quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' The research was carried out within the framework of the initiative scientific topic 0122U200549 (“National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine is the customer).' 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van der Waals model of multicomponent hadron gas in the grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 2019 arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='08441v1 [hep-ph];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Krivenko- Emetov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Interparticle attractive forces account of the multicomponent hadron gas in the grand canonical ensenble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Book of abstract of the 24th Annual Scientific Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' of Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' for Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Research, Kyiv, Ukraine, April 10-13, 2017 (Kyiv, 2017) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Vovchenko at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Multicomponent van der Waals equation of state: Applications in nuclear and hadronic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' C 96 (2017) 045202.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' of Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' for Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Research, Kyiv, Ukraine, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 27 – Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 01, 2021 (Kyiv, 2021) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 27.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' for Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Research, Kyiv, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 26 – 30, 2022, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='21-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (Ukr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Feynman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Statistical Mechanics: a set of lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Advanced Book Classics (2 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Perseus Books, Reading, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=', 1998) 354 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fedorchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Quantum mechanics, thermodynamics and statistical physics (Kyiv: Vyshcha shkola, 1993) 415 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (Ukr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Fedoruk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Saddle point method (Moskva, 1977) 368 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' (Rus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Я.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Кривенко-Еметов* Інститут ядерних досліджень НАН України, Київ, Україна Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», Київ, Україна Відповідальний автор: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='kryvenko-emetov@kpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='ua;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' krivemet@ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content='net БАГАТОКОМПОНЕНТНА МОДЕЛЬ ВАН ДЕР ВАЛЬСА ЯДЕРНОГО ФАЄРБОЛУ НА СТАДІЇ ФРІЗАУТУ Двокомпонентна газова модель Ван-дер-Ваальсу запропонована для опису адронних етапів еволюції ядерного фаєрболу у стадії охолодження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Для першого етапу адронізації, коли домінують мезони, запропонована двокомпонентна мезонна модель( 0 \uf070 - та \uf02b \uf070 -мезонів) з ефективним двочастинковим потенціалом взаємодії прямокутної ями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Для останнього етапу, коли майже усі мезони розпались, запропонована двокомпонентна нуклонна модель протонів та нейтронів з відповідним ефективним прямокутним нуклонним потенціалом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' При аналітичних розрахунках статистичної суми використовувався методу перевалу, що дозволило єдиним чином отримати аналітичні вирази, як для тиску та щільності з урахуванням скінченних розмірів системи, так і вирази для хімічних потенціалів.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Очікується, що запропоновані моделі й отримані формули можуть бути використані для аналізу експериментальних даних щодо кількісних характеристик виходу частинок різних сортів у кінцевому стані від адронних стадій еволюції ядерного фаєрболу, а також для визначення критичних параметрів системи у ядро-ядерних зіткненнях за високих енергій.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'} +page_content=' Ключові слова: фаєрбол, фрізаут, рівняння Ван-дер Ваальса, ефективний ядерний потенціал, Великий канонічний ансамбль, флуктуація тиску, кварк-глюонна плазма, експериментальні дані.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQf1vn1/content/2301.00742v1.pdf'}