Shear viscosity to entropy density ratio of nuclear matter by transport model

1. Shear viscosity to entropy density ratio of nuclear matter by transport model Deqing Fang, Yugang Ma, Shaoxin Li, Chenlong Zhou Shanghai Institute of Applied Physics, Chinese Academy of Sciences

2. Outline • Backgound and motivation • Method introduction • Calculation results • Summary

3. Background and Motivation Intermediate energy heavy ion collisions have been extensively studied both experimentally and theoretically for obtaining information about the properties of nuclear matter under a wide range of density and temperature Due to van der Waals nature of the nucleon-nucleon force, liquid-gas phase transition (LGPT) exhibits around hundred MeV/nucleon. Multifragmentation and LGPT have become the most important subjects in heavy ion collision at intermediate energies in the past years. In ultra-relativistic heavy ion collision, hydrodynamic model has been used to study the QGP phase and critical phenomenon. It is found that QGP has small viscosity and behaves like a perfect fluid. Only few studies has been devoted to studying the viscosity of nuclear matter formed in intermediate energy heavy ion collision. (P. Danielewicz, PLB146, 168 (1984), L. Shi and P. Danielewicz, PRC68, 064604 (2003).)

4. Lacey et al., PRL 98, 092301 (2007) • Empirical observation of temperature dependence of shear viscosity to entropy density ratio exhibits a minimum at the critical point of phase transition • A lower bound of the ratio (/s>1/4π) is speculated to be valid universally according to certain gauge theory (Kovtun-Son-Starinets (KSS) bound) Based on a transport model (BUU) simulation, we have studies the transport coefficients, like the viscosity of nuclear matter formed during heavy ion collision at intermediate energies. S.X. Li, D. Q. Fang, Y. G. Ma, C.L. Zhou, Phys Rev C 84, 024607 (2011) 50-100AMeV Au+Au, central collsion (b=0 fm) BUU + Green-kubo method

5. BUU model Boltzmann-Uehling-Uhlenbeck(BUU) model is a one body microscopic transport model based on the Boltzmann equation: 1) Mean field 2) Two-body collisions 3) Pauli blocking soft EOS with K=200 (a=-356MeV, b=303MeV, =7/6) hard EOS with K=380 (a=-124MeV, b=70.5MeV, =2) G. Bertsch et al., PRC29, 673 (1984),

6. Shear viscosity of fluid Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. Put simply, the less viscous the fluid is, the greater its ease of movement (fluidity). Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. With the exception of superfluids, all real fluids have some resistance to stress and therefore are viscous, but a fluid which has no resistance to shear stress is known as an ideal fluid. Shear viscosity:

7. Green-kubo method The Green-Kubo formalism relates linear transport coefficients to near-equilibrium correlations of dissipative fluxes and treats dissipative fluxes as perturbations to local thermal equilibrium. In this fluctuation-dissipation theotry, shear viscosity is determined by the stress tensor correlations: volume size: r=5 fm where If the nucleons are uniformly distributed in a fixed space, theshear viscosity could be expressed as with determined by R. Kubo, Rep. Prog. Phys. 29 (1966) 255; A. Muronga, Phys. Rev. C 69, 044901 (2004)‏

8. Temperature Temperature of the system is derived from the momentum fluctuation of nucleons in the center-of-mass frame of the fragmenting source. The variance is obtained from the Qz distribution through: Relation between temperature and the variance is: S. Wuenschel et al., Nuclear Physics A 843(2010)1.

9. System Equilibrium The stopping parameter is used to measure the degree of equilibration reached in a heavy-ion reaction, which is defined as

10. Entropy density Thermodynamic quantities Energy density, Pressure With With the chemical potential =20 MeV

11. Shear viscosity volume size: r=5 fm

12. Shear viscosity / entropy density Relaxation time approach: J. Xu, Phys. Rev. C 84(2011)064603.

13. /s and LGPT • Fisher droplet model predicted that there is an affective minimum power-law exponent eff , from the fragment mass distributions around the critical point of liquid gas phase transition. Eg. In Ar-like data: Ref: Y. G. Ma et al., PRC 71, 054606 (2005) QMD simulation

14. IQMD framwork Isospin-dependent quantum molecular dynamics (IQMD) model The propagation in the effective potential Hartnack, C. et al. Eur. Phys. J., 1998, A1, 151-169; Nucl. Phys., 1989, A495, 303c-320c

15. The formula of extract thermal properties • The hot Thomas-Fermi formulism is used to extract the thermal properties, e.g. temperature, entropy density, chemical potential. • This method treat the colliding nucleus as two piece of penetrating nuclear matter which obeys Fermi-Dirac distribution, and all the thermal values are expressed as a function of nuclear matter density and kinetic energy density Dao et al..nuclear physics A 542, 671-698 (1992)

16. By using the Green-Kubo method, we studied thermodynamic variables as well as viscosity and entropy density for nuclear matter formed in intermediate-energy heavy-ion collisions within the framework of BUU model. It is found that η/s decreases very quickly before 70A MeV and then drops slowly toward a smaller value of η/s around 0.5 at higher energy. • However, no obvious minimum η/s value occurs at intermediate energy range in BUU model. This may indicates that there is no liquid-gas phase transition in the BUU model which lacks dynamical fluctuation and correlation effect of NN interaction. • Further investigation by using QMD model is in progress. Summary