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3.3 BBGKY Hierarchy ABSTRACT CHAO, JINGYI. Transport Properties of Strongly Interacting Quantum Fluids: From CFL Quark Matter to Atomic Fermi Gases. (Under the direction of Thomas Sch¨afer.) Kinetic theory is a theoretical approach starting from the first principle, which is par- ticularly suit to study the transport coefficients of the dilute fluids. Under the framework of kinetic theory, two distinct topics are explored in this dissertation. CFL Quark Matter We compute the thermal conductivity of color-flavor locked (CFL) quark matter. At temperatures below the scale set by the gap in the quark spec- trum, transport properties are determined by collective modes. We focus on the contribu- tion from the lightest modes, the superfluid phonon and the massive neutral kaon. We find ∼ × 26 8 −6 −1 −1 −1 that the thermal conductivity due to phonons is 1:04 10 µ500 ∆50 erg cm s K ∼ × 21 4 1=2 −5=2 −1 −1 −1 and the contribution of kaons is 2:81 10 fπ;100 TMeV m10 erg cm s K . Thereby we estimate that a CFL quark matter core of a compact star becomes isothermal on a timescale of a few seconds. Atomic Fermi Gas In a dilute atomic Fermi gas, above the critical temperature, Tc, the elementary excitations are fermions, whereas below Tc, the dominant excitations are phonons. We find that the thermal conductivity in the normal phase at unitarity is / T 3=2 but is / T 2 in the superfluid phase. At high temperature the Prandtl number, the ratio of the momentum and thermal diffusion constants, is 2=3. The ratio increases as the temperature is lowered. As a consequence we expect sound attenuation in the normal phase just above Tc to be dominated by shear viscosity. Finally, We calculate the viscosity spectral function of the dilute atomic Fermi gas in three different channels. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is !0 ' (2")=(3η) where " is the energy density and η is the shear viscosity. At finite momentum the spectral function approaches the 1=2 collisionless limit and the width is of order !0 ∼ k(T=m) . ⃝c Copyright 2011 by Jingyi Chao All Rights Reserved Transport Properties of Strongly Interacting Quantum Fluids: From CFL Quark Matter to Atomic Fermi Gases by Jingyi Chao A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Physics Raleigh, North Carolina 2011 APPROVED BY: John Blondin Chueng Ji Dean Lee Wenbin Lu (ST) Thomas Sch¨afer Chair of Advisory Committee BIOGRAPHY The author was born in Yifeng, Jiangxi, China. ii ACKNOWLEDGEMENTS First and foremost I would like to express my deepest gratitude to my advisor Dr. Thomas Schaefer, for his thoughtful guidance during my graduate study. I would like to thank Dr. Schaefer, who led me to experience research from intuitive physics pictures to solid mathematic skills. I thank him for his patience, encouragement and support. I also thank him for the financial support as graduate student. I would like to thank Dr. Dean Lee for helping me practice the presentation. I would also like to thank Dr. John Blondin, Dr. Chueng Ji, Dr. Dean Lee and Dr. Wenbin Lu for serving on my oral defense committee and my dissertation reading committee. I greatly appreciate their time and effort. I would like to thank Dr. Matt Braby for discussions and close collaborations. Thank all my officemates. It would be a lonely office without you guys. Thank all my friends for helping me go through the hardest time in my life; for helping me have a regular dinner schedule. Thank my sisters for believing and bearing, and for everything that taken for me. Last, but most important, I thank my father for him letting me know how to be a tough person; I thank my mother for her caring, worrying and considering. Thank my parents for their unconditional love and endless support. iii TABLE OF CONTENTS List of Tables ..................................... vii List of Figures .................................... viii Chapter 1 Introduction .............................. 1 1.1 Quantum Liquids . 1 1.1.1 Fermi Liquids, Electron Gases . 3 1.1.2 Nearly Perfect Fluid, Strongly Coupled QGP . 5 1.1.3 Helium . 7 1.2 Response to Gradient Perturbations . 10 1.2.1 Shear Viscosity . 11 1.2.2 Bulk Viscosity . 12 1.2.3 Thermal Conductivity . 13 1.3 Organization of the Thesis . 14 Chapter 2 Mathematical Models ........................ 15 2.1 Hydrodynamic Equation . 15 2.2 Navier-Stokes Equations . 18 2.2.1 Energy Dissipation from N-S equations . 20 2.3 Israel-Stewart Equations . 22 2.4 A Word at the End . 25 Chapter 3 Kinetic Theory ............................. 26 3.1 Introduction . 26 3.2 Quasi-particle Excitation . 27 3.3 BBGKY Hierarchy . 27 3.3.1 Liouville's Equation . 27 3.3.2 The BBGKY Hierarchy . 28 3.4 The Boltzmann Transport Equation . 30 3.4.1 Collisional Invariants . 30 3.4.2 Macroscopic Variables and Conservation Equations . 32 3.4.3 Grad's Momentum Expanding Method . 34 3.4.4 Chapman-Enskog Expansions Method . 36 3.5 Transport Coefficients . 37 3.5.1 Constrains on the Corrected Distribution Function . 37 3.5.2 The Solutions of Transport Coefficients . 39 iv Chapter 4 Quark Matter in the CFL Phase .................. 44 4.1 Introduction . 44 4.2 QCD Matter . 46 4.2.1 Overview of the Quark Matter Phase Diagram . 46 4.3 Color Super-Conductivity . 48 4.3.1 Attracting Forces in QCD . 48 4.3.2 Pairing Symmetry . 50 4.3.3 Gap Equation . 51 4.3.4 Two Flavors Color Superconductivity . 53 4.4 Color-Flavor-Locked Phase . 54 4.4.1 Properties of CFL . 55 4.4.2 Effective Field Theory . 56 4.4.3 Kaon Condensation . 61 4.5 Neutron star . 63 4.6 Thermal Conductivity and Transport Theory . 65 4.6.1 Estimations from Mean Free Path . 65 4.6.2 Subtle Results from Boltzmann Equation . 66 4.7 Variational Solution to the Boltzmann Equation . 73 4.8 Collision Terms . 75 4.8.1 Superfluid Phonon . 75 4.8.2 Massive Kaon . 78 4.9 Results . 79 4.10 Conclusions . 83 Chapter 5 Trapped Atomic gases ........................ 86 5.1 Introduction . 86 5.2 Unitarity Fermi Gas . 89 5.2.1 Feshbach Resonances . 89 5.2.2 The Crossover from BCS to BEC . 92 5.2.3 Effective Field Theory at Superfluid Phase . 93 5.3 Elementary Excitations . 97 5.4 Thermal Conductivity . 100 5.4.1 Normal Phase . 101 5.4.2 Superfluid Phase . 104 5.5 Shear Viscosity and the Prandtl Number . 107 5.5.1 Shear Viscosity . 107 5.5.2 Prandtl Number . 109 5.6 Sound Attenuation . 110 5.7 Conclusions . 113 v Chapter 6 Viscosity Spectral Functions of the Dilute Fermi Gas in Ki- netic Theory .............................. 115 6.1 Introduction . 115 6.2 Preliminaries . 117 6.3 Hydrodynamics . 118 6.4 Kinetic Theory in Curved Space . 121 6.4.1 Transverse Channel . 123 6.4.2 Longitudinal Case . 128 6.5 Summary and Discussion . 131 Chapter 7 Outlook ................................. 133 7.1 Three Body Interaction . 133 7.2 Higher Order of Dissipative Fluid Dynamics . 136 7.2.1 Results from Baier-Romatschke-Son-Starinets-Stephanov . 136 7.2.2 Results from York-Moore . 138 7.2.3 Results from Denicol-Koide-Rischke . 142 7.3 Conclusion . 144 References ....................................... 146 Appendices ...................................... 159 Appendix A . 160 A.1 Streaming Terms in the Boltzmann Equation . 160 A.1.1 Superfluid Phonon . 160 A.1.2 Massive Kaon . 162 A.2 Orthogonal Polynomials . 165 Appendix B . 167 B.1 Trial Functions . 167 B.2 Thermodynamic Properties . 168 B.2.1 Virial Expansion . 168 B.2.2 Specific Heat . 170 vi LIST OF TABLES Table 1.1 Phases in 3He............................. 9 vii LIST OF FIGURES Figure 1.1 Elliptic flow v2 for different hadron species [1, 2, 3], plotted as a function of transverse momentum, compared with ideal hydrody- namics predictions [4, 5] . 6 Figure 1.2 Helium Three, the phase diagram plotted as a function of temper- ature and pressure[6]. 8 Figure 1.3 Helium Four, the phase diagram plotted as a function of temper- ature and pressure[6]. 10 Figure 4.1 Sketched phase diagram of QCD matter as the function of tem- perature and density [7]. 46 Figure 4.2 The one-gluon exchange vertex. 49 Figure 4.3 The graphical representation of the Schwinger-Dyson equation in the improved rainbow approximation. The quark propagator in the Nambu-Gorkov basis and the gluon propagator are denoted by the solid lines and the wavy lines, respectively. 52 Figure 4.4 One-loop diagrams. External dashed lines represent the NG boson field; solid lines are fermion propagators. 59.
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