Hadronic Transport Coefficients from a Microscopic Transport Model

1. Hadronic Transport Coefficients from a Microscopic Transport Model Nasser Demir, Steffen A. Bass Duke University April 22, 2007

2. Overview • Motivation: “Low Viscosity Matter” at RHIC & Consequences • Theory: Kubo Formalism for Transport Coefficients • Analysis/Results: Equilibriation, Results for Viscosity • Summary/Outlook: Time-dependence of Transport Coefficients!

3. Low Viscosity Matter at RHIC large viscosity QGP and hydrodynamic expansion initial state freeze-out low viscosity pre-equilibrium hadronic phase QGP-like phase at RHIC observed to behave very much like ideal fluid: ideal hydro treatment of QGP phase works well – but what about hadronic phase? • Why study hadronic phase? • Need to know hadronic • viscosity to constrain QGP • viscosity. • Viscosity changes as function • of time in a heavy ion collision!

4. Two Questions re: “low viscosity” PRL 94. 111601 (2005) Kovtun, Son, Starinets • How low? (AdS/CFT: η/s≥1/4π? KSS bound) • If there is a minimum, where is it? Near Tc? Csernai, Kapusta, McLerran: nucl-th/0604032 PRL 97. 152303 (2006) Pert. Theory N/A here.

5. What do we know thus far? • Determining hadronic viscosity necessary to constrain viscosity of QGP. • Perturbative methods not well trusted near Tc on hadronic side  microscopic transport model can help here! Next Question: How do we compute transport coefficients?

6. Linear Transport Coefficients & Green-Kubo Relations Phenomenological Transport Equation: thermodynamic/mechanical flux linearly proportional to applied field in small field limit. Examples of transport coefficients: thermal conductivity, diffusion,shear viscosity. Shear Viscosity Coefficient: y x Vx= v2 y=a Pyx y=0 Vx= v1 Green-Kubo: compute linear transport coefficients by examining near-equilibrium correlations!

7. Green Kubo Relations: Near-Equilibrium Stat. Mech Green Kubo tells us we can compute linear transport coefficients by examining near-equilibrium fluctuations. < … > indicate ensemble averaging once equilibrium has been reached. Suggests technique of molecular dynamics (MD) simulations. OK, how to model the hadronic medium?

8. Modeling the Hadronic Medium:UrQMD (Ultrarelativistic Quantum Molecular Dynamics) - Transport model based on Boltzmann Equation: -Hadronic degrees of freedom. -Particles interact only through scattering. ( cascade ) -Classical trajectories in phase space. -Interaction takes place only if: (dmin is distance of closest approach between centers of two hadrons) - Values for σ of experimentally measurable processes input from experimental data. • 55 baryon- and 32 meson species, among those 25 N*, Δ* resonances and 29 hyperon/hyperon resonance species • Full baryon-antibaryon and isospin symmetry: - i.e. can relate nn cross section to pp cross section.

9. “Box Mode” for Infinite Hadronic Matter & Equilibriation • Strategy: PERIODIC BOUNDARY CONDITIONS! • Force system into equilibrium, and PREVENT FREEZEOUT. Equilibrium Issues : - Chemicalequilibrium:DISABLE multibody decays/collisions. RESPECT detailed balance! - KineticEquilibrium: ComputeTEMPERATUREby fitting toBoltzmanndistribution!

10. What about Kinetic Equilibrium? T=168.4 MeV ε= 0.5 GeV/fm3 ρB =ρ0 ε= 0.5 GeV/fm3 ρB =ρ0

11. Calculating Correlation Functions NOTE: correlation function found to empirically obey exponential decay. Ansatz also used in Muronga, PRC 69:044901,2004

12. Entropy Considerations Method I: Gibbs formula for entropy: (extract μB for our system from SHAREv2, P and ε known from UrQMD.) Denote as sGibbs. • SHARE v2: Torrieri et.al.,nucl-th/0603026 • Tune particles/resonances to those in UrQMD. Method II: Weight over specific entropies of particles, where s/n is a function of m/T & μB/T! Denote as sspecific

13. Entropy Scaling For system with fixed volume in equilibrium:

14. Summarizing our technology • Use UrQMD in box mode to describe infinite equilibriated hadronic matter. • Apply Green-Kubo formalism to extract transport coefficients. • Calculate entropy by counting specific entropies of particles.  Perform analysis of η, η/s as a function of T and baryon # density for a hadron gasIN EQUILIBRIUM.

15. Preliminary Results for η and η/s • Viscosity increases with Temperature. • Viscosity decreases with finite baryon number density.

16. Where is the minimum viscosity? • - η/s decreases w. finite μB. • Minimum hadronic η/s≈ 1.7/(4π) • Is minimum η/s near Tc? Need μ=0 results for T<100 MeV to answer this question with certainty. (IN PROGRESS)

17. η increasing as function of T: Think specific binary collisions! η ~p/σ: p increases w. T, and mean total CM energy shifts further to right of resonance peak. T increases E/V =0.3 GeV/cubic fm E/V =1.0 GeV/cubic fm σ decreases

18. η decreasing w. finite μB: Think specific binary collisions! η ~p/σ: Resonant πN crosssxns larger than ππ. Increasing μB! ε=0.5 GeV/fm3 ε=0.2 GeV/fm3

19. Full 3-d Hydrodynamics • QGP evolution UrQMD Hadronization hadronic rescattering TC TSW t fm/c Summary/Outlook • Can apply Green-Kubo formalism to hadronic matter in equilibrium: • Use UrQMD to model hadronic matter. • Use box mode to ensure equilibrium. Calculated entropy via 2 different methods (microscopic and macroscopic pictures self-consistent). • Preliminary results: • Hadronic η /s satisfies viscosity bound from AdS/CFT (at least 1.7 times above bound). • η notably reduced at finite μB. In progress: Analyzing μ=0 mesonic matter for T<100 MeV. • Outlook: - Describe time-evolution of transport coefficient in relativistic heavy-ion reaction.

20. Backup Slides

21. String theory to the rescue? A nice conjecture on viscosity. Strong coupling limit for η/s in QCD can’t be calculated! Duality Idea: For a class of string theories, a black hole solution to a string theory (AdS5) equivalent to finite temperature solution for its dual field theory (N=4 SUSY YM). Csernai, Kapusta, McLerran: nucl-th/0604032 PRL 97. 152303 (2006) Kovtun, Son, Starinets: hep-th/0405231 PRL 94. 111601 (2005)

22. Kovtun, Son, Starinets: hep-th/0405231 PRL 94. 111601 (2005)

23. New η/s measurement for ultra-cold atoms cond-mat.other/arXiv:0707.2574v1

24. UrQMD EoS comparison with Statistical Model

25. Another computation of η/s from a cascade Muroya, Sasaki ; Prog. Theor. Phys. 113, 2 (2005) “A Calculation of the Viscosity to Entropy Ratio of a Hadronic Gas” Note: Muroya et. al have factor of 2 coefficient in viscosity formula, whereas we don’t.

26. Preliminary Results for Baryon Diffusion (Units in fm)

27. A previous study of diffusion Sasaki, Nonaka, et al. Europhys. Lett., 54 (1) (2004)

28. Idea : Compute Time-Evolution of Viscosity of System Losing Equilibrium • Full 3-d Hydrodynamics • QGP evolution UrQMD Hadronization Cooper-Frye formula hadronic rescattering Monte Carlo TC TSW t fm/c PREMISE TO BE ESTABLISHED:Timescale over which η is extracted << timescale over which system alters macroscopic properties. <πxy(0) πxy (t)> < (πxy (0) )2 > yielding η(t + kΔt) = < πxy(0) πxy(Δt )> corres. to η(t + (k-1)Δt) . Recursion Relation: