1. Transport Coefficients of Interacting Hadrons Anton Wiranata & Madappa Prakash (Advisor) Madappa Prakash (Advisor) Department of Physics and Astronomy Ohio University, Athens, OH

2.  Motivation  Shear & bulk viscosities in the Chapman-Enskog approximation  Non relativistic limit for shear and bulk viscosities  Bulk viscosity and the speed of sound  Inelastic collisions & transport coefficients  Shear & bulk viscosities of mixtures

3.  Transport coefficients (bulk & shear viscosities) are important inputs to viscous hydrodynamic simulations of relativistic heavy-ion collisions.  Collective motion with viscosity influences (reduces) the magnitude of elliptic flow relative to ideal (non-viscous) hydrodynamic motion.

4. “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 (e.g., bulk & shear viscosities) are then calculable from relativistic kinetic theory.” Deviation function Equilibrium distribution function μ (x) : Chemical potential u(x) : Flow velocity T(x) : Temperature

5. the solution (deviation function) has the general structure The collision integral Bulk viscosity Heat conductivityShear Viscosity

6. Reduced enthalpy Relativistic Omega Integrals Relativity parameter Ratio of specific heats Transport cross section Thermal weightRelative momentum dependent Relative momentum : g = mc sinh ψ Total momentum : P = 2mc cosh ψ

7. Thermal variable Contains collision cross sections

8. Shear viscosity Non-relativistic case : z=m/kT >> 1 Non-relativistic omega integral g : Dimensionless relative velocity Note the g 7 – dependence in the kernel, which favors high relative velocity particles in the heat bath (energy density & pressure carry a g 4 – dependence). Note also the importance of the relative velocity and angle dependences of the cross section. The quantities c i,j contain omega integrals

9. Hard sphere cross section –

10. Hard-sphere cross section – It is desirable to reproduce these results in alternative approaches, such as variational & Green-Kubo calculations (in progress with collaborators from Minneapolis & Duke).

11. Bulk viscosity Non relativistic case : z = m/kT >> 1 Non-relativistic omega integral g : Dimensionless relative velocity The quantities a i,j contain omega integrals

12. Hard sphere cross section –

14. Illustration Continued : Interacting Pions (Experimental Cross Sections) Parametrization from Bertsch et al., PR D37 (1988) 1202.

17. 1. The convergence of successive approximations to shear viscosity is significantly better than that for the bulk viscosity. 2. However, bulk viscosity is about 10 -3 x shear viscosity, so its influence on the collision dynamics would be minimal, except possibly near the transition temperature.

18. Chapman-Enskog 1 st approximation Adiabatic speed of sound Features thermodynamic variables The omega integral contains transport cross-section Utilizing

19. 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 η v

20. 2 nd order bulk viscosity Feature thermodynamic variables 3 rd order bulk viscosity Terms that feature speed of sound dependences

22. Inelastic collisions can induce transitions to excited stated or result in new species of particles. For the general formalism, see e.g., Kapusta(2008). In the nonrelativistic case (applicable to heavy resonances) of i + j  k + l (Wang et al., 1964), Energy of particle i Integration variable Energy difference Inelastic terms In the limit of the small Delta epsilon, inelastic collisions do not affect shear viscosity

23. Internal excitation and creation of new species of particles contribute to bulk viscosity c int & c v are the internal heat capacity & heat capacity at fixed V per molecules Inelastic term In non-relativistic limit (z = m/kT >>1), inelastic part of the cross section Contributes the most for bulk viscosity.

24. Thermodynamics terms The same kind of particle collision Different kind of particle collision Relative momentum weight Thermal weight Transport cross section Relativistic Omega Integrals

25. Bulk viscosity of two type of spherical particles

26. Result for N – species at p th order of approximation Solubility conditions (assures 4-momentum conservation in collisions) Coefficients to be determined Omega integral Involves ratios of specific heats

27. Coefficients to be determined

28.  Calculation of η s & η s /s for a mixture of interacting hadrons with masses up to 2 GeV.  Development of an approach to calculate the needed differential cross-sections for hadron-hadron interactions including resonances up to 2 GeV( In collaboration with Duke Univeristy).  Comparison of the Chapman-Enskog results with those of the Green-Kubo approach( In collaboration with Duke Univeristy).  Inclusion of decay processes (In collaboration with Minessota University  All the above for bulk viscosity η v and η v /s