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

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

Outline

Backgound and motivation

Method introduction

Calculation results

Summary

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).)

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• 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)

Lacey et al., PRL 98, 092301 (2007)

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

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

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.

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 blockingsoft EOS with K=200 (a=-356MeV, b=303MeV, =7/6)

hard EOS with K=380 (a=-124MeV, b=70.5MeV, =2)

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),

Shear viscosity of fluid

y

ux

Shear viscosity:

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.

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:

If the nucleons are uniformly distributed in a fixed space, the shear viscosity could be expressed as

2)0(ijT

V

where tpxfp

ppdtrT

jiij ,,,

03

0

3 ,0,01

equil

ijij trdtrdT

iiijijij TT

3

1

with determined by)

1exp()()0(

Ptijij

R. Kubo, Rep. Prog. Phys. 29 (1966) 255; A. Muronga, Phys. Rev. C 69, 044901 (2004)

volume size: r=5 fm

Temperature

S. Wuenschel et al., Nuclear Physics A 843(2010)1.

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:

2222 4 TAmRelation between temperature and the variance is:

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

R

RR p

||2

Entropy density

T

uPs

Energy density, Pressure

With the chemical potential =20 MeV

Thermodynamic quantities

iEV

1

22iii pmE

i

i

E

p

VP

2

3

1

With

Shear viscosity

2)0(ijT

Vvolume size: r=5 fm

Shear viscosity / entropy density

Relaxation time approach:

J. Xu, Phys. Rev. C 84(2011)064603.

/s and LGPT

Fisher droplet model predicted that there Fisher droplet model predicted that there is an affective minimum power-law is an affective minimum power-law exponent exponent eff eff , from the fragment mass , from the fragment mass distributions around the critical point of distributions around the critical point of liquid gas phase transition. Eg. In Ar-like liquid gas phase transition. Eg. In Ar-like data:data:

Ref: Y. G. Ma et al., PRC 71, 054606 (2005)

1 2 3 4 5 6 7 8 92

3

5.85.96.06.16.26.36.46.5

eff

E*/A (MeV)

QMD simulationQMD simulation

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IQMD framwork

Isospin-dependent quantum molecular dynamics (IQMD) modelIsospin-dependent quantum molecular dynamics (IQMD) model

Hartnack, C. et al. Eur. Phys. J., 1998, A1, 151-169; Nucl. Phys., 1989, A495, 303c-320c

The propagation in the effective potentialThe propagation in the effective potential

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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)

Summary 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.