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 initial state pre-equilibrium QGP and hydrodynamic expansion hadronic phase freeze-out 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? low viscosity large viscosity Why study hadronic phase? 1)Need to know hadronic viscosity to constrain QGP viscosity. 2)Viscosity changes as function of time in a heavy ion collision!

4. Two Questions re: “low viscosity” 1)How low? (AdS/CFT: η/s≥1/4π? KSS bound) 2)If there is a minimum, where is it? Near T c ? PRL 94. 111601 (2005) Kovtun, Son, Starinets 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 T c on hadronic side  microscopic transport model can help here! Next Question: How do we compute transport coefficients?

6. Phenomenological Transport Equation: thermodynamic/mechanical flux linearly proportional to applied field in small field limit. Examples of transport coefficients: thermal conductivity, diffusion, shear viscosity. y x y=a y=0 P yx V x = v 1 V x = v 2 Shear Viscosity Coefficient: Green-Kubo: compute linear transport coefficients by examining near-equilibrium correlations! Linear Transport Coefficients & Green-Kubo Relations

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. OK, how to model the hadronic medium? Suggests technique of molecular dynamics (MD) simulations.

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: (d min 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 : - Kinetic Equilibrium: Compute TEMPERATURE by fitting to Boltzmann distribution! - Chemical equilibrium: DISABLE multibody decays/collisions.  RESPECT detailed balance!

10. What about Kinetic Equilibrium? ε= 0.5 GeV/fm 3 ρ B =ρ 0 ε= 0.5 GeV/fm 3 ρ B =ρ 0 T=168.4 MeV

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 μ B for our system from SHAREv2, P and ε known from UrQMD.) Denote as s Gibbs. Method II: Weight over specific entropies of particles, where s/n is a function of m/T & μ B /T! Denote as s specific SHARE v2: Torrieri et.al.,nucl-th/0603026 -Tune particles/resonances to those in UrQMD.

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 gas IN EQUILIBRIUM.

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

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

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 σ decreases E/V =0.3 GeV/cubic fm E/V =1.0 GeV/cubic fm

18. η decreasing w. finite μ B : Think specific binary collisions! η ~p/σ: Resonant πN crosssxns larger than ππ. Increasing μ B ! ε=0.2 GeV/fm 3 ε=0.5 GeV/fm 3

19. 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 μ B. In progress : Analyzing μ=0 mesonic matter for T<100 MeV. Outlook: - Describe time-evolution of transport coefficient in relativistic heavy-ion reaction. Full 3-d Hydrodynamics QGP evolution UrQMD t fm/c hadronic rescattering Hadronization TCTC T SW

20. Backup Slides

21. String theory to the rescue? A nice conjecture on viscosity. Kovtun, Son, Starinets: hep-th/0405231 PRL 94. 111601 (2005) Csernai, Kapusta, McLerran: nucl-th/0604032 PRL 97. 152303 (2006) 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 (AdS 5 ) equivalent to finite temperature solution for its dual field theory (N=4 SUSY YM).

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 Cooper-Frye formula UrQMD t fm/c hadronic rescattering Monte Carlo Hadronization TCTC T SW PREMISE TO BE ESTABLISHED: Timescale over which η is extracted << timescale over which system alters macroscopic properties. yielding η(t + kΔt) = corres. to η(t + (k-1)Δt). Recursion Relation: