Viscous Coefficients of a Hot Pion Gas

Sourav Sarkar

doi:10.1155/2013/627137

Abstract

The steps essentially involved in the evaluation of transport coefficients in linear response theory using Kubo formulas are to relate the defining <i >retarded</i> correlation function to the corresponding <i >time-ordered</i> one and to evaluate the latter in the conventional perturbation expansion. Here we evaluate the viscosities of a pion gas carrying out both the steps in the <i >real-time</i> formulation. We also obtain the viscous coefficients by solving the relativistic transport equation in the Chapman-Enskog approximation to leading order. An in-medium <svg style="vertical-align:-0.1638pt;width:20.2125px;" id="M1" height="8.1499996" version="1.1" viewBox="0 0 20.2125 8.1499996" width="20.2125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.888)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43
t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" /></g><g transform="matrix(.017,-0,0,-.017,10.109,7.888)"><use xlink:href="#x1D70B"/></g> </svg> cross-section is used in which spectral modifications are introduced in the propagator of the exchanged <svg style="vertical-align:-3.56265pt;width:9.2875004px;" id="M2" height="12.175" version="1.1" viewBox="0 0 9.2875004 12.175" width="9.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D70C" d="M516 277q0 -126 -140 -231q-36 -27 -75.5 -42.5t-63.5 -15.5q-38 0 -73 43q-10 -44 -18 -106q-18 -134 -32 -151q-12 -16 -40 -25t-46 -10l-5 14q22 38 84 406q16 92 44.5 146.5t88.5 95.5q68 47 136 47q63 0 101.5 -47.5t38.5 -123.5zM432 259q0 61 -27 101t-79 40
t-82 -49t-49 -148l-24 -127q42 -36 88 -36q79 0 126 61.5t47 157.5z" /></g> </svg>.

Highlights

One of the most interesting results from experiments at the Relativistic Heavy Ion Collider (RHIC) is the surprisingly large magnitude of the elliptic flow of the emitted hadrons
We review the formulation of the nonequilibrium density operator and obtain the expressions for the viscosities in terms of equilibrium two-point functions
We review the real-time formulation of equilibrium thermal field theory leading to the spectral representations of bosonic two-point functions [16]. This formulation begins with a comparison between the time evolution operator e−iH(t2−t1) of quantum theory and the Boltzmann weight e−βH = e−iH(τ−iβ−τ) of statistical physics, where we introduce τ as a complex variable

Summary

Introduction

One of the most interesting results from experiments at the Relativistic Heavy Ion Collider (RHIC) is the surprisingly large magnitude of the elliptic flow of the emitted hadrons. One is the kinetic theory method in which the nonequilibrium distribution function which appears in the transport equation is expanded in terms of the gradients of the flow velocity field The coefficients of this expansion which are related to the transport coefficients are perturbatively determined using various approximation methods. The other approach is based on response theory in which the nonequilibrium transport coefficients are related by Kubo formulas to equilibrium correlation functions They are perturbatively evaluated using the techniques of thermal field theory. We calculate the viscous coefficients in a kinetic theory framework by solving the transport equation in the Chapman-Enskog approximation to the leading order This approach being computationally more efficient [12] has been mostly used in the literature to obtain the viscous coefficients.

Viscous Coefficients in the Linear

Viscous Coefficients in the Kinetic Theory Approach

Findings

Summary and Conclusion