Determining the temperature in heavy-ion collisions with multiplicity distribution
DDetermining the temperature in heavy-ion collisions with multiplicity distribution
Yi-Dan Song, Rui Wang, ∗ Yu-Gang Ma, † Xian-Gai Deng, and Huan-Ling Liu Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai , China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai , China (Dated: January 27, 2021)By relating the charge multiplicity distribution and the temperature of a de-exciting nucleusthrough a deep neural network, we propose that the charge multiplicity distribution can be used asa thermometer of heavy-ion collisions. Based on an isospin-dependent quantum molecular dynamicsmodel, we study the caloric curve of reaction
Pd + Be with the apparent temperature determinedthrough the charge multiplicity distribution. The caloric curve shows a characteristic signatureof nuclear liquid-gas phase transition around the apparent temperature T ap = 6 . PACS numbers: 24.10.Ai, 25.70.Gh, 25.70.Mn, 27.60.+j
I. INTRODUCTION
Understanding the properties of nuclear matter is oneof the major objectives in nuclear physics. At zero tem-perature, the properties of nuclear matter have beenstudied extensively, and its equation of state (EOS), in-cluding its isospin dependence, i.e., symmetry energy, hasbeen determined relatively well [1–3], while its propertiesat finite temperature are relatively little touched upon.Among these properties, two noticeable examples are thenuclear liquid-gas phase transition [4–16] and the temper-ature dependence of the ratio of shear viscosity to entropydensity ( η/s ) [17–22]. The latter is also connected to thenuclear giant dipole resonance at finite temperature [23–25], since both of them are related to the two-body dis-sipation of nucleons.The difficulties of studying the finite temperatureproperties of nuclear matter mainly come from the prepa-ration of a finite temperature nuclear system, as well asthe determination of its temperature. Heavy-ion colli-sions (HICs) at intermediate-to-low energies provide apossible venue of investigating the finite temperatureproperties of nuclear matter [26]. During the reaction,a transient excited system is formed, and commonly itcan be regarded as a (near)-equilibrium state [27, 28],since the evolution of its constituent nucleons is suffi-ciently short comparing with the global evolution. Itstemperature can be accessed by, e.g., energy spectrathrough moving source fitting [26], excited state popula-tions [29, 30], (double)-isotope ratios [31–33], or quadru-ple momentum fluctuation [34] etc. For a reliable ther-mometer of HICs, we require it is insensitive to both thecollective effects and the secondary decay of unstable nu-clei after the system disintegrates, which is commonlyhard to achieve. Besides that, because of the difficulty ∗ Electronic address: [email protected] † Electronic address: [email protected] of examining the accuracy of the apparent temperatureobtained through these thermometers, it is not a trivialtask to propose different ways of determining the appar-ent temperature, and thus provide more opportunities ofcrosscheck.Machine-learning techniques [35, 36], which have beenapplied extensively in physics [37] due to their ability ofrecognizing and characterizing complex sets of data, pro-vide an alternative and peculiar way of determining theapparent temperature of HICs at intermediate-to-low en-ergies. Besides the common uses like particle identifica-tion and tagging in experiments, machine-learning tech-niques have various novel applications in physics, e.g.,solving quantum many-body problem [38, 39], analyz-ing strong gravitational lenses [40], classifying phases ofmatter [41–44], extrapolating the cross section of nuclearreactions [45], and instructing single crystal growth [46]and optimizing experimental control [47]. In a recentwork of studying nuclear liquid-gas phase transition withmachine-learning techniques [48], it has been shown thatmachine-learning techniques can capture the essentialfeatures of HICs directly from experimental final-statecharge multiplicity distribution.In the present letter, we propose that, with the help ofmachine-learning technique, the charge multiplicity dis-tribution can be applied to determine the apparent tem-perature of HICs at intermediate-to-low energies. Ourbasic methodology is as follows. We first obtain the mul-tiplicity of fragments of an excited nuclear source withgiven temperature based on a theoretical model, e.g.,transport models [49, 50], statistical models [51, 52] ortheir hybrid [53–55]. We then relate the final-state chargemultiplicity distribution with the temperature of thesource via a deep neural network (DNN). This relationcan be employed to determine the apparent temperatureof a certain transient state during HICs at intermediate-to-low energies through their final-state charge multiplic-ity distribution. We use the above method to determinethe apparent temperature of a fragmentation reaction
Pd + Be. We further test the viability of the present a r X i v : . [ nu c l - t h ] J a n method by comparing it with the momentum fluctuationthermometer [34], and by studying the characteristic sig-nature of the caloric curve with the apparent tempera-ture determined through the above method. To adoptthe above method to realistic HICs analyses with ex-perimental charge multiplicity distribution relies on theprecise determination of the fragment multiplicities fromtheoretical model. Nevertheless, as a viability quest, inthe present work, we employ an isospin-dependent quan-tum molecular dynamics (IQMD) model [56] to simulatethe de-excitation of the finite temperature nuclear source,and do not require it to describe precisely the experimen-tal fragment multiplicities. More accurate description ofthe experimental fragment multiplicities can be achievedthrough, e.g., combining certain transport model withstatistical model of multi-fragmentation [55], and it isbeyond the scope of the present work. II. METHODOLOGY
In the present work, we focus on fragmentation reac-tions [57, 58], i.e., central
Pd + Be collision with in-cident energies ranging from 20 to 400 A MeV, and try todetermine their apparent temperature through its final-state charge multiplicity distribution M c ( Z cf ), where Z cf represents the charge number of the final charged frag-ments, ranging from 1 to the charge number of the reac-tion system. In the process of intermediate-to-low energyHICs, especially for fragmentation reactions, the two in-cident nuclei collide and then form a compound-like sys-tem. This compound-like system is regarded as a (near-)equilibrium system, and we can mimic approximatelythis transient state of the reaction by a nuclear source ora finite nucleus at given temperature, with mass number A and proton number Z , which are the same as those inthe reaction Pd + Be [59]. This finite temperaturenuclear source has been employed to study finite-size scal-ing phenomenon [60]. We first simulate the evolution ofthe nuclear source
Sn ( A = 112 and Z = 50) at differ-ent initial temperature T model , and obtain their M c ( Z cf )based on the IQMD model. These simulations provideus M c ( Z cf ) from a nuclear source with a given tempera-ture, which is difficult to obtain directly from HICs. Wethen establish a relation between the source temperature T model and M c ( Z cf ) through a DNN, which recasts thecomplex relation into a non-linear map through its neu-rons. Based on this relation, the apparent temperature ofthe reaction Pd + Be can be determined through itsfinal-state M c ( Z cf ). In Fig. 1, we show the basic proce-dure of the proposed method to determine apparent tem-perature of fragmentation reaction Pd + Be throughthe nuclear source
Sn.
FIG. 1: The basic procedure of determining the T ap of Pd+ Be through its final-state M c ( Z cf ) with DNN. A. IQMD model
The IQMD model, a many-body theory to describe thedynamics of HICs, can be derived from a time-dependentHartree theory with Gaussian single-particle wave func-tion φ i ( (cid:126)r, t ), φ i ( (cid:126)r, t ) = 1(2 πL ) / exp (cid:104) − [ (cid:126)r − (cid:126)r i ( t )] L + i(cid:126)p i ( t ) · (cid:126)r (cid:126) (cid:105) , (1)with its spatial center (cid:126)r i ( t ) and momentum center (cid:126)p i ( t )as variational parameters. Other quantities can be ob-tained subsequently through (cid:126)r i ( t ) and (cid:126)p i ( t ). In the aboveequation, L is the square of the width of Gaussian wavepacket and is set to be 2.18 fm . Through a productof single-particle wave functions φ i ( (cid:126)r, t ), we can get thesystem wave function, ψ ( (cid:126)r , ...(cid:126)r n , t ) = A (cid:89) i =1 φ i ( (cid:126)r, t ) , (2)where A is the mass number of the system. The potentialenergy U in the IQMD model includes Skyrme, Yukawa,symmetry, momentum-dependent and Coulomb terms, U = U Sky + U Yuk + U sym + U MDI + U Coul . (3)Detailed descriptions of the IQMD model, including thepotential and equations of motion of (cid:126)r i ( t ) and (cid:126)p i ( t ), canbe found in Ref. [56, 61]. Numerous applications ofIQMD have been made for different observable, and somerecent ones can be found in Ref. [62–66].In the IQMD model, the initial nucleons are generatedthrough the local density approximation. For a ground-state nucleus, we generate A Gaussian wave-packets, withtheir spatial center (cid:126)r i ( t = 0) sampled randomly withina sphere of radius given by r A / with r = 1 .
12 fm,and their momentum center (cid:126)p i ( t = 0) sampled follow-ing the zero-temperature Fermi-Dirac distribution. A fi-nite nucleus at a given temperature T can be generatedby sampling (cid:126)p i ( t = 0) according to finite temperatureFermi-Dirac distribution [19], which is given by, f ( (cid:15) ) = 1exp( (cid:15) − µT ) + 1 , (4)where (cid:15) = (cid:112) p + m + U is the single particle energy,and µ is the chemical potential. The parameters p , m and U are momentum, mass and potential energy, respec-tively. For simplicity, we have omitted the contributionfrom momentum-dependent part in the above equation. B. Deep neural network (a) bxWz n x x W n W )( zfy (b) )( cfc ZM )(2 cfc ZM )( cfc ZZM ... ... ...
Input ...... ......
Hidden layers
DNNap T Output ...... ......
FIG. 2: (a). A single artificial neuron, with n inputs labelledas x through x n and an output y . The output of the neuronis computed by applying the activation function f ( z ) to theproduct of x , weights W , and biases b , e.g., z = W · x + b . (b).The feed-forward neural network used in the present work,consisting of an input layer, an output layer and four hiddenlayers. The input of the network is the charge-weighted chargemultiplicity distribution ZM c ( Z cf ). In the present work, we adopt a feed-forward DNN toestablish the relation between the temperature T Model ofthe de-exciting nuclear source generated in IQMD and itsfinal-state charge multiplicity distribution M c ( Z cf ). Wetreat M c ( Z cf ) as the input image, while its correspondingtemperature as its label. The DNN contains successivelyone input layer, several hidden layers, and one outputlayer. Each layer generates its output z through a ma-trix multiplication of its input x , i.e., z = W · x + b .The elements in the matrix W are known as weights andin the vector b as biases. In a normal full-connectedneural network these parameters are single values. Theneuron is then followed by an activation function f ( z ),which turns a linear transform to a non-linear one. Com-monly used activation functions are sigmoid , tanh , and ReLU (rectified linear unit). f ( z ) is then used as theinput of the next layer. In the present work, the neuralnetwork can be treated as a functional T DNNap = g (cid:2) M c ( Z cf ); W , b (cid:3) , (5)which relates non-linearly a given input M c ( Z cf ), a vec-tor with 50 elements in our case, to an output predictedapparent temperature T DNNap . We employ four hiddenlayers, with each consists of 32 artificial neurons. The input layer and the first three hidden layers are followedby
ReLU , while the last hidden layer and the output pre-dicted T DNNap are connected directly by the matrix mul-tiplication. The sketch of the DNN used in the presentwork can be seen in Fig. 2.We train the network based on a data set (cid:8) M c ( Z cf ) , T Model (cid:9) , to minimize the cost function, thedifference between the given temperature and the DNNprediction, i.e., ( T model − T DNNap ) , by adjusting its pa-rameters W and b . The optimization is fulfilled by the Adam [67] package in
Tensorflow . During training thenetwork, we use an exponential decreasing learning rate α = 10 − + (10 − − − ) exp( − i/ i thetraining epoch and it is equal to 10,000. To preventthe network from over fitting the data set, we includea standard l regularization term, i.e., a term propor-tional to the norm of the weight W and the bias b , l ( (cid:107) W (cid:107) / (cid:107) b (cid:107) / l a positive number, in thecost function of the neural network. The l regulariza-tion prevents the weights and biases from increasing toarbitrary large values during the optimization. III. RESULTS AND DISCUSSIONA. Apparent temperature − − −
10 110 M u l t i p li c i t y = 2 MeV model T = 5 MeV model
T = 8 MeV model
T = 11 MeV model
T = 14 MeV model
T = 17 MeV model
TSn
Nuclear source A=112,Z=50
FIG. 3: The charge distribution of fragments from excitednuclear source
Sn at several T model . For each T model , weshow the simulated data of one run consisting of 2,000 eventsfrom the IQMD model. Based on the IQMD model, we simulate the de-excitation process of a
Sn nucleus at different T model .The Sn nucleus has the same mass and charge num-ber with the
Pd + Be reaction, and it can be re-garded as a nuclear source to mimic the transient ex-cited state of the reaction. We perform 50 runs for each T model , with each run consists of 2 ,
000 events. T model ranges from 0 to 20 MeV with 1 MeV interval. Fig. 3displays charge distributions of fragments from hot nu-clear source Sn at several T model with the data sampleof one run (2 ,
000 events) at each temperature. Thesefragment charge distributions exhibit typical changes ofdisassembly mechanism of hot nuclei with temperature[9, 14], i.e., from evaporation mechanism at lower tem-perature (e.g. T model = 2 MeV), to multifragmentation atmedium temperature (e.g. T model = 8 MeV), till vapor-ization at higher temperature (e.g. T model = 17 MeV).According to the shapes of these charge distributionsversus temperature, a nuclear liquid-gas phase transitionshall begin to happen at a certain moderate temperaturefor this system [14, 48, 60, 68].We can train the DNN once we get the above-mentioned charge multiplicity distributions M c ( Z cf ). Inthe present work, the total 50 ×
21 charge multiplicitydistributions M c ( Z cf ), and their corresponding T model ,are treated as the images and labels, respectively, andthey form the data set (cid:8) M c ( Z cf ) , T model (cid:9) of the DNN.The data set is further divided into training set and test-ing set, each contains half of the total data set. After wetrained the DNN with the training set, i.e., determiningthe parameters W and b in Eq. (5) to best reproducethe given T model , one can predict the apparent temper-ature with a given M c ( Z cf ) through Eq. (5). We showin Fig.4 the histogram of the DNN’s predicting error σ T ,i.e., the difference between the original T model and itsDNN prediction T DNNap , of the testing set. The standarderror of the DNN prediction is about 0 .
62 MeV, which issmall enough for further analyses based on the apparenttemperature obtained through the present way. -4 -3 -2 -1 0 1 2 3 4050100150200250 counts s T = T model -T DNNap
A=112,Z=50Nuclear source(
Sn) < s T2 > XX represents range of ZX=1,Z[1,5] X=2,Z[1,10] X=3,Z[1,20] X=4,Z[1,30] X=5,Z[1,40] X=6,Z[1,50]
FIG. 4: The histogram of errors σ T between the predicted ap-parent temperature from DNN T DNNap and the original T model given in the IQMD simulations for the excited nuclear source(finite temperature Sn). The red line is the Gaussian fittingof the histogram. The inset shows the testing accuracy (cid:104) σ (cid:105) for different charge multiplicity distributions. X representsdifferent ranges of charge multiplicity distribution. In the inset of Fig. 4, we show the dependence of thetesting accuracy on the range of charge multiplicity dis-tribution, i.e., only the charge multiplicity distributionswithin the given range (represented by X in the inset)are used in training and testing the DNN. We notice fromthe inset that the light fragments, i.e., Z ∈ [1 ,
5] play ma-jor role in determining the apparent temperature, whileincluding heavier fragments do help to increase the accu-racy. In another perspective, this feature actually indi- cates the superiority of the present method to the tradi-tional isotopic ratio thermometer, since for the later onlythe information of light fragments is taken into consid-eration. At present, we only use the charge multiplicitydistribution to predict the apparent temperature by theDNN. In principle, the information in momentum spacecan be included in the present method for better accu-racy.After establishing the relation between the apparenttemperature and M c ( Z cf ) through training the DNN, weturn to determine the apparent temperature of Pd + Be fragmentation reactions. We first examine the reac-tion dynamics of
Pd + Be within the IQMD model.We simulate the reactions
Pd + Be with incident en-ergy E lab ranging from 20 A MeV to 400 A MeV, and foreach incident energy we employ 2 ,
000 events. As theincident energy increases, the reaction becomes more vi-olent, and the apparent temperature of the reaction in-creases. In fragmentation reaction, the projectile andtarget nuclei initially form a compound-like system afterthey collide each other. In the early stage of the reac-tion, only a small number of nucleons evaporate or eject,so the mass of the compound-like system approximatelyequals to the sum of the projectile and the target. Weexhibit in Fig. 5 the time evolution of the central densityof the heaviest fragment formed in
Pd + Be reactionat different incident energies from the IQMD model. Thedensity in the IQMD model is obtained through the sumof the single-particle density, ρ ( (cid:126)r ) = A (cid:88) i =1 ρ i = A (cid:88) i =1 πL ) / exp (cid:26) − (cid:2) (cid:126)r − (cid:126)r i ( t ) (cid:3) L (cid:27) . (6)We notice from the figure that the central density ofthe heaviest fragment exhibits some oscillations at thebeginning. This reflects the breathing mode caused bythe initial compression of the system, since before thecompound-like system dismantling, the largest fragmentis the compound-like system itself. The black dotted linerepresents ρ = 0 .
156 fm − , which is the initial centraldensity of the nuclear source ( Sn) at finite temper-ature we generate for training the DNN. Therefore, theinitial compound-like system with density around nuclearsaturation density can be mimicked reasonably by an ex-cited nuclear source with certain given temperature.With the final-state charge multiplicity distribution M c ( Z cf ) of the reaction Pd + Be simulated with theIQMD, we obtain their apparent temperature throughthe trained DNN in Eq. (5). We plot the obtained ap-parent temperature T ap of the reaction Pd + Be atdifferent incident energies E lab in Fig. 6. Since the fi-nite temperature nuclear source generated to train theDNN is initialized at around nuclear saturation density, T ap obtained through the present method actually re-flects the apparent temperature of the early stage of thecompound system in reaction Pd + Be. In Fig. 6,the black symbols are the predicted T ap of compoundsystem at the early stage of the reaction by DNN. We lab = 20A MeVE lab = 40A MeV E lab = 60A MeVE lab =100A MeV E lab =150A MeVE lab =200A MeV E lab =400A MeV t(fm/c) r (fm -3 ) Pd+ Be r =0.156 fm -3 FIG. 5: Time evolution of the central density of the heaviestfragment in reaction
Pd + Be with the IQMD simulation.Different lines denote the results from different incident ener-gies from 20 A MeV to 400 A MeV. The black dashed line rep-resents the initial central density of the nuclear source
Sn. further test the effects of imperfect acceptance and effi-ciency on the obtained T ap , by applying an acceptanceand efficiency cut, respectively, in the IQMD simulations.It shows that these effects are negligible, which indicatesthe robustness of the present method on imperfect ex-perimental acceptance and efficiency.
50 100 150 200 250 300 350 400246810
Pd+ Be T ap by DNN T ap by MFT of d T ap (MeV) E lab (A MeV) FIG. 6: The apparent temperature T ap of the reaction Pd+ Be at different E lab , predicted by DNN (black dots) andMFT (red dots). The solid lines are their fittings to guide forthe eye. In order to provide a crosscheck of the presentmethod, we employ a momentum fluctuation thermome-ter (MFT) [34] to determine the apparent temperatureof the reaction
Pd + Be simulated by IQMD. InMFT, the distribution of a certain species of light frag-ment (deuteron in the present work) is assumed to beMaxwellian, and the temperature of the system is relatedto the variance of its quadrupole moment σ through σ = 4 A m T , (7)where m is the mass of a nucleon and A is the mass number of the fragment. We add in Fig. 6 the appar-ent temperature predicted by MFT with red dots. Wenotice that the T ap of the two methods are very close,which increased the reliability of the extracted apparenttemperature. B. Caloric curve
Pd+ Be E * /A by IQMD model Fitting curve T ap (MeV) E * /A(MeV) c c T ap (MeV) FIG. 7: Caloric curve of the reaction
Pd + Be. Blackopen squares represent the result based on the IQMD model,with T ap determined by DNN using M c ( Z cf ). Blue dashedline is its polynomial fit. The inset shows the specific heatcapacity ˜ c derived from the fitted formula. The caloric curve, i.e., the apparent temperature asa function of excitation energy per nucleon E ∗ /A ofHICs has been considered as an important probe tothe existence of the nuclear liquid-gas phase transition[7, 12, 13, 68]. In order to examine the validity of T ap determined by DNN using the charge multiplicity distri-bution M c ( Z cf ), we study the caloric curve of the reaction Pd + Be. In the IQMD model, the excitation energyof the compound system at the early stage of the reaction
Pd + Be can be obtained by E ∗ = U + E k − E , (8)where U , E k and E are the potential energy, kinetic en-ergy and experimental binding energy [69], respectively.To properly account the energy deposited in the system,the kinetic energy of emitted or evaporated nucleons,should be counted when calculating the excitation en-ergy.We exhibit in Fig. 7 the caloric curve of the Pd + Be reaction from the IQMD simulation with the appar-ent temperature determined by DNN using M c ( Z cf ) ofthe reaction. As shown in the figure, the increase of T ap slows down when E ∗ /A reaches to about 8 MeV. Tradi-tionally, this characteristic behavior of the caloric curveis explained that, as the excitation energy increases, thesystem is driven to a spinodal region, in which part of theexcitation energy begins to transfer to latent heat. Tocharacterize this feature of caloric curve quantitatively,the specific heat capacity of the system [10] is defined˜ c ≡ d ( E ∗ /A ) dT ap . (9)Note that it is different from c p or c v because it is notpracticable to maintain the external condition on pres-sure or volume during the reaction. The apparent tem-perature corresponding to the maximum of ˜ c is calledlimiting temperature T lim , which can be used to deducethe critical temperature of nuclear matter [13]. We fur-ther obtain ˜ c through a polynomial fit (red dashed linein Fig. 7) of the obtained caloric curve, as shown in theinset of Fig. 7. Based on T ap obtained by DNN from M c ( Z cf ), the obtained T lim of the reaction Pd + Beis about 6 . T lim follows the generaltrend of the Natowitz’s limiting-temperature dependenceto the system size [12], and thus indicates the validityof determining the apparent temperature through chargemultiplicity distribution presented in this article. IV. SUMMARY AND OUTLOOK
In the present work, we have examined the possibil-ity of determining the apparent temperature T ap of HICsat intermediate-to-low energies through their final-statecharge multiplicity distribution M c ( Z cf ). Based on theIQMD simulations of de-exciting nuclear sources ( Sn)at given temperatures, we have established a relation be-tween the final-state M c ( Z cf ) of a nuclear source, andits corresponding temperature, through training a DNN.The trained DNN can predict the apparent temperaturewithin an error of 0 .
62 MeV, which is small enough forapplying it to analyze the reaction dynamics. We havethen employed the above method to obtain the apparent temperature of the
Pd + Be reactions at different in-cident energies simulated by the IQMD, and subsequentlythe caloric curve. The caloric curve shows a characteris-tic behavior at, i.e., T lim = 6 . Declaration of competing interest
The authors declare that they have no known com-peting financial interests or personal relationships thatcould have appeared to influence the work reported inthis paper.
Acknowledgments
This work is partially supported by the National Nat-ural Science Foundation of China under Contracts No.11890710, 11890714 and No. 11625521, the Key ResearchProgram of Frontier Sciences of the CAS under Grant No.QYZDJ-SSW-SLH002, the Strategic Priority ResearchProgram of the CAS under Grants No. XDB34000000,Guangdong Major Project of Basic and Applied BasicResearch No. 2020B0301030008, and the PostdoctoralInnovative Talent Program of China under Grants No.BX20200098. [1] J. Lattimer and M. Prakash, Phys. Rep. , 109 (2007).[2] M. Baldo and G. Burgio, Prog. Part. Nucl. Phys. , 203(2016).[3] X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. ,96 (2018).[4] J. E. Finn, S. Agarwal, A. Bujak, J. Chuang, L. J. Gutay,A. S. Hirsch, R. W. Minich, N. T. Porile, R. P. Scharen-berg, B. C. Stringfellow, et al., Phys. Rev. Lett. , 1321(1982).[5] P. J. Siemens, Nature , 410 (1983).[6] A. D. Panagiotou, M. W. Curtin, H. Toki, D. K. Scott,and P. J. Siemens, Phys. Rev. Lett. , 496 (1984).[7] J. Pochodzalla, T. M¨ohlenkamp, T. Rubehn,A. Sch¨uttauf, A. W¨orner, E. Zude, M. Begemann-Blaich, T. Blaich, H. Emling, A. Ferrero, et al., Phys.Rev. Lett. , 1040 (1995).[8] Y. G. Ma, A. Siwek, J. P´eter, F. Gulminelli, R. Dayras,L. Nalpas, B. Tamain, E. Vient, G. Auger, C. O. Bacri, et al., Phys. Lett. B , 41 (1997).[9] Y. G. Ma, Phys. Rev. Lett. , 3617 (1999).[10] P. Chomaz, V. Duflot, and F. Gulminelli, Phys. Rev.Lett. , 3587 (2000).[11] J. Richert and P. Wagner, Phys. Rep. , 1 (2001).[12] J. B. Natowitz, R. Wada, K. Hagel, T. Keutgen, M. Mur-ray, A. Makeev, L. Qin, P. Smith, and C. Hamilton, Phys.Rev. C , 034618 (2002).[13] J. B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin,R. Wada, and J. Wang, Phys. Rev. Lett. , 212701(2002).[14] Y. G. Ma, J. B. Natowitz, R. Wada, K. Hagel, J. Wang,T. Keutgen, Z. Majka, M. Murray, L. Qin, P. Smith,et al., Phys. Rev. C , 054606 (2005).[15] C.-W. Ma and Y.-G. Ma, Prog. Part. Nucl. Phys. , 120(2018).[16] B. Borderie and J. Frankland, Prog. Part. Nucl. Phys. , 82 (2019). [17] N. Auerbach and S. Shlomo, Phys. Rev. Lett. ,172501 (2009).[18] N. D. Dang, Phys. Rev. C , 034309 (2011).[19] D. Q. Fang, Y. G. Ma, and C. L. Zhou, Phys. Rev. C ,047601 (2014).[20] X. G. Deng, Y. G. Ma, and M. Veselsk´y, Phys. Rev. C , 044622 (2016).[21] D. Mondal, D. Pandit, S. Mukhopadhyay, S. Pal, B. Dey,S. Bhattacharya, A. De, S. Bhattacharya, S. Bhat-tacharyya, P. Roy, et al., Phys. Rev. Lett. , 192501(2017).[22] C. Q. Guo, Y. G. Ma, W. B. He, X. G. Cao, D. Q. Fang,X. G. Deng, and C. L. Zhou, Phys. Rev. C , 054622(2017).[23] A. Bracco, J. J. Gaardhøje, A. M. Bruce, J. D. Garrett,B. Herskind, M. Pignanelli, D. Barn´eoud, H. Nifenecker,J. A. Pinston, C. Ristori, et al., Phys. Rev. Lett. , 2080(1989).[24] P. F. Bortignon, A. Bracco, D. Brink, and R. A. Broglia,Phys. Rev. Lett. , 3360 (1991).[25] O. Wieland, A. Bracco, F. Camera, G. Benzoni, N. Blasi,S. Brambilla, F. Crespi, A. Giussani, S. Leoni, P. Mason,et al., Phys. Rev. Lett. , 012501 (2006).[26] R. Wada, D. Fabris, K. Hagel, G. Nebbia, Y. Lou, M. Go-nin, J. B. Natowitz, R. Billerey, B. Cheynis, A. Demeyer,et al., Phys. Rev. C , 497 (1989).[27] D. G. d’Enterria, L. Aphecetche, A. Chbihi, H. Dela-grange, J. D´ıaz, M. J. van Goethem, M. Hoefman, A. Ku-gler, H. L¨ohner, G. Mart´ınez, et al., Phys. Rev. Lett. ,022701 (2001).[28] B. Borderie, D. Durand, F. Gulminelli, M. Parlog, M. F.Rivet, L. Tassan-Got, G. Auger, C. O. Bacri, J. Benlliure,E. Bisquer, et al., Phys. Lett. B , 224 (1996).[29] C. B. Chitwood, C. K. Gelbke, J. Pochodzalla, Z. Chen,D. J. Fields, W. G. Lynch, R. Morse, M. B. Tsang, D. H.Boal, and J. C. Shillcock, Phys. Lett. B , 27 (1986).[30] C. Schwarz, W. G. Gong, N. Carlin, C. K. Gelbke, Y. D.Kim, W. G. Lynch, T. Murakami, G. Poggi, R. T. deSouza, M. B. Tsang, et al., Phys. Rev. C , 676 (1993).[31] M. B. Tsang, W. G. Lynch, H. Xi, and W. A. Friedman,Phys. Rev. Lett. , 3836 (1997).[32] V. Serfling, C. Schwarz, R. Bassini, M. Begemann-Blaich, S. Fritz, S. J. Gaff, C. Groß, G. Imm´e, I. Iori,U. Kleinevoß, et al., Phys. Rev. Lett. , 3928 (1998).[33] S. Albergo, S. Costa, E. Costanzo, and A. Rubbino, ILNuov. Cim. A , 1 (1985).[34] S. Wuenschel, A. Bonasera, L. W. May, G. A. Souliotis,R. Tripathi, S. Galanopoulos, Z. Kohley, K. Hagel, D. V.Shetty, K. Huseman, et al., Nucl. Phys. A , 1 (2010).[35] Y. LeCun, Y. Bengio, and G. Hinton, Nature , 436(2015).[36] M. I. Jordan and T. M. Mitchell, Science , 255 (2015).[37] G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld,N. Tishby, L. Vogt-Maranto, and L. Zdeborov´a, Rev.Mod. Phys. , 045002 (2019).[38] G. Carleo and M. Troyer, Science , 602 (2017).[39] Z.-W. Zhang, S. Yang, Y.-H. Wu, C.-X. Liu, Y.-M. Han,C.-H. Lee, Z. Sun, G.-J. Li, and X. Zhang, Chin. Phys.Lett. , 018401 (2020).[40] Y. D. Hezaveh, L. P. Levasseur, and P. J. Marshall, Na- ture , 555 (2017).[41] J. Carrasquilla and R. G. Melko, Nat. Phys. , 431(2017).[42] E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber,Nat. Phys. , 435 (2017).[43] J. F. Rodriguez-Nieva and M. S. Scheurer, Nat. Phys. , 790 (2019).[44] W.-J. Rao, Chin. Phys. Lett. , 080501 (2020).[45] C.-W. Ma, D. Peng, H.-L. Wei, Z.-M. Niu, Y.-T. Wang,and R. Wada, Chin. Phys. C , 014104 (2020).[46] T.-S. Yao, C.-Y. Tang, M. Yang, K.-J. Zhu, D.-Y. Yan,C.-J. Yi, Z.-L. Feng, H.-C. Lei, C.-H. Li, L. Wang, et al.,Chin. Phys. Lett. , 068101 (2019).[47] Y. Wu, Z. Meng, K. Wen, C. Mi, J. Zhang, and H. Zhai,Chin. Phys. Lett. , 103201 (2020).[48] R. Wang, Y.-G. Ma, R. Wada, L.-W. Chen, W.-B. He,H.-L. Liu, and K.-J. Sun, Phys. Rev. Research , 043202(2020).[49] G. Bertsch and S. Das Gupta, Phys. Rep. , 189(1988).[50] J. Aichelin, Phys. Rep. , 233 (1991).[51] J. P. Bondorf, A. Botvina, A. Iljinov, I. Mishustin, andK. Sneppen, Phys. Rep. , 133 (1995).[52] R. Charity, M. McMahan, G. Wozniak, R. McDon-ald, L. Moretto, D. Sarantites, L. Sobotka, G. Guarino,A. Pantaleo, L. Fiore, et al., Nucl. Phys. A , 371(1988).[53] T. Gaitanos, H. Lenske, and U. Mosel, Prog. Part. Nucl.Phys. , 439 (2009).[54] Z.-F. Zhang, D.-Q. Fang, and Y.-G. Ma, Nucl. Sci. Tech. , 78 (2018).[55] A. Ono, Prog. Part. Nucl. Phys. , 139 (2019).[56] Y. G. Ma, Y. B. Wei, W. Q. Shen, X. Z. Cai, J. G. Chen,J. H. Chen, D. Q. Fang, W. Guo, C. W. Ma, G. L. Ma,et al., Phys. Rev. C , 014604 (2006).[57] K. S¨ummerer and B. Blank, Phys. Rev. C , 034607(2000).[58] Y.-D. Song, H.-L. Wei, C.-W. Ma, and J.-H. Chen, Nucl.Sci. Tech. , 96 (2018).[59] R. J. Charity, Phys. Rev. C , 014610 (2010).[60] H. L. Liu, Y. G. Ma, and D. Q. Fang, Phys. Rev. C ,054614 (2019).[61] C. Hartnack, R. K. Puri, J. Aichelin, J. Konopka, S. Bass,H. St¨ocker, and W. Greiner, Eur. Phys. J. A , 151(1998).[62] P.-C. Li, Y.-J. Wang, Q.-F. Li, and H.-F. Zhang, Nucl.Sci. Tech. , 177 (2018).[63] Z.-Q. Feng, Nucl. Sci. Tech. , 40 (2018).[64] T.-Z. Yan, S. Li, Y.-N. Wang, F. Xie, and T.-F. Yan,Nucl. Sci. Tech. , 15 (2019).[65] T.-Z. Yan and S. Li, Nucl. Sci. Tech. , 43 (2019).[66] H. Yu, D.-Q. Fang, and Y.-G. Ma, Nucl. Sci. Tech. ,61 (2020).[67] D. P. Kingma and J. Ba, ArXiv14126980 Cs (2017),1412.6980.[68] R. Wada, W. Lin, P. Ren, H. Zheng, X. Liu, M. Huang,K. Yang, and K. Hagel, Phys. Rev. C , 024616 (2019).[69] M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi,and X. Xu, Chin. Phys. C41