Probing the density dependence of the symmetry energy with central heavy ion collisions

Probing the density dependence of the symmetry energy with central heavy ion collisions 1 2 3 first-author 1 2 4 corresp Corresponding author. E-mail address: fszhang@bnu.edu.cn Corresponding author. E-mail address: fszhang@bnu.edu.cn fszhang@bnu.edu.cn The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China Department of Physics, Yuncheng University, Yuncheng 044000, China Department of Physics, Yuncheng University, Yuncheng 044000, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China An improved isospin dependent Boltzmann Langevin model, in which the inelastic channels and momentum dependent interactions are incorporated, is used to investigate the high-density behavior of nuclear symmetry energy. By taking several forms of nuclear symmetry energy, we calculate the time evolutions of neutron over proton ratio, π multiplicity and <math id="M1"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio, and the kinetic energy and transverse momentum spectra of <math id="M2"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio in the heavy ion collisions at 400 A MeV. It is found that the neutron over proton ratio and <math id="M3"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio are very sensitive to the nuclear symmetry energy, and the π – is more sensitive to the nuclear symmetry energy than the π + . A supersoft symmetry energy results in a larger <math id="M4"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio. Improved isospin dependent Boltzmann-Langevin model Nuclear symmetry energy Neutron over proton ratio Pion production 1 Introduction 1 Constraints on the nuclear symmetry energy <math id="M5"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> have been an very interesting subject in the last few years. The <math id="M6"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> which is an estimate of the energy cost to convert all protons to neutrons at the fixed density in the symmetric nuclear matter, is very important both in nuclei, such as the binding energy, and in neutron stars, such as the nature and stability of the phases within the star [ 1 1 , 2 2 ] . The energy per nucleon in asymmetric nuclear matter can usually be expressed as <math display="block" id="M7"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>δ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mi>δ</mi><mn>4</mn></msup><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where <math id="M8"><mrow><mi>δ</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>n</mi></msub><mo>−</mo><msub><mi>ρ</mi><mi>p</mi></msub><mo stretchy="false">)</mo></mrow><mo>/</mo><mi>ρ</mi></mrow></mrow></math> and <math id="M9"><mrow><msub><mi>ρ</mi><mi>n</mi></msub></mrow></math> , <math id="M10"><mrow><msub><mi>ρ</mi><mi>p</mi></msub></mrow></math> and <math id="M11"><mi>ρ</mi></math> are the neutron, proton and total nucleon densities, respectively. The first term in Eq.(1) is the symmetric contribution and the second term is the symmetry energy contribution. The higher-order terms <math id="M12"><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>δ</mi><mn>4</mn></msup><mo stretchy="false">)</mo></mrow></math> in <math id="M13"><mi>δ</mi></math> are commonly negligible [ 3 3 ] . Generally, there are two forms of <math id="M14"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> predicted by various theoretical models. The one is that the <math id="M15"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> increases with the increasing nucleon density and the other is that <math id="M16"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> increases up to the saturation density <math id="M17"><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></math> and then decreases with the increasing density. At supra-saturation densities ( <math id="M18"><mrow><mi>ρ</mi><mo>></mo><msub><mi>ρ</mi><mn>0</mn></msub></mrow></math> ), the form of the <math id="M19"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> is very controversial. Even the trend of the <math id="M20"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> with increasing density is not constrained yet. A lot of probes have been proposed to constrain the <math id="M21"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> at supra-saturation density, such as the neutron over proton ratio of squeezed-out nucleons [ 4 4 ] , the neutron-proton differential flow [ 5 5 ] , the <math id="M22"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio [ 6 6 , 7 7 , 8 8 , 9 9 , 10 10 ] , the <math id="M23"><mrow><mrow><mrow><msup><mi>K</mi><mn>0</mn></msup></mrow><mo>/</mo><mrow><msup><mi>K</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio [ 10 10 ] , the <math id="M24"><mrow><mrow><mrow><msup><mi>Σ</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>Σ</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio [ 11 11 ] and so on. Among above probes, the <math id="M25"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio as a promising one is motivated by the <math id="M26"><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mn>1232</mn><mo stretchy="false">)</mo></mrow></math> resonance model [ 12 12 ] , in which a primordial relation between <math id="M27"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio and <math id="M28"><mrow><mi>N</mi><mo>/</mo><mi>Z</mi></mrow></math> is predicted. That is <math display="block" id="M29"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mn>5</mn><msup><mi>N</mi><mn>2</mn></msup><mo>+</mo><mi>N</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mn>5</mn><msup><mi>Z</mi><mn>2</mn></msup><mo>+</mo><mi>N</mi><mi>Z</mi></mrow></mrow><mo stretchy="false">)</mo><mo>≈</mo><msup><mrow><mrow><mo>(</mo><mrow><mrow><mi>N</mi><mo>/</mo><mi>Z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>,</mo></mrow></math> where the <math id="M30"><mi>N</mi></math> and <math id="M31"><mi>Z</mi></math> are the neutron and proton numbers in the participant region of the reaction. The <math id="M32"><mrow><mi>N</mi><mo>/</mo><mi>Z</mi></mrow></math> is determined by the <math id="M33"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> by the dynamical isospin fractionation [ 6 6 ] . Therefore, one can use the <math id="M34"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio to constrain the <math id="M35"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . Using the <math id="M36"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio as a probe of the <math id="M37"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> , a larger uncertainty exists. By comparing with the FOPI data [ 15 15 ] , a very soft <math id="M38"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> was suggested by the isospin-dependent Boltzmann Uehling Uhlenbeck model (IBUU04) [ 7 7 ] and improved isospin dependent Boltzmann Langevin model (ImIBL) [ 9 9 ] , and a hard <math id="M39"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> was suggested by the improved isospin dependent quantum molecular dynamics model (ImIQMD) [ 8 8 ] . At sub-saturation densities ( <math id="M40"><mrow><mi>ρ</mi><mo><</mo><msub><mi>ρ</mi><mn>0</mn></msub></mrow></math> ), the trend of the <math id="M41"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> with increasing density is almost constrained but the quantitative results are not obtained yet. There are also a lot of probes to constrain the <math id="M42"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> in sub-saturation density region, such as the isospin diffusion [ 14 14 , 15 15 ] , the neutron skin thickness [ 16 16 ] , the single and double neutron over proton ratios [ 14 14 , 17 17 , 18 18 , 19 19 , 20 20 ] , the heavy fragment mass distribution [ 21 21 ] and so on. The IBUU04 model [ 15 15 ] suggested an almost linear <math id="M43"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> by analyzing the isospin diffusion data. Also the constrained molecular dynamics II model (CoMDII) suggested a linear one by calculating the heavy fragment mass distribution [ 21 21 ] . However, the improved quantum molecular dynamics model (ImQMD) [ 14 14 ] and the isospin dependent quantum molecular dynamics model (IQMD) [ 17 17 ] predicted a soft <math id="M44"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> by analyzing the double neutron over proton ratio. Recently, a quantum statistical approach predicted the <math id="M45"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> at low temperature and low density limit [ 22 22 ] . As a kind of ensemble-averaged theory, the usual BUU model have been very successful in describing the various phenomena in heavy ion collisions. However, the fluctuations, which are very important in preequilibrium particle emission, subthreshold particle production and multifragmentation are not included in the BUU model. In the QMD-like model which is a kind of many-body theory, the reactions are simulated event by event. Though the fluctuations are realized in the QMD simulations, the correlations among the events are uncertain. Moreover, the QMD model fails to explain the subthreshold kaon production [ 23 23 ] . In this paper, we will use the ImIBL model to investigate the <math id="M46"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mi>（</mi><mi>ρ</mi><mi>）</mi></mrow></math> using the neutron over proton ratio and <math id="M47"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio in the 48 Ca+ 48 Ca collisions at 400 A MeV. The paper is organized as follows. We introduce our model in Section 2. The calculated results are given in Section 3. Finally, we will give some conclusions. 2 Theoretical model 1 The ImIBL model used in this paper is an updated version of the Boltzmann-Langevin model(BL) [ 24 24 ] . According to the BL model, the fluctuating phase space single particle density fulfills the following equation [ 24 24 ] : <math display="block" id="M48"><mtable columnalign="left"><mtr><mtd><mrow><mo>(</mo><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>m</mi></mfrac><mo>•</mo><msub><mo>∇</mo><mi>r</mi></msub><mo>−</mo><msub><mo>∇</mo><mi>r</mi></msub><mi>U</mi><mrow><mo>(</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow><mo>•</mo><msub><mo>∇</mo><mi>p</mi></msub></mrow><mo>)</mo></mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>K</mi><mrow><mo>(</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow><mo>+</mo><mi>δ</mi><mi>K</mi><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> The left-hand side describes the Vlasov propagation determined by nuclear mean field <math id="M49"><mrow><mi>U</mi><mrow><mo>(</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow></mrow></math> . <math id="M50"><mrow><mi>K</mi><mrow><mo>(</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow></mrow></math> is the collision term of the usual BUU form but is characterized by the fluctuating density <math id="M51"><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math> . <math id="M52"><mrow><mi>δ</mi><mi>K</mi><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math> is the fluctuating collision term, which can be explained as a stochastic force acting on the <math id="M53"><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math> , and is characterized by the following correlation function: <math display="block" id="M54"><mtable columnalign="left"><mtr><mtd><mrow><mo>〈</mo><mrow><mi>δ</mi><mi>K</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>1</mn></msub><mo>,</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mi>δ</mi><mi>K</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>2</mn></msub><mo>,</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>〉</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>C</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>1</mn></msub><mo>,</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>−</mo><msub><mi>t</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr></mtable></math> where the angle brackets denote a local average, performed over fluctuating densities generated during a time interval δt . The correlation function <math id="M55"><mrow><mi>C</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>1</mn></msub><mo>,</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math> can be reduced and expressed as in the weak-coupling limit and determined by the one-body properties of the locally averaged distribution as indicated in Ref.[ 25 25 ]. The numerical simulation method used in BL model is the projection method [ 24 24 , 25 25 ] which projects the fluctuations on a set of low order local multipole moments <math id="M56"><mrow><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub></mrow></math> of order <math id="M57"><mi>L</mi></math> with magnetic quantum numbers <math id="M58"><mi>M</mi></math> of the momentum distribution. The fluctuations of these multipole moments can be characterized by a diffusion matrix and can be deduced from the correlation matrix <math id="M59"><mrow><mi>C</mi><mrow><mo>(</mo><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math> as [ 25 25 ] <math display="block" id="M60"><mtable columnalign="left"><mtr><mtd><msub><mi>C</mi><mrow><mi>L</mi><mi>M</mi><msup><mi>L</mi><mo>′</mo></msup><msup><mi>M</mi><mo>′</mo></msup></mrow></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mi>d</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>d</mi><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>′</mo></msup><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mrow><mo>(</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>)</mo></mrow><msub><mi>Q</mi><mrow><msup><mi>L</mi><mo>′</mo></msup><msup><mi>M</mi><mo>′</mo></msup></mrow></msub><mrow><mo>(</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>′</mo></msup><mo>)</mo></mrow><mi>C</mi><mrow><mo>(</mo><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>,</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mi>d</mi><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>1</mn></msub><mi>d</mi><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msub><mi>d</mi><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>3</mn></msub><mi>d</mi><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>4</mn></msub><mi>Δ</mi><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mi>Δ</mi><msub><mi>Q</mi><mrow><msup><mi>L</mi><mo>′</mo></msup><msup><mi>M</mi><mo>′</mo></msup></mrow></msub></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>×</mo><mi>W</mi><mrow><mo>(</mo><mrow><mn>12</mn><mo>,</mo><mn>34</mn></mrow><mo>)</mo></mrow><msub><mi>f</mi><mn>1</mn></msub><msub><mi>f</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>f</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>f</mi><mn>4</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math> with <math id="M61"><mrow><mi>Δ</mi><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mo>=</mo><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>Q</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>4</mn></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> According to the Refs.[24-27], the stochastic component of the dynamics can be reduced to the following two equations: <math display="block" id="M62"><mtable columnalign="left"><mtr><mtd><msub><mover accent="true"><mi>Q</mi><mo>⌢</mo></mover><mrow><mn>20</mn></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi><mo>+</mo><mi>δ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Q</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi><mo>+</mo><mi>δ</mi><mi>t</mi></mrow><mo>)</mo></mrow><mo>+</mo><msqrt><mrow><mi>δ</mi><mi>t</mi><msub><mi>C</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></msqrt><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo></mtd></mtr><mtr><mtd><msub><mover accent="true"><mi>Q</mi><mo>⌢</mo></mover><mrow><mn>30</mn></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi><mo>+</mo><mi>δ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Q</mi><mn>3</mn></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi><mo>+</mo><mi>δ</mi><mi>t</mi></mrow><mo>)</mo></mrow><mo>+</mo><msqrt><mrow><mi>δ</mi><mi>t</mi><msub><mi>C</mi><mn>3</mn></msub><mrow><mo>(</mo><mrow><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></msqrt><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo></mtd></mtr></mtable></math> where <math id="M63"><mrow><msub><mi>W</mi><mn>2</mn></msub></mrow></math> and <math id="M64"><mrow><msub><mi>W</mi><mn>3</mn></msub></mrow></math> are Gaussian random numbers. The <math id="M65"><mrow><msub><mover accent="true"><mi>Q</mi><mo>⌢</mo></mover><mrow><mn>20</mn></mrow></msub></mrow></math> and <math id="M66"><mrow><msub><mover accent="true"><mi>Q</mi><mo>⌢</mo></mover><mrow><mn>30</mn></mrow></msub></mrow></math> are the fluctuating quadrupole and octupole moments. The fluctuations are inserted into the momentum space through a scaling procedure as indicated in Ref.[ 25 25 ]. In Fig.1 Fig.1 , we show the time evolution of the ensemble-averaged total quadrupole moment Q 20 , <math id="M71"><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub><mo>±</mo><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub></mrow></math> and <math id="M72"><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub><mo>±</mo><mn>2</mn><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub></mrow></math> of the momentum distribution and the associated variance <math id="M73"><mrow><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub></mrow></math> from central 40 Ca+ 48 Ca collisions at 100 A MeV in the ImIBL and BUU models, where <math id="M74"><mrow><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub><mo>=</mo><msqrt><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>Q</mi><mrow><mn>20</mn></mrow><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow><mo>−</mo><msup><mrow><mrow><mo>〈</mo><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub></mrow><mo>〉</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math> is the standard deviation function. For a Gaussian distribution, <math id="M75"><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub><mo>±</mo><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub></mrow></math> and <math id="M76"><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub><mo>±</mo><mn>2</mn><msub><mi>σ</mi><mrow><mn>20</mn></mrow></msub></mrow></math> correspond to 84.3 and 99.5 percent of the number of events respectively [ 27 27 ] . From the upper panel of Fig.1 Fig.1 , we can see that the <math id="M77"><mrow><msub><mi>Q</mi><mrow><mn>20</mn></mrow></msub></mrow></math> value of the ImIBL model has a larger range of variation as compared with the BUU model. From the bottom panel of Fig.1 Fig.1 , there is a bump in the early collision stage for BL model. But this phenomenon does not happen in the BUU simulations. This phenomenon suggests that there is a larger difference between the real quadrupole moment and averaged quadrupole moment, especially in the early collision stage. We think this phenomenon is important in the dynamical evolution process. In the Vlasov part of Eq.(3), we work with an isospin and momentum dependent single nucleon potential, which reads <math display="block" id="M78"><mtable columnalign="left"><mtr><mtd><msub><mi>U</mi><mi>τ</mi></msub><mrow><mo>(</mo><mrow><mi>ρ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover></mrow><mo>)</mo></mrow><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><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mi>γ</mi></msup><mo>+</mo><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><msup><mi>δ</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mfrac><mrow><mo>∂</mo><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow><mrow><mo>∂</mo><mi>ρ</mi></mrow></mfrac><mi>ρ</mi><msup><mi>δ</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mi>ρ</mi><mfrac><mrow><mo>∂</mo><msup><mi>δ</mi><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msub><mi>ρ</mi><mi>τ</mi></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><msub><mi>U</mi><mrow><mtext>MDI</mtext></mrow></msub><mo>,</mo></mtd></mtr></mtable></math> the bulk parameters <math id="M79"><mi>α</mi></math> , <math id="M80"><mi>β</mi></math> , and <math id="M81"><mi>γ</mi></math> taken here are −390 MeV, 320 MeV and 1.14 for the soft EOS plus MDI as SM and −130 MeV, 59 MeV and 2.09 for the hard EOS plus MDI as HM [ 28 28 ] . The compressibilities K are 200 and 380 MeV for the SM and HM, respectively. The <math id="M82"><mrow><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></math> , which is the local part of the <math id="M83"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mi>（</mi><mi>ρ</mi><mi>）</mi></mrow></math> , taken here are the following two forms <math display="block" id="M84"><mrow><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>C</mi><mrow><mtext>sym</mtext></mrow></msub><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><mrow><msub><mi>γ</mi><mi>s</mi></msub></mrow></msup></mrow></math> and <math display="block" id="M85"><mrow><msubsup><mi>E</mi><mrow><mtext>sym</mtext></mrow><mrow><mtext>loc</mtext></mrow></msubsup><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>b</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><mn>2</mn></msup><mo>+</mo><mi>c</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><mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></mrow></msup><mo>.</mo></mrow></math> In Eq.(8), the <math id="M86"><mrow><msub><mi>γ</mi><mi>s</mi></msub></mrow></math> =0.5, 1.0 and 2.0 correspond to the soft, linear and hard symmetry energy respectively. The parameters <math id="M87"><mrow><msub><mi>C</mi><mrow><mtext>sym</mtext></mrow></msub></mrow></math> , <math id="M88"><mi>a</mi></math> , <math id="M89"><mi>b</mi></math> and <math id="M90"><mi>c</mi></math> are 33.4, 38.9, –18.9, and –3.8 MeV, respectively. Eq.(9) gives the supersoft symmetry energy. The curvature parameters <math id="M91"><mrow><msub><mi>K</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mo>=</mo><msub><mrow><mrow><mrow><mn>9</mn><msubsup><mi>ρ</mi><mn>0</mn><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mrow><mrow><msup><mo>∂</mo><mn>2</mn></msup><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub></mrow><mo>/</mo><mrow><mo>∂</mo><msup><mi>ρ</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></msub></mrow></math> are 275.2, –25.4, –62.9, –403.6 MeV for the hard, linear, soft, and supersoft symmetry energy, respectively. Shown in Fig.2 Fig.2 is the comparison of the <math id="M92"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mi>（</mi><mi>ρ</mi><mi>）</mi></mrow></math> for different cases from Eqs.(8) and (9), and also from the Refs.[8,14]. The momentum dependent potential can be expressed as [ 28 28 ] <math display="block" id="M93"><mrow><msub><mi>U</mi><mrow><mtext>MDI</mtext></mrow></msub><mo>=</mo><mn>1.57</mn><msup><mrow><mi>ln</mi></mrow><mn>2</mn></msup><mrow><mo>[</mo><mrow><mn>0.0005</mn><msup><mrow><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>i</mi></msub><mo>−</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>]</mo></mrow><mfrac><mi>ρ</mi><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo>.</mo></mrow></math> The inelastic channels and the parameterized cross sections of each channel are taken from Ref.[ 29 29 ]. We take the free space and in-medium cross sections for the elastic and inelastic channels, respectively. The in-medium effects are the medium correction of <math id="M94"><mi>ρ</mi></math> mass [ 30 30 ] . 3 Results and discussion 1 3.1 Neutron over proton ratio 2 Figure 3 Figure 3 shows the time evolution of the free neutron over proton ratio and the corresponding central point density in central 48 Ca+ 48 Ca collisions at 400 A MeV. The calculations are performed by using the supersoft and hard symmetry energies, respectively. The solid and dashed lines denote the results in the angle interval <math id="M95"><mrow><msup><mrow><mn>70</mn></mrow><mtext>o</mtext></msup><mo><</mo><mi>θ</mi><mo><</mo><msup><mrow><mn>110</mn></mrow><mtext>o</mtext></msup></mrow></math> and in the rapidity interval <math id="M96"><mrow><mo>−</mo><mn>0.2</mn><mo><</mo><msup><mi>y</mi><mtext>o</mtext></msup><mo><</mo><mn>0.2</mn></mrow></math> . The dotted and dotted-dashed lines show the results without any cuts. From Fig.3 Fig.3 , we can see that the supersoft behavior of the symmetry energy leads to a low neutron emission in the high-density region. In low-density region, an inversion of this trend is observed because of contributions to the nucleon emission in the expansion phase while fragments are forming. Our results are consistent with the one of Ref.[ 31 31 ]. 3.2 Pion multiplicity 2 In the left panel of Fig.4 Fig.4 , we show the time evolution of <math id="M97"><mrow><msup><mi>π</mi><mo>±</mo></msup></mrow></math> multiplicity in central 48 Ca+ 48 Ca collisions at 400 A MeV. The calculations are performed by taking four different forms of the <math id="M98"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . From the left panel of Fig.4 Fig.4 and the bottom panel of Fig.3 Fig.3 , one can see that the <math id="M99"><mi>π</mi></math> is mainly produced in the high density region. Therefore, the <math id="M100"><mi>π</mi></math> production will take useful information of the high density phase space. Most interestingly, the produced <math id="M101"><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow></math> is more sensitive to the <math id="M102"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mi>（</mi><mi>ρ</mi><mi>）</mi></mrow></math> than the <math id="M103"><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></math> . In order to further shed light on this point, we show in the right panel of Fig.4 Fig.4 the <math id="M104"><mrow><msup><mi>π</mi><mo>±</mo></msup></mrow></math> multiplicity as a function of the curvature parameter <math id="M105"><mrow><msub><mi>K</mi><mrow><mtext>sym</mtext></mrow></msub></mrow></math> . From the right panel of Fig.5 Fig.5 , we can see that the <math id="M106"><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow></math> yield strongly varies with different <math id="M107"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> from soft to stiff case. This is because the <math id="M108"><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow></math> is mainly produced in neutron-neutron collisions. In the high density region a softer <math id="M109"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> will result in a low neutron emission, and then there will be more neutron rich as compared with the case of a hard <math id="M110"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . We can conclude that the <math id="M111"><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow></math> yield play an important role in investigating the <math id="M112"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . 3.3 π – /π + ratio 2 To further understand the dependence of the <math id="M113"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> on the <math id="M114"><mi>π</mi></math> production, we show in Fig.5 Fig.5 the differential <math id="M115"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratios as functions of the kinetic energy and transverse momentum in the 48 Ca+ 48 Ca collisions at 400 A MeV with the impact parameter <math id="M116"><mi>b</mi></math> =1 fm. One can see that in the low energy and low transverse momentum regions the <math id="M117"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio is very sensitive to the <math id="M118"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . Therefore, we do not need to measure the high energy and high transverse momentum pions to constrain the density dependence of the symmetry energy. This will be useful in designing the experimental facilities with an aim of investigating the density dependent symmetry energy. From Fig.5 Fig.5 , it is found that a supersoft symmetry energy results in a larger <math id="M119"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio, which is consistent with the results of Refs.[7,9] and inconsistent with the one of Refs.[8,10] . 4 Conclusion 1 In summary, the <math id="M126"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> has been investigated by using the free neutron over proton ratio and the <math id="M127"><mi>π</mi></math> production in the 48 Ca+ 48 Ca collisions at 400 A MeV in the framework of ImIBL model. It is found that both the free neutron over proton ratio and the <math id="M128"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio are sensitive to the <math id="M129"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . A soft <math id="M130"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> results in a low emission of neutron component in the high density region. The <math id="M131"><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow></math> yield is strongly dependent on the <math id="M132"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> and the <math id="M133"><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></math> yield is almost independent on the <math id="M134"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . The <math id="M135"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio in the low kinetic energy and low transverse momentum region is sufficient to constrain the <math id="M136"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> . The <math id="M137"><mrow><mrow><mrow><msup><mi>π</mi><mo>−</mo></msup></mrow><mo>/</mo><mrow><msup><mi>π</mi><mo>+</mo></msup></mrow></mrow></mrow></math> ratio becomes lager when the <math id="M138"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> varies from the stiff to the soft. Meson production as a probe of the <math id="M139"><mrow><msub><mi>E</mi><mrow><mtext>sym</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> is still an open question. It is known that the threshold effect is important in the <math id="M140"><mi>K</mi></math> meson production. For <math id="M141"><mi>π</mi></math> meson, the threshold effect, which is not concluded in the present calculations, may also play an important role in the whole evolution stage. We will incorporate the threshold effect into the ImIBL model in the future work. 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