Transport Coefficients of Interacting Hadrons Anton Wiranata & MadappaPrakash (Advisor) Department of Physics and Astronomy Ohio University, Athens, OH
Topics • 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
Motivation • 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.
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 (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
The Boltzmann equation The collision integral Bulk viscosity Heat conductivity Shear Viscosity the solution (deviation function) has the general structure
Bulk Viscosity1st approximation Relativity parameter Reduced enthalpy Relativistic Omega Integrals Ratio of specific heats Relative momentum dependent Thermal weight Transport cross section Relative momentum : g = mc sinhψ Total momentum : P = 2mc coshψ
Shear Viscosity1st approximation Thermal variable Contains collision cross sections
Shear viscosity: Limiting Situations Shear viscosity The quantities ci,j contain omega integrals Non-relativistic case : z=m/kT >> 1 g : Dimensionless relative velocity Non-relativistic omega integral Note the g7 – dependence in the kernel, which favors high relative velocity particles in the heat bath (energy density & pressure carry a g4 – dependence). Note also the importance of the relative velocity and angle dependences of the cross section.
Shear viscosity - Illustrations Hard sphere cross section –
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).
Bulk Viscosity : Limiting Situations Bulk viscosity The quantities ai,j contain omega integrals Non relativistic case : z = m/kT >> 1 g : Dimensionless relative velocity Non-relativistic omega integral
Bulk Viscosity : Illustrations Hard sphere cross section –
Illustration Continued : Interacting Pions (Experimental Cross Sections) Parametrization from Bertsch et al. , PR D37 (1988) 1202.
Convergence of Successive Approximations 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.
Bulk viscosity & the speed of sound Adiabatic speed of sound Chapman-Enskog 1st approximation Features thermodynamic variables The omega integral contains transport cross-section Utilizing
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
Higher order bulk viscosity 3rd order bulk viscosity 2nd order bulk viscosity Feature thermodynamic variables Terms that feature speed of sound dependences
Inelastic collisions & shear viscosity 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
The role of inelastic collisions in bulk viscosity Internal excitation and creation of new species of particles contribute to bulk viscosity cint & cv 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.
Multi-component system (binary mixture) The same kind of particle collision Thermodynamics terms Different kind of particle collision Relativistic Omega Integrals Relative momentum weight Thermal weight Transport cross section
Binary mixture of constant cross section Bulk viscosity of two type of spherical particles
N-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
1st order calculation Coefficients to be determined
Ongoing work and future plans • Calculation of ηs & ηs /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( 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 ηv and ηv /s