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Rapidity Dependence of Transverse Momentum Correlations from Fluctuating Hydrodynamics Rajendra Pokharel 1, Sean Gavin 1 and George Moschelli 2 1 Wayne State University 2 Frankfurt Institute of Advanced Studies Winter Workshop on Nuclear Dynamics Feb 3-10 Squaw Valley, CA

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Outlines o Motivation o Hydrodynamics of Fluctuations and Viscosity o Diffusion of p t correlations o Results o Summary WWND 2013 Rajendra Pokharel 2/4/13

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Motivation o Modification of transverse momentum fluctuations by viscosity o Transverse momentum fluctuations have been used as an alternative measure of viscosity o Estimate the impact of viscosity on fluctuations using best information on EOS, transport coefficients, and fluctuating hydrodynamics WWND 2013 Rajendra Pokharel 2/4/13

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Quantity of interest Experiments measure p t correlations and find C. Theory calculates it from the quantity r. Our quantity of interest is C, given by r : two-particle transverse momentum correlation function: Sean Gavin & Mohamed Abdel-Aziz, Phys. Rev. Lett. 97 (2006) WWND 2013 Rajendra Pokharel 2/4/13

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Transverse momentum fluctuations Linearized Navier-Stoke equation for momentum density: Small fluctuation in transverse flow Results in shear viscosity Helmholtz decomposition: sound waves (damped by viscosity) Longitudinal modes: viscous diffusion Transverse modes: We are interested on transverse modes WWND 2013 Rajendra Pokharel 2/4/13

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Regular diffusion of transverse flow fluctuations. dissipative ideal Relativistic viscous hydro and diffusion of flow fluctuations Local conservation of energy-momentum First order (Navier-Stokes) hydro: Linearized Navier-Stokes for transverse component for flow fluctuation Problem with this regular diffusion equation - violates causality !

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Second order hydro and diffusion of two-particle correlations Second order (Israel-Stewart) hydro: A. Muronga, Phys.Rev. C69, (2004) We ignore bulk viscosity. Linearized Israel-Stewart: Saves causality satisfies the diffusion equation r satisfies satisfies the causal diffusion equation r satisfies WWND 2013 Rajendra Pokharel 2/4/13

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Temperature dependent η/s Diffusion of Δr using Bjorken flow and (τ, η) coordinates Entropy production Ideal First order Second order A. Muronga, Phys.Rev. C69, (2004) Viscosity T. Hirano and M. Gyulassy, Nucl. Phys. A769, 71(2006), nucl-th/ WWND 2013 Rajendra Pokharel 2/4/13

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Temperature dependent η/s Entropy density EOS II standard EOS I Lattice (s95p-v1) Lattice: P. Huovinen and P. Petreczky, Nucl.Phys. A837, 26(2010), T. Hirano and M. Gyulassy, Nucl. Phys. A769, 71(2006), nucl-th/ Temperature and dependent diffusion coefficient WWND 2013 Rajendra Pokharel 2/4/13

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Wave vs diffusion WWND 2013 Rajendra Pokharel 2/4/13

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Results Relaxation time: τ π = 5-6, AMY, Phys. Rev. D79, (2009), τ π = 6.3, J. Hong, D. Teaney, and P. M. Chesler (2011), STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467 R. Pokharel, S. Gavin, G. Moschelli in preparation WWND 2013 Rajendra Pokharel 2/4/13

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Results How about other centralities ? R. Pokharel, S. Gavin, G. Moschelli in preparation STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467 STAR (other centralities): M. Sharma’s presentation, WWND 2011, Winter Park, CO Bumps in a few most central cases both in data and second order diffusion calculations WWND 2013 Rajendra Pokharel 2/4/13 This occurs at the same centralities in (although the comparison is not great) We claim that this a second order diffusion effect

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Results WWND 2013 Rajendra Pokharel 2/4/13 First order vs second order

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Results WWND 2013 Rajendra Pokharel 2/4/13 STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467 Computation: R. Pokharel, S. Gavin, G. Moschelli in preparation NeXSPheRIO: Sharma et al., Phys.Rev. C84 (2011) Width of correlation NeXSPHeRIO (= NEXUS + SPHERIO) uses ideal hydro for the evolution of initial correlation. It reproduces most qualitative features of correlation (e.g., the “ridge”). However, it does not reproduce the increasing width with centralities. Except for the a few most central cases, first order diffusion does not reproduce the data Second order does! Also, very small difference due to EOS I and EOS II. Order of entropy production makes almost no change in the results.

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Summary o The observable C has the second order “bump in the hump”. Experimental data shows the effect for the same centralities. o Theory the bumps is clear: pronounced effect of wave part of the causal diffusion equation. o NeXSPheRIO (ideal hydro + correlation) does not produce broadening width, and therefore does not agree with width data. o First order viscous hydro calculations does not reproduce data except for a few most central collisions o Second order viscous hydrodynamic calculation of width fits the data. WWND 2013 Rajendra Pokharel 2/4/13

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Thank You ! Contact: WWND 2013 Rajendra Pokharel 2/4/13

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Backups Constant, first order vs second order WWND 2013 Rajendra Pokharel 2/4/13

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Backups WWND 2013 Rajendra Pokharel 2/4/13

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Backups STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467 WWND 2013 Rajendra Pokharel 2/4/13

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Backups M. Sharma WWND 2011 presentation WWND 2013 Rajendra Pokharel 2/4/13

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