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Bulk viscosity of interacting Hadrons

Anton Wiranata and Madappa Prakash Department of Physics and Astronomy Ohio University, Athens, OH. Bulk viscosity of interacting Hadrons. Quark Matter 2009, March 30 – April 4, Knoxville, TN. Topics. Overview of results Bulk viscosity in the Chapman– Enskog approximation

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Bulk viscosity of interacting Hadrons

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  1. Anton Wiranata and Madappa Prakash Department of Physics and Astronomy Ohio University, Athens, OH Bulk viscosity of interacting Hadrons Quark Matter 2009, March 30 – April 4, Knoxville, TN

  2. Topics • Overview of results • Bulk viscosity in the Chapman–Enskog approximation • Bulk viscosity and speed of sound • Analysis in limiting situations • Numerical results of ηv and ηv /s • Chiralpions • Massive interacting pions • Multi-component mixtures • Binary and N-species systems • Ongoing work and future plans Quark Matter 2009, March 30 – April 4, Knoxville, TN

  3. Bulk viscosity / entropy density vs T P.Arnold, et.al JHEP05, 051(2003) Kharzeev & Tuchin JHEP09, 093(2008) Chen and Wang, Arxiv:0711.4824v1 Wiranata & Prakash. Kapusta, arxiv:0809.3746v2 Quark Matter 2009, March 30 – April 4, Knoxville, TN

  4. Chapman-Enskog Approximation “For small deviations from equilibrium, the distribution function can be expressed in terms of hydrodynamic variables ( f(x,p) μ(x), u(x), T(x) ) and their gradients. Transport coefficients (viscosities, thermal conductivity, etc. ) are then calculable from relativistic kinetic theory.” φ(1) : Deviation function Equilibrium distribution function μ(x) : Chemical potential u(x) : Flow velocity T(x) : Temperature Quark Matter 2009, March 30 – April 4, Knoxville, TN

  5. Beginning with the relativistic Boltzman equation and inserting f(0) the solution (deviation function) has the general structure Quark Matter 2009, March 30 – April 4, Knoxville, TN

  6. Bulk Viscosity Ratio of specific heats First approximation : Reduced enthalpy : (ε + P) / nmc2 Relativistic Omega Integrals Relative momentum weight Thermal weight Transport cross section Relative momentum : g = mc sinh ψ Total momentum : P = 2mc cosh ψ Quark Matter 2009, March 30 – April 4, Knoxville, TN

  7. Bulk Viscosity & the Speed of Sound Chapman-Enskog 1st approximation Features thermodynamic variables The omega integral contains transport cross-section Utilizing Quark Matter 2009, March 30 – April 4, Knoxville, TN

  8. Analysis for limiting cases Nearly massless particles or very high temperatures (z << 1) : Massive particles such that z >> 1 : Lesson : For a given T, intermediate mass particles contribute significantly to ηv Quark Matter 2009, March 30 – April 4, Knoxville, TN

  9. Variational approximation Sean Gavin, Nucl. Phy. A 435, 826 (1985) Gavin’s result can be rewritten as Kapusta , arxiv:0809.3746v2 A similar analysis leads to the result that intermediate mass particles control the magnitude of ηv Quark Matter 2009, March 30 – April 4, Knoxville, TN

  10. Numerical results (interacting pions) Parametrization from Bertsch et al. , PR D37 (1988) 1202. Quark Matter 2009, March 30 – April 4, Knoxville, TN

  11. Entropy Density Non-interacting part Entropy Density Fugacity β =1/kT : Inverse Temperature Energy Density Pressure ``Venugopalan & Prakash   Nucl. Phys. A546 (1992) 718'' Interacting part (up to 2nd virial expansion) Kinematic Factor Thermal weight Interactions Quark Matter 2009, March 30 – April 4, Knoxville, TN

  12. Results (Interacting Pions) Prakash et al. , Phys. Rep. 227 (1993) 321 Bulk &Shear Viscosity Entropy Density ηv 3rd appro x 103 2nd 1st Total [GeV][fm-2][c-1] [fm-3] Non-interacting ηs 1st appro Interacting 2nd T[MeV] T[MeV] Quark Matter 2009, March 30 – April 4, Knoxville, TN

  13. Bulk viscosity / entropy density Chen and Wang, Arxiv:0711.4824v1 Wiranata & Prakash QGP sum rule Massless pions Lattice For chiral pions ηv/s increases with T in contrast to ηv /s for massive interacting pions Quark Matter 2009, March 30 – April 4, Knoxville, TN

  14. Multi-Component system Begin with binary mixture Ratio of specific heats Relative momentum weight Thermal weight Transport cross section W.A. Leeuwen, et.al Quark Matter 2009, March 30 – April 4, Knoxville, TN

  15. Binary mixture Results for constant cross-sections Quark Matter 2009, March 30 – April 4, Knoxville, TN

  16. Multi-component system Result for N – species at pth order of approximation Omega integral Coefficients to be determined Solubility conditions (assures 4-momentum conservation in collisions) Involves ratios of specific heats Quark Matter 2009, March 30 – April 4, Knoxville, TN

  17. 1st order calculation Coefficients to be determined Quark Matter 2009, March 30 – April 4, Knoxville, TN

  18. Ongoing work and future plans • Calculation of ηv & ηv /s for a mixture of interacting hadrons with masses up to 2 GeV. • Development of an approach to calculate the needed differentialcross-sections for hadron-hadron interactions including resonances up to 2 GeV. • Comparison of the Chapman-Enskog results with those of the Green-Kubo approach. • Inclusion of decay processes. • All the above for shear viscosity ηv and ηv /s Quark Matter 2009, March 30 – April 4, Knoxville, TN

  19. Thank you Quark Matter 2009, March 30 – April 4, Knoxville, TN

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