1. Symmetry energy and pion production in the Boltzmann-Langevin approach
Wen-Jie Xie, Jun Su, Long Zhu, Feng-Shou Zhang
谢文杰，苏军，祝龙，张丰收
Supervisor: Prof. Feng-Shou Zhang
Beijing Normal University
北京师范大学核科学与技术学院

2. Contents 1. Introduction
2.The isospin- and momentum-dependent Boltzmann-Langevin Model (IBL)
3. The calculated results on the pion production using the IBL model
(1) the pion multiplicity
(2) the ratio
4. Conclusions

3. Introduction
The form of symmetry energy is important in both nuclear physics and astrophysics and not well constrained up to now.
At subnormal densities:
The symmetry energy increases with increasing the density.
At supernormal densities:
The trend of the symmetry energy with increasing density is not constrained.
Supersoft : IBUU04, Z. G. Xiao, PRL102 (2009)
Superstiff: ImIQMD, Z. Q. Feng, PLB 683 (2010) 140
Among the probes of the symmetry energy, the is promising.
The resonance model predicts a ratio of
where the is determined by the symmetry energy.
R. Stock, Phys. Rep. 135 (1986) 259

4. 1. The Boltzmann-Langevin equation :
The IBL model
1. The Boltzmann-Langevin equation :
2. The evolution equation :
3. The effective potential :
4. The collision term:
5. The fluctuation term:

5. The fluctuation term
The fluctuating collision term can be interpreted as a stochastic force acting on and is charaterized by a correlation funtion,
After using an approximate treatment, the local momentum distribution is projected on a set of low-order multipole moments of order with magnetic quantum numbers And the can be reduced and written as:
F. S. Zhang et al. Phys. Rev. C 51(1995) 3201
E. Suraud et al. Nucl. Phys. A542 (1992) 141
Y. ABE, S. Ayik, et, al. Phys. Rep. 275 (1996)49

6. The scaling procedure
The scaling procedure used to rescale the local momentum distribution to the local values of and is given in the following
The are solved in terms of the following equations:
F. S. Zhang et al. Phys. Rev. C 51(1995) 3201

7. The evolution process of IBL
1. Starting with a definite density at time , the first step is to determine the local average evolution from to , which yields and the elements of the diffusion matrix
2. The diffusion matrix is diagonalized and the fluctuations are calculated.
3. The fluctuations are inserted into the single-particle density. The above three steps are repeated at each time step.

8. The differences between the IBL and usual BUU models
Vlasov: 演化过程只受哈密顿方程的约束，每次事件模拟的状态确定，不需做多次事件模拟。
BUU： Vlasov+ 随机碰撞，尽管碰撞具有随机性，但平均路径是一样的，只能讨论单体观测量。
BLE：BUU+ 涨落，涨落的效果是在每一步演化的过程中加一随机量，使得系统每次演化的路径不一样，使得模型可以讨论除单体观测量意外的其他观测量
A. Ono, J. Randrup, Eur. Phys. J. A 30 (2006) 109

9. The differences between the IBL and usual BUU models
BLE：由于涨落的存在，动量四极距值有较大的变化范围
The is the standard deviation function.

10. The forms of the symmetry energy
Z. Q. Feng, G. M. Jin, Phys. Lett. B 683 (2010) 140

11. Pion multiplicity W. Reisdorf, et, al., Nucl. Phys. A 781(2007) 459
Z. G. Xiao, et, al. ,Phys. Rev. Lett. 102 (2009)

12. The dependence of the ratio on the N/Z at 400A MeV
W. Reisdorf, et, al, Nucl. Phys. A 781(2007) 459
R. Stock, Phys. Rep. 135 (1986) 259

13. The excitation function of from central Au+Au collisions

14. Conclusions
The pion multiplicity is dependent on the EOS and independent on the symmetry energy.
The pion multiplicity and ratio calculated by the IBL mdoel are larger than those obtained by the usual BUU model at lower incident energies, especially for below the pion threshold value.
Calculations with a supersoft symmetry energy describe well the experimental data.

15. Thank you for your attention！