Density fluctuations in intermediate-energy heavy-ion collisions

Density fluctuations in intermediate-energy heavy-ion collisions 1 2 first-author 3 4 corresp xiangai_deng@fudan.edu.cn xiangai_deng@fudan.edu.cn xiangai_deng@fudan.edu.cn https://orcid.org/0000-0003-4061-8529 3 4 corresp mayugang@fudan.edu.cn mayugang@fudan.edu.cn mayugang@fudan.edu.cn https://orcid.org/0000-0002-0233-9900 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China Within the framework of the isospin-dependent quantum molecular dynamics model, we simulate 129 Xe + 119 Sn collisions in an incident energy range of 20 to 190 MeV/nucleon and discuss the liquid-gas phase transition with density fluctuations. For comparison, we also extract the effective Fisher parameter τ eff , multiplicity of intermediate-mass fragments (IMFs), and information entropy. It is found that the Fisher parameter and maximum information entropy of collisions have peak values at E beam = 50–80 MeV/nucleon. In addition, the maximum multiplicity of IMFs has a plateau around 70 MeV/nucleon. For higher-order density moments in a larger central region of [-5,5] 3 fm 3 in the collision system, a maximum plateau also appears as function of beam energy at energies exceeding 70 MeV/nucleon. These observables, which are consistent with each other, indicate a liquid-gas phase transition around 70 MeV/nucleon for the 129 Xe + 119 Sn system. Heavy-ion collisions Liquid-gas phase transition IQMD Density moment Intermediate mass fragment Information entropy Fisher-law parameter Introduction 1 Exploration of the phase change and equation of state (EOS) of nuclear matter during heavy-ion collisions is an important topic in nuclear physics [ 1 1 - 6 6 1-6 ]. It is also closely related to the nuclear transport properties [ 7 7 - 15 15 7-15 ], nuclear landscape, and nuclear synthesis [ 16 16 - 23 23 16-23 ]. In the late 20th century, the relationship between the temperature and excitation energy of the system were found to exhibit behavior similar to the macroscopic phenomenon of the liquid-gas phase transition (LGPT) of classical fluid, and it was proposed that a first-order LGPT, and even a second-order phase transition at the critical point, occur in nuclear matter [ 24 24 ]. One expects that in a finite-size system of heavy-ion collisions, the LGPT could appear. The LGPT has been investigated in intermediate-energy heavy-ion collisions, both theoretically and experimentally [ 25 25 - 34 34 25-34 ]. In recent decades, many probes have been proposed for determining whether the LGPT has occurred, for example, the caloric curve [ 25 25 - 28 28 25-28 ], intermediate-mass fragments (IMFs) [ 35 35 , 36 36 ], Fisher's power-law exponent of the fragment distribution [ 35 35 - 39 39 35-39 ], the charge fluctuation of the largest fragment [ 27 27 , 40 40 - 42 42 40-42 ], the negative heat capacity [ 43 43 ], the information entropy [ 44 44 - 46 46 44-46 ], and the nuclear Zipf law of Ma [ 44 44 , 47 47 , 48 48 ]. Furthermore, recent research has focused on density fluctuations with the goal of investigating the phase transition theoretically and experimentally as in Refs. [ 49 49 - 51 51 49-51 ]. The normalized (net) baryon density moments, which are related to the relative production yield of composite baryons, are observationally relevant and may be useful for exploring phase transitions by enhanced production of composite particles [ 49 49 - 52 52 49-52 ]. The density moment is expected to be an alternative way of identifying the phase transition in heavy-ion collisions. Because density fluctuations could be associated with the LGPT, the density moments are calculated to investigate the energy dependence and identify the transition. In this study, we explore the dependence of the density moments on the incident energy in intermediate-energy heavy-ion collisions. Within the framework of the isospin-dependent quantum molecular dynamics (IQMD) model, central 129 Xe + 119 Sn nuclear collisions are simulated, and the density moments are calculated. For verification, the power-law fits of the charge distribution of fragments, fragment multiplicities, and information entropy are presented. This paper is organized as follows. In Sect. 2, the IQMD model is introduced briefly, and some formulas are presented. In Sect. 3, different LGPT probes are calculated, the maximum values of the density moments are extracted, and the results are discussed. In Sect. 4, a conclusion is presented. MODEL AND FORMALISM 1 IQMD Model 2 The QMD model is essentially a quantum extension of the classical molecular dynamics approach, which is widely applied in chemistry and astrophysics; it is designed to describe fragment formation [ 54 54 - 58 58 54-58 ]. This n -body approach uses a microscopic framework that treats the dynamics of colliding nuclei directly and simulates heavy-ion collisions on an event-by-event basis [ 54 54 ]. The descriptions of mean positions and momenta are purely classical, and particles are considered to be distinguishable in the QMD model [ 55 55 ]. In this work, we use an improved version of the QMD model that incorporates isospin-dependent interactions and the Pauli exclusion principle. In this model, each nucleon is treated as a Gaussian wave packet in a coherent state [ 54 54 , 55 55 , 59 59 ]: <math display="block" id="M1"><mrow><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>L</mi><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mfrac><mtext>e</mtext><mi>x</mi><mi>p</mi><mo stretchy="false">[</mo><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>−</mo><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>L</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>i</mi><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover></mrow><mi>ℏ</mi></mfrac><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math id="M2"><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></math> and <math id="M3"><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></math> represent the position and momentum of the i -th nucleon, respectively. L is the square of the Gaussian wave packet width, which is set to 2.16 fm 2 . The following interaction terms are included in the IQMD model: <math display="block" id="M4"><mrow><msub><mi>V</mi><mrow><mtext>tot</mtext></mrow></msub><mo>=</mo><msub><mi>V</mi><mrow><mtext>sky</mtext></mrow></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>yuk</mtext></mrow></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>sym</mtext></mrow></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>MDI</mtext></mrow></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>Coul</mtext></mrow></msub><mo>;</mo></mrow></math> they correspond to Skyrme, Yukawa, symmetry, momentum-dependent, and Coulomb interactions, respectively. The detailed forms of these interaction terms are given in Refs. [ 54 54 , 55 55 ]. The Skyrme potential component of the EOS, which is associated with the Skyrme interaction, is given by <math display="block" id="M5"><mrow><msub><mi>U</mi><mrow><mtext>sky</mtext></mrow></msub><mo>=</mo><mi>α</mi><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>+</mo><mi>β</mi><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mi>γ</mi></msup><mn>.</mn></mrow></math> In this work, we set α = -129 MeV, β = 59 MeV, and γ = 2.09, which is called the hard EOS, and α = -390 MeV, β = 320 MeV, and γ = 1.14, which is called the soft EOS. Formulas 2 The following probes are calculated on the basis of information on the phase space and fragments: the Fisher parameter τ eff , fragment multiplicity, information entropy ( H ), and density moments. The effective Fisher parameter τ eff can be obtained from the charge distribution by power-law fitting as follows: <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>d</mtext><mi>N</mi><mo>/</mo><mtext>d</mtext><mi>Z</mi><mo>∼</mo><msup><mi>Z</mi><mrow><mo>−</mo><msub><mi>τ</mi><mrow><mtext>e</mtext><mi>f</mi><mi>f</mi></mrow></msub></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Here the charge distribution is taken at 300 fm/ c when the system has reached the kinetic freeze-out stage. These charge distributions can be fitted well in the range 3 ≤ Z ≤ 7, as shown in Fig. 1 Fig. 1 . The charge range is selected to avoid the lightest fragments, with Z = 1 and 2, which could be affected by the evaporation and decay process, as well as heavier residual fragments. The information entropy H of all multiplicity events, which was presented for the first time by Ma [ 44 44 ] and introduced into the study of LGPTs of nuclei, is given by <math display="block" id="M7"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>H</mi><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><msub><mi>f</mi><mi>i</mi></msub><mtext>ln</mtext><msub><mi>f</mi><mi>i</mi></msub><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where f i , which is calculated for the event space, is the normalized event probability that i particles are produced, and <math id="M8"><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>i</mi></msub></mstyle><msub><mi>f</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow></math> . The sum is taken over the entire distribution of f i [ 44 44 ]. The density moments can be calculated from the phase space data computed in an IQMD simulation. First, from the phase space data computed by the IQMD simulation, the nuclear matter densities can be calculated at each coordinate space point and at every time as follows: <math display="block" id="M9"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>A</mi><mtext>T</mtext></msub><mo>+</mo><msub><mi>A</mi><mtext>P</mtext></msub></mrow></munderover></mstyle><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>L</mi><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mtext>e</mtext><mi>x</mi><mi>p</mi><mo stretchy="false">[</mo><mfrac><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>−</mo><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>L</mi></mrow></mfrac><mo stretchy="false">]</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where the summation is taken over all nucleons. Then, the numerical value of the nuclear matter density can be used to calculate the density moments via the formula described in Ref. [ 49 49 - 51 51 49-51 ], that is, <math display="block" id="M10"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>〈</mo><msup><mi>ρ</mi><mi>N</mi></msup><mo>〉</mo><mo>≡</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>ρ</mi><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow><mi>N</mi></msup><mi>ρ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo stretchy="false">)</mo><msup><mtext>d</mtext><mn>3</mn></msup><mi>r</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where A = <math id="M11"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>ρ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo stretchy="false">)</mo><msup><mtext>d</mtext><mn>3</mn></msup><mi>r</mi></mrow></math> . The normalized density moments are given by <math id="M12"><mrow><mrow><mo>〈</mo><mrow><msup><mi>ρ</mi><mi>N</mi></msup></mrow><mo>〉</mo></mrow><mo>/</mo><msup><mrow><mrow><mo>〈</mo><mi>ρ</mi><mo>〉</mo></mrow></mrow><mi>N</mi></msup></mrow></math> , which is thus unity for N = 1 [ 49 49 - 51 51 49-51 ]. These quantities have observational relevance because of their intimate relationship with the relative production yield of fragments. CALCULATIONS AND DISCUSSION 1 Simulations were performed using the IQMD model for central collisions of 129 Xe + 119 Sn at various beam energies ranging from 20 to 190 MeV/nucleon. Both the hard and soft EOSs were considered. The calculated time range was 0–800 fm/ c . The density moments were calculated within central regions consisting of [-3,3] 3 and [-5,5] 3 fm 3 cubic boxes, whose centers are located at the center of mass of the collision system, to check the central area dependence of the observables. Time evolution of properties 2 Figure 2 Figure 2 shows the time evolution of the IMF multiplicity ( N IMF ) and information entropy ( H ) at various incident energies. In Fig. 2 (a) Fig. 2 (a) , the IMF multiplicity increases, reaches a maximum, and then decreases with increasing time. Here IMFs are defined as fragments with charge number ( Z ) greater than or equal to 3 and smaller than the charge number of the source. As the energy increases, the IMF multiplicity changes more rapidly. Thus, at higher energies, the collision system generally has a higher excitation energy and breaks more rapidly into numerous fragments. Using the distribution { f i } calculated from all events, one can obtain the information entropy ( H ), as shown in Fig. 2 (b) Fig. 2 (b) . The pattern of H evolution is slightly different from that of the IMF multiplicity. With increasing H , the system becomes more chaotic. In the compression stage, the information entropy increases rapidly. With increasing time, more fragments are created, and the multiplicity probability distribution f i becomes more diverse. As time increases further, the information entropy decreases slightly because long-term binding of nucleons in clusters is difficult under the IQMD Hamiltonian. Thus, the number of particles produced will increase with time, which will result in a change in the probability distribution f i and finally a slight decrease in H . Figure 3 Figure 3 shows the time evolution of the normalized density moments ( <math id="M13"><mrow><mrow><mo>〈</mo><mrow><msup><mi>ρ</mi><mi>N</mi></msup></mrow><mo>〉</mo></mrow><mo>/</mo><msup><mrow><mrow><mo>〈</mo><mi>ρ</mi><mo>〉</mo></mrow></mrow><mi>N</mi></msup></mrow></math> ), which is the major focus of this work, in the central region of [-3,3] 3 fm 3 at various incident energies for N = 2 and 6. The time evolution of the normalized density moments is similar to that of the IMF multiplicity. It increases with time and then shows near-saturation or a slight decrease. At a given energy, for higher-order density moments, the density moment values are larger. In particular, high-order moments show cleaner structure as a function of beam energy, that is, a clearer broad peak structure at a certain energy, although there are larger fluctuations, as shown in Fig. 3 (b) Fig. 3 (b) , because obtaining higher-order density moments with the same precision requires better statistics. In addition, higher-order density moments are more sensitive to density fluctuations. Discussion of liquid-gas phase transition 2 To discuss the LGPT in collisions, we extracted the effective Fisher parameter τ eff ( Fig. 1 Fig. 1 ), maximum IMF multiplicity ( Fig. 2 (a) Fig. 2 (a) ), and maximum information entropy ( Fig. 2 (b) Fig. 2 (b) ) as a function of incident energy, as shown in Fig. 4 Fig. 4 . Both τ eff and the maximum information entropy exhibit non-monotonic behavior, and the peak values are approximately 50–80 MeV/nucleon. For the maximum IMF multiplicity, a plateau appears at energies above 70 MeV/nucleon. All of these results seem consistent with each other, and they indicate that the LGPT for this system could occur in this energy region. The hard and soft EOSs are indicated by black lines with circles and blue lines with squares, respectively, in Fig. 4 Fig. 4 . The turning energies are not sensitive to the EOS for all τ eff , <math id="M14"><mrow><msubsup><mi>N</mi><mrow><mtext>IMF</mtext></mrow><mrow><mtext>max</mtext></mrow></msubsup></mrow></math> , and H max . In addition, only the value of <math id="M15"><mrow><msubsup><mi>N</mi><mrow><mtext>IMF</mtext></mrow><mrow><mtext>max</mtext></mrow></msubsup></mrow></math> is higher for the soft EOS than for the hard EOS, as shown in Fig. 4 (b) Fig. 4 (b) . We also deduced the temperature using the fluctuation in the proton transverse momentum [ 60 60 ], which is defined as <math id="M16"><mrow><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mo>〈</mo><msubsup><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow><mn>2</mn></msubsup><mo>〉</mo><mo>−</mo><msup><mrow><mo>〈</mo><msub><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>〉</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><msup><mi>T</mi><mn>2</mn></msup></mrow></math> , where <math id="M17"><mrow><msub><mi>Q</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup></mrow></math> . When temperature is used as a variable instead of incident energy, as shown in Fig. 5 Fig. 5 , the turning temperature is shifted to slightly lower temperature for all τ eff , <math id="M18"><mrow><msubsup><mi>N</mi><mrow><mtext>IMF</mtext></mrow><mrow><mtext>max</mtext></mrow></msubsup></mrow></math> , and H max for the soft EOS. Thus, the soft EOS can reduce the phase transition temperature of a collision system. Further, we give the maximum normalized density moments for different orders, as shown in Fig. 6 Fig. 6 . To present all the orders of the normalized density moment in a single figure, these lines are scaled by different factors. In addition, because statistical fluctuations exist, these lines are fitted by polynomial functions. For the order N = 2 [black dotted line in Fig. 6 (a) Fig. 6 (a) ], the line seems flat. As the order of the density moment increases to N = 6, the maximum values appear around E = 90 MeV/nucleon, as shown in Fig. 6 (b) Fig. 6 (b) . Thus, the collision system has the maximum density fluctuation around E = 90 MeV/nucleon, indicating that the LGPT could occur here when the system enters spinodal instability [ 61 61 , 53 53 ]. However, if we choose a larger central region of [-5,5] 3 fm 3 , as shown in Fig. 7 Fig. 7 , the maximum normalized density moments reach maximum values around 70 MeV/nucleon and have a plateau at higher incident energies, like the maximum IMF multiplicity shown in Fig. 4 (b) Fig. 4 (b) . This energy is also close to the energy given by the effective Fisher parameter τ and the maximum information entropy, as shown in Fig. 4 Fig. 4 . Note that the turning energy varies with the central region. That is, the turning point has regional dependence. In fact, this can be understood from the central densities of the selected regions. We found that the [-3,3] 3 fm 3 region can reach a higher average density than the [-5,5] 3 fm 3 region. Thermodynamically, the temperature, pressure, and density are correlated; therefore, the turning point of the energy depends on the density. Actually, as reported in our previous work [ 62 62 ], the temperature extracted from heavy-ion collisions is lower for a smaller central region, where a higher beam energy would be required to reach the same temperature in a smaller central region. Thus, the calculated density moments for a region of a certain size correspond to a certain density and temperature. The [-5,5] 3 fm 3 is generally a more reasonable option, because our other observables, that is, the fragment distributions and their effective Fisher parameters, the IMF multiplicities, and the information entropy, are those of the entire space. CONCLUSION 1 Heavy-ion collisions of 129 Xe + 119 Sn were simulated by the IQMD model. We calculated the fragment charge distribution, IMF multiplicity, and information entropy. Turning points were found at E = 50–80 MeV/nucleon from the effective Fisher parameter τ eff of the fragment charge distribution, maximum IMF multiplicity, and maximum information entropy as a function of incident energy, which are associated with the LGPT. In addition, the turning energy extracted using τ eff seems to be smaller than that obtained using the IMFs. For both the hard and soft EOSs, the turning beam energies from all of the above observables of the phase change are not sensitive to the EOS, but a soft EOS could reduce the phase transition temperature of this collision system. Furthermore, we analyzed the density fluctuations in heavy-ion collisions with density moments of different orders. The obtained turning points are close to those given by effective Fisher parameter, IMF multiplicity, and information entropy. For higher-order normalized density moments, there are also peaks or saturation regions versus beam energy, which have the same properties as those obtained using the IMF multiplicity and information entropy. However, they depend on the size of the region; that is, they are pressure- or density-dependent. References B. Borderie and J.D. Frankland , Liquid-Gas phase transition in nuclei . Prog. Part. Nucl. Phys. 105 , 82 - 138 ( 2019 ). doi: 10.1016/j.ppnp.2018.12.002 http://doi.org/10.1016/j.ppnp.2018.12.002 . Liquid-Gas phase transition in nuclei B. A. Li , L. W. Chen , and C. M. Ko , Recent progress and new challenges in isospin physics with heavy-ion reactions . Phys. Rep. 464 , 113 - 281 ( 2008 ). doi: 10.1016/j.physrep.2008.04.005 http://doi.org/10.1016/j.physrep.2008.04.005 . Recent progress and new challenges in isospin physics with heavy-ion reactions Y. G. Ma and W. Q. Shen , Recent progress of nuclear liquid gas phase transition . Nucl. Sci. Tech. 15 , 4 - 29 ( 2004 ). Recent progress of nuclear liquid gas phase transition C.-W. Ma and Y.-G. Ma , Shannon information entropy in heavy-ion collisions . Prog. Part. Nucl. Phys. 99 , 120 ( 2018 ). doi: 10.1016/j.ppnp.2018.01.002 http://doi.org/10.1016/j.ppnp.2018.01.002 . Shannon information entropy in heavy-ion collisions B. Borderie and M. F. Rivet , Nuclear multifragmentation and phase transition for hot nuclei . Prog. Part. Nucl. Phys. 61 , 551 ( 2008 ). doi: 10.1016/j.ppnp.2008.01.003 http://doi.org/10.1016/j.ppnp.2008.01.003 . Nuclear multifragmentation and phase transition for hot nuclei G. Giuliani , H. Zheng , and A. Bonasera , The many facets of the (non-relativistic) Nuclear Equation of State . Prog. Part. Nucl. Phys. 76 , 116 ( 2014 ). doi: 10.1016/j.ppnp.2014.01.003 http://doi.org/10.1016/j.ppnp.2014.01.003 . The many facets of the (non-relativistic) Nuclear Equation of State P. Danielewicz , Transport properties of excited nuclear matter and the shock-wave profile . Phys. Lett. B 146 , 168 ( 1984 ). doi: 10.1016/0370-2693(84)91010-4 http://doi.org/10.1016/0370-2693(84)91010-4 . Transport properties of excited nuclear matter and the shock-wave profile L. Shi and P. Danielewicz , Nuclear isospin diffusivity . Phys. Rev. C 68 , 064604 ( 2003 ). doi: 10.1103/PhysRevC.68.064604 http://doi.org/10.1103/PhysRevC.68.064604 . Nuclear isospin diffusivity R.A. Lacey , N.N. Ajitanand , J.M. Alexander et al. , Has the QCD critical point been signaled by observations at the BNL relativistic heavy ion collider? Phys. Rev. Lett. 98 , 092301 ( 2007 ). doi: 10.1103/PhysRevLett.98.092301 http://doi.org/10.1103/PhysRevLett.98.092301 . Has the QCD critical point been signaled by observations at the BNL relativistic heavy ion collider? S.X. Li , D.Q. Fang , Y.G. Ma et al. , Shear viscosity to entropy density ratio in the Boltzmann-Uehling-Uhlenbeck model . Phys. Rev. C 84 , 024607 ( 2011 ). doi: 10.1103/PhysRevC.84.024607 http://doi.org/10.1103/PhysRevC.84.024607 . Shear viscosity to entropy density ratio in the Boltzmann-Uehling-Uhlenbeck model C.L. Zhou , Y.G. Ma , D.Q. Fang et al. , Thermodynamic properties and shear viscosity over entropy-density ratio of the nuclear fireball in a quantum-molecular dynamics model . Phys. Rev. C 88 , 024604 ( 2013 ). doi: 10.1103/PhysRevC.88.024604 http://doi.org/10.1103/PhysRevC.88.024604 . Thermodynamic properties and shear viscosity over entropy-density ratio of the nuclear fireball in a quantum-molecular dynamics model J. Xu , L.W. Chen , C.M. Ko et al. , Shear viscosity of neutron-rich nucleonic matter near its liquid?gas phase transition . Phys. Lett. B 727 , 244 ( 2013 ). doi: 10.1016/j.physletb.2013.10.051 http://doi.org/10.1016/j.physletb.2013.10.051 . Shear viscosity of neutron-rich nucleonic matter near its liquid?gas phase transition D.Q. Fang , Y.G. Ma and C.L. Zhou , Shear viscosity of hot nuclear matter by the mean free path method . Phys. Rev. C 89 , 047601 ( 2014 ). doi: 10.1103/PhysRevC.89.047601 http://doi.org/10.1103/PhysRevC.89.047601 . Shear viscosity of hot nuclear matter by the mean free path method C.Q. Guo , Y. G. Ma , W.B. He et al. , Isovector dipole resonance and shear viscosity in low energy heavy-ion collisions . Phys. Rev. C 95 , 054622 ( 2017 ). doi: 10.1103/PhysRevC.95.054622 http://doi.org/10.1103/PhysRevC.95.054622 . Isovector dipole resonance and shear viscosity in low energy heavy-ion collisions X.G. Deng , Y. G. Ma , and Y. X. Zhang , Green-Kubo formula for Boltzmann and Fermi-Dirac statistics . Eur. Phys. J. A 57 , 242 ( 2021 ). doi: 10.1140/epja/s10050-021-00550-4 http://doi.org/10.1140/epja/s10050-021-00550-4 . Green-Kubo formula for Boltzmann and Fermi-Dirac statistics H.L. Liu , D.D. Han , Y.G. Ma et al. , Network structure of thermonuclear reactions in nuclear landscape . Sci. China-Phys. Mech. Astron. 63 , 112062 ( 2020 ). doi: 10.1007/s11433-020-1552-2 http://doi.org/10.1007/s11433-020-1552-2 . Network structure of thermonuclear reactions in nuclear landscape H.L. Liu , D.D. Han , P. Ji et al. , Reaction rate weighted multilayer nuclear reaction network . Chin. Phys. Lett. 37 , 112601 ( 2020 ). doi: 10.1088/0256-307X/37/11/112601 http://doi.org/10.1088/0256-307X/37/11/112601 . Reaction rate weighted multilayer nuclear reaction network C. A. Bertulani , Topology of nuclear reaction networks of interest for astrophysics . Sci. China-Phys. Mech. Astron. 63 , 112063 ( 2020 ). doi: 10.1007/s11433-020-1553-6 http://doi.org/10.1007/s11433-020-1553-6 . Topology of nuclear reaction networks of interest for astrophysics Y. H. Ge , Y. Zhang , J.N. Hu , Effects of pairing correlation on neutron drop . Sci. China-Phys. Mech. Astron. 63 , 242011 ( 2020 ). doi: 10.1007/s11433-019-1455-9 http://doi.org/10.1007/s11433-019-1455-9 . Effects of pairing correlation on neutron drop T.L. Ma , B. Guo , D.Y. Pang et al. , New determination of astrophysical 14C(n, γ )15C reaction rate from the spectroscopic factor of 15 C . Sci. China-Phys. Mech. Astron. 63 , 212021 ( 2020 ). doi: 10.1007/s11433-019-9411-2 http://doi.org/10.1007/s11433-019-9411-2 . New determination of astrophysical 14C(n, γ)15C reaction rate from the spectroscopic factor of 15C C.J. Jiang , Y. Qiang , D.-W. Guan et al. , From finite nuclei to neutron stars: The essential role of high-order density dependence in effective forces . Chin. Phys. Lett. 38 , 052101 ( 2021 ). doi: 10.1088/0256-307X/38/5/052101 http://doi.org/10.1088/0256-307X/38/5/052101 . From finite nuclei to neutron stars: The essential role of high-order density dependence in effective forces J. Xu , Constraining isovector nuclear interactions with giant dipole resonance and neutron skin in 208 Pb from a Bayesian approach . Chin. Phys. Lett. 38 , 042101 ( 2021 ). doi: 10.1088/0256-307X/38/4/042101 http://doi.org/10.1088/0256-307X/38/4/042101 . Constraining isovector nuclear interactions with giant dipole resonance and neutron skin in 208Pb from a Bayesian approach C.-W. Ma , H.L. Wei , X.Q. Liu et al. , Nuclear fragments in projectile fragmentation reactions . Prog. Part. Nucl. Phys. 121 , 103911 ( 2021 ). doi: 10.1016/j.ppnp.2021.103911 http://doi.org/10.1016/j.ppnp.2021.103911 . Nuclear fragments in projectile fragmentation reactions J. Pochodzalla , T. Möhlenkamp , T. Rubehn et al. (ALADIN Collaboration), Probing the nuclear liquid-gas phase transition . Phys. Rev. Lett. 75 , 1040 ( 1995 ). doi: 10.1103/PhysRevLett.75.1040 http://doi.org/10.1103/PhysRevLett.75.1040 . Probing the nuclear liquid-gas phase transition Y. G. Ma , A. Siwek , J. Péter et al. (INDRA Collaboration), Surveying the nuclear caloric curve . Phys. Lett. B 390 , 41 ( 1997 ). doi: 10.1016/S0370-2693(96)01372-X http://doi.org/10.1016/S0370-2693(96)01372-X . Surveying the nuclear caloric curve J.B. Natowitz , R. Wada , K. Hagel et al. , Caloric curves and critical behavior in nuclei . Phys. Rev. C 65 , 034618 ( 2002 ). doi: 10.1103/PhysRevC.65.034618 http://doi.org/10.1103/PhysRevC.65.034618 . Caloric curves and critical behavior in nuclei Y.G. Ma , J.B. Natowitz , R. Wada et al. , Critical behavior in light nuclear systems: Experimental aspects . Phys. Rev. C 71 , 054606 ( 2005 ). doi: 10.1103/PhysRevC.71.054606 http://doi.org/10.1103/PhysRevC.71.054606 . Critical behavior in light nuclear systems: Experimental aspects A. Rios , Effective interaction dependence of the liquid-gas phase transition in symmetric nuclear matter . Nucl. Phys. A 845 , 58 - 87 ( 2010 ). doi: 10.1016/j.nuclphysa.2010.05.057 http://doi.org/10.1016/j.nuclphysa.2010.05.057 . Effective interaction dependence of the liquid-gas phase transition in symmetric nuclear matter R. Wang , Y. G. Ma , R. Wada et al. , Nuclear liquid-gas phase transition with machine learning . Phys. Rev. Research 2 , 043202 ( 2020 ). doi: 10.1103/PhysRevResearch.2.043202 http://doi.org/10.1103/PhysRevResearch.2.043202 . Nuclear liquid-gas phase transition with machine learning S. Yang , X.D. Sun , J. Geng et al. , Liquid-gas phase transition of thermal nuclear matter and the in-medium balance between nuclear attraction and repulsion . Phys. Rev. C 103 , 014304 ( 2021 ). doi: 10.1103/PhysRevC.103.014304 http://doi.org/10.1103/PhysRevC.103.014304 . Liquid-gas phase transition of thermal nuclear matter and the in-medium balance between nuclear attraction and repulsion S.S. Wang , Y.G. Ma , X.G. Cao et al. , Hard-photon production and its correlation with intermediate-mass fragments in a framework of a quantum molecular dynamics model . Phys. Rev. C 102 , 024620 ( 2020 ). doi: 10.1103/PhysRevC.102.024620 http://doi.org/10.1103/PhysRevC.102.024620 . Hard-photon production and its correlation with intermediate-mass fragments in a framework of a quantum molecular dynamics model O. Savchuk , V. Vovchenko , R. V. Poberezhnyuk et al. , Traces of the nuclear liquid-gas phase transition in the analytic properties of hot QCD . Phys. Rev. C 101 , 035205 ( 2020 ). doi: 10.1103/PhysRevC.101.035205 http://doi.org/10.1103/PhysRevC.101.035205 . Traces of the nuclear liquid-gas phase transition in the analytic properties of hot QCD W. Lin , P. Ren , H. Zheng et al. , Solidarity of signal of measures for the liquid-gas phase transition in the statistical multifragmentation model . Phys. Rev. C 99 , 054616 ( 2019 ). doi: 10.1103/PhysRevC.99.054616 http://doi.org/10.1103/PhysRevC.99.054616 . Solidarity of signal of measures for the liquid-gas phase transition in the statistical multifragmentation model R. Wada , W. Lin , P. Ren et al. , Experimental liquid-gas phase transition signals and reaction dynamics . Phys. Rev. C 99 , 024616 ( 2019 ). doi: 10.1103/PhysRevC.99.024616 http://doi.org/10.1103/PhysRevC.99.024616 . Experimental liquid-gas phase transition signals and reaction dynamics M.E. Fisher , The theory of equilibrium critical phenomena . Rep. Prog. Phys. 30 , 615 ( 1969 ). doi: 10.1088/0034-4885/30/2/306 http://doi.org/10.1088/0034-4885/30/2/306 . The theory of equilibrium critical phenomena Y.G. Ma and W.Q. Shen , Onset of multifragmentation in intermediate energy light asymmetrical collisions . Phys. Rev. C 51 , 710 ( 1995 ). doi: 10.1103/PhysRevC.51.710 http://doi.org/10.1103/PhysRevC.51.710 . Onset of multifragmentation in intermediate energy light asymmetrical collisions M. Huang , R. Wada , Z. Chen et al. , Power law behavior of the isotope yield distributions in the multifragmentation regime of heavy ion reactions . Phys. Rev. C 82 , 054602 ( 2010 ). doi: 10.1103/PhysRevC.82.054602 http://doi.org/10.1103/PhysRevC.82.054602 . Power law behavior of the isotope yield distributions in the multifragmentation regime of heavy ion reactions J.B. Elliott , L.G. Moretto , L. Phair et al. , Liquid to vapor phase transition in excited nuclei . Phys. Rev. Lett. 88 , 042701 ( 2002 ). doi: 10.1103/PhysRevLett.88.042701 http://doi.org/10.1103/PhysRevLett.88.042701 . Liquid to vapor phase transition in excited nuclei Y.G. Ma , R. Wada , K. Hagel et al. , Evidence of critical behavior in the disassembly of nuclei with A-36 . Phys. Rev. C 69 , 031604(R) ( 2004 ). doi: 10.1103/PhysRevC.69.031604 http://doi.org/10.1103/PhysRevC.69.031604 . Evidence of critical behavior in the disassembly of nuclei with A-36 C.O. Dorso , V.C. Latora , and A. Bonasera , Signals of critical behavior in fragmenting finite systems . Phys. Rev. C 60 , 034606 ( 1999 ). doi: 10.1103/PhysRevC.60.034606 http://doi.org/10.1103/PhysRevC.60.034606 . Signals of critical behavior in fragmenting finite systems B. Borderie , Dynamics and thermodynamics of the liquid-gas phase transition in hot nuclei studied with the INDRA array . J. Phys. G 28 , 217(R) ( 2002 ). doi: 10.1088/0954-3899/28/8/201 http://doi.org/10.1088/0954-3899/28/8/201 . Dynamics and thermodynamics of the liquid-gas phase transition in hot nuclei studied with the INDRA array J. Su , L. Zhu , and E.X. Xiao , Fluctuations of the largest fragment charge in projectile fragmentation and its nonequilibrium effect . Phys. Rev. C 105 , 024608 ( 2022 ). doi: 10.1103/PhysRevC.105.024608 http://doi.org/10.1103/PhysRevC.105.024608 . Fluctuations of the largest fragment charge in projectile fragmentation and its nonequilibrium effect Ph. Chomaz , V. Duflot , and F. Gulminelli , Caloric curves and energy fluctuations in the microcanonical liquid-gas phase transition . Phys. Rev. Lett. 85 , 3587 ( 2000 ). doi: 10.1103/PhysRevLett.85.3587 http://doi.org/10.1103/PhysRevLett.85.3587 . Caloric curves and energy fluctuations in the microcanonical liquid-gas phase transition Y. G. Ma , Application of information theory in nuclear liquid gas phase transition . Phys. Rev. Lett. 83 , 3617 ( 1999 ). doi: 10.1103/PhysRevLett.83.3617 http://doi.org/10.1103/PhysRevLett.83.3617 . Application of information theory in nuclear liquid gas phase transition C.W. Ma , Y.P. Liu , H.L. Wei et al. , Determination of neutron-skin thickness using configurational information entropy . Nucl. Sci. Tech. 33 , 6 ( 2022 ). doi: 10.1007/s41365-022-00997-0 http://doi.org/10.1007/s41365-022-00997-0 . Determination of neutron-skin thickness using configurational information entropy F. Li and G. Chen , The evolution of information entropy components in relativistic heavy-ion collisions . Eur. Phys. J. A 56 , 167 ( 2020 ). doi: 10.1140/epja/s10050-020-00169-x http://doi.org/10.1140/epja/s10050-020-00169-x . The evolution of information entropy components in relativistic heavy-ion collisions Y.G. Ma , Zpif’s law in the liquid gas phase transition of nuclei . Eur. Phys. J. A 6 , 367 ( 1999 ). doi: 10.1007/s100500050355 http://doi.org/10.1007/s100500050355 . Zpif’s law in the liquid gas phase transition of nuclei Y.G. Ma , Moment analysis and Zipf law . Eur. Phys. J. A 30 , 227 ( 2006 ). doi: 10.1140/epja/i2006-10119-4 http://doi.org/10.1140/epja/i2006-10119-4 . Moment analysis and Zipf law J. Steinheimer and J. Randrup , Spinodal amplification of density fluctuations in fluid-dynamical simulations of relativistic nuclear collisions . Phys. Rev. Lett. 109 , 212301 ( 2012 ). doi: 10.1103/PhysRevLett.109.212301 http://doi.org/10.1103/PhysRevLett.109.212301 . Spinodal amplification of density fluctuations in fluid-dynamical simulations of relativistic nuclear collisions J. Steinheimer and J. Randrup , Spinodal density enhancements in simulations of relativistic nuclear collisions . Phys. Rev. C 87 , 054903 ( 2013 ). doi: 10.1103/PhysRevC.87.054903 http://doi.org/10.1103/PhysRevC.87.054903 . Spinodal density enhancements in simulations of relativistic nuclear collisions J. Steinheimer and J. Randrup , Spinodal amplification and baryon number fluctuations in nuclear collisions at NICA . Eur. Phys. J. A 52 , 239 ( 2016 ). doi: 10.1140/epja/i2016-16239-2 http://doi.org/10.1140/epja/i2016-16239-2 . Spinodal amplification and baryon number fluctuations in nuclear collisions at NICA X.G. Deng , P. Danielewicz , Y. G. Ma et al. , submitted to Phys. Rev. C. Phys. Rev. C. F. Li and C. M. Ko , Spinodal instabilities of baryon-rich quark matter in heavy ion collisions . Phys. Rev. C 95 , 055203 ( 2017 ). doi: 10.1103/PhysRevC.95.055203 http://doi.org/10.1103/PhysRevC.95.055203 . Spinodal instabilities of baryon-rich quark matter in heavy ion collisions J. Aichelin , “Quantum” molecular dynamics—a dynamical microscopic n-body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions . Phys. Rep. 202 , 233 ( 1991 ). doi: 10.1016/0370-1573(91)90094-3 http://doi.org/10.1016/0370-1573(91)90094-3 . “Quantum” molecular dynamics—a dynamical microscopic n-body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions Ch. Hartnack , et al. , Modelling the many-body dynamics of heavy ion collisions: Present status and future perspective . Eur. Phys. J. A 1 , 151 ( 1998 ). doi: 10.1007/s100500050045 http://doi.org/10.1007/s100500050045 . Modelling the many-body dynamics of heavy ion collisions: Present status and future perspective C. C. Guo , J. Su , and L. Zhu , Secondary decay effects of the isospin fractionation in the projectile fragmentation at GeV/nucleon . Nucl. Sci. Tech. 31 , 123 ( 2020 ). doi: 10.1007/s41365-020-00832-4 http://doi.org/10.1007/s41365-020-00832-4 . Secondary decay effects of the isospin fractionation in the projectile fragmentation at GeV/nucleon H. Yu , D.Q. Fang , and Y. G. Ma , Investigation of the symmetry energy of nuclear matter using isospin-dependent quantum molecular dynamics . Nucl. Sci. Tech. 31 , 61 ( 2020 ). doi: 10.1007/s41365-020-00766-x http://doi.org/10.1007/s41365-020-00766-x . Investigation of the symmetry energy of nuclear matter using isospin-dependent quantum molecular dynamics Y. J. He , C. C. Guo , J. Su et al. , Study on deuteron formation mechanism in nucleon-induced reactions . Nucl. Sci. Tech. 31 , 84 ( 2020 ). doi: 10.1007/s41365-020-00788-5 http://doi.org/10.1007/s41365-020-00788-5 . Study on deuteron formation mechanism in nucleon-induced reactions H.L. Liu , Y.G. Ma , A. Bonasera et al. , Mean free path and shear viscosity in central 129 Xe + 119 Sn collisions below 100 MeV/nucleon . Phys. Rev. C 96 , 064604 ( 2017 ). doi: 10.1103/PhysRevC.96.064604 http://doi.org/10.1103/PhysRevC.96.064604 . Mean free path and shear viscosity in central 129Xe + 119Sn collisions below 100 MeV/nucleon S. Wuenschel , A. Bonasera , L.W. May et al. , Measuring the temperature of hot nuclear fragments . Nucl. Phys. A 843 ( 2010 ) 1 - 13 . doi: 10.1016/j.nuclphysa.2010.04.013 http://doi.org/10.1016/j.nuclphysa.2010.04.013 . Measuring the temperature of hot nuclear fragments C. M. Ko and F. Li , Density fluctuations in baryon-rich quark matter . Nucl. Sci. Tech. 27 , 140 ( 2016 ). doi: 10.1007/s41365-016-0141-3 http://doi.org/10.1007/s41365-016-0141-3 . Density fluctuations in baryon-rich quark matter X. G. Deng , Y. G. Ma and M. Veselský , Thermal and transport properties in central heavy-ion reactions around a few hundred MeV/nucleon . Phys. Rev. C 94 , 044622 ( 2016 ). doi: 10.1103/PhysRevC.94.044622 http://doi.org/10.1103/PhysRevC.94.044622 . Thermal and transport properties in central heavy-ion reactions around a few hundred MeV/nucleon 10.1007/s41365-022-01040-y This work was partial supported by the National Natural Science Foundation of China (Nos. 11890714, 11891070, 12075061, and 12147101), Key Research Program of the CAS (No. XDB34000000), China Postdoctoral Science Foundation (No. 2019M661332), Postdoctoral Innovative Talent Program of China (No. BX20200098) ; Shanghai Natural Science Foundation (No. 20ZR1404100), and Guangdong Major Project of Basic and Applied Basic Research (No. 2020B0301030008) . 2022-03-29 2022-04-18 2022-04-23 2022-06-15 2022-07-26 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01040-y http://dx.doi.org/10.1007/s41365-022-01040-y . 2022 Nuclear Science and Techniques 05 33