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Measurement of Shear Viscosity in Lattice Gauge Theory without Kubo Formula

Lattice2008 Jul. 14, 2008. Measurement of Shear Viscosity in Lattice Gauge Theory without Kubo Formula. Masakiyo Kitazawa with M. Asakawa, B. Muller, C. Nonaka. Transport Coefficients of the QGP. One of the hottest topics!. Success of ideal hydrodynamic models to describe RHIC data.

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Measurement of Shear Viscosity in Lattice Gauge Theory without Kubo Formula

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  1. Lattice2008 Jul. 14, 2008 Measurement of Shear Viscosityin Lattice Gauge Theorywithout Kubo Formula Masakiyo Kitazawa with M. Asakawa, B. Muller, C. Nonaka

  2. Transport Coefficients of the QGP One of the hottest topics! • Success of ideal hydrodynamic models to describe RHIC data. • h/s=1/4p from AdS/CFT – lower bound? Analyses of viscosities on the lattice Karsch, Wyld 1987; Nakamura, Sakai 1997,2004; Meyer 2007, 2008, talk on Fri. ; Pica talk on Thu. • based on the Kubo formula • - problem in analytic contituation • Is spatial volume large enough?

  3. Our idea: Create the spatially inhomogeneous flow on the lattice and measure viscosities “experimentally”. Spatially inhomogeneous system: Gopie, Ogilvie PRD59,034009(1999) Experimental Measurement of h v F L

  4. z u3(x) is linear. u3(x) x x Velocity Distribution v F L :const.

  5. Direct Measurements of Viscosities If we can create a static “hydrodynamic” flow on the lattice, transport coefficients can be determined by measuring Tmn’s. The energy-momentum tensor microscopically: hydrodynamic: directly observed on the lattice long range and course gained

  6. z y x L 0 L/2 L cf.) grand canonical: Lagrange multiplier Path integral representation imaginary  sign problem Momentum Source Statistical average of an observable O: Put external sources to Hamiltonian

  7. L L/2 x Microscopic dynamics governs short range behavior. The hydro. mode forming the linear behavior will survive at long range. Momentum Flow with Source z y x

  8. never gives rise to a linear func. at long range • not responsible for the hydrodynamic flow Teaney, PRD74,045025(2006) Meyer, arXiv:0806.3914 Taylor Expansion • 0th order: • 1st order: 2-point functions

  9. Taylor Expansion • We need higher-order correlation functions including both • sources at x=0 and L/2 to create the hydrodynamic modes. • 2nd order term should vanish, since l.h.s. is an odd function of l. • The hydrodynamic mode can appear at least from 3rd order.

  10. Numerical Simulation parameter determined by Bielefeld group pure gauge: b = 6.499, a = 0.049fm, Nt=6(T = 2.5Tc) lattice size: 64x322x6 – Lx= 3.13fm Nconf ~ 20k 128x322x6 – Lx= 6.27fm Nconf ~ 27k 192x322x6 – Lx= 9.41fm Nconf ~ 13k • each 20~60 steps of HB+OR4 • on bluegene@KEK (128nodes) • one week simulation n m Clover term for field strength

  11. exp. damping no structure ~0.4fm Numerical results 128x322x6 Nconf ~ 27k 1st order x L/2 source 3.1fm

  12. Numerical results 128x322x6 Nconf ~ 27k 3rd order x source L/2 • No structure is mesuared except for near the source…

  13. Summary • We tried to create a system having hydrodynamic flow • by introducing the momentum source to the Hamiltonian. • Evaluating its effect by Taylor expansion up to 3rd order, • any signals for the flow is not observed thus far. What’s wrong? • Just a problem of statistics? • Microscopic dynamics forbids a generation of hydro. flow? • Is Taylor expansion available for this problem? L/2 L x

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