Rapidity Dependence of Transverse Momentum Correlations from Fluctuating Hydrodynamics
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Rapidity Dependence of Transverse Momentum Correlations from Fluctuating Hydrodynamics
Rajendra Pokharel1, Sean Gavin1 and George Moschelli21 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
oMotivation
o Hydrodynamics of Fluctuations and Viscosity
o Diffusion of pt correlations
o Results
o Summary
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Motivationo 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
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Quantity of interest
Experiments measure pt 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) 162302
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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 modesWWND 2013 Rajendra Pokharel 2/4/13
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Regular diffusion of transverse flow fluctuations.
dissipativeideal
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, 034903 (2004)
We ignore bulk viscosity. Linearized Israel-Stewart:
Saves causality
satisfies the diffusion equation r satisfies
satisfies the causal diffusion equation r satisfies
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Temperature dependent η/sDiffusion of Δr using Bjorken flow and (τ, η) coordinates
Entropy production
Ideal First order Second order
A. Muronga, Phys.Rev. C69, 034903 (2004)
Viscosity
T. Hirano and M. Gyulassy, Nucl. Phys. A769, 71(2006), nucl-
th/0506049.
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Temperature dependent η/s
Entropy density
EOS IIstandard
EOS ILattice (s95p-v1)Lattice: P. Huovinen and P. Petreczky, Nucl.Phys.
A837, 26(2010), 0912.2541
T. Hirano and M. Gyulassy, Nucl. Phys. A769,
71(2006), nucl-th/0506049.
Temperature and dependent diffusion coefficient
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Wave vs diffusion
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Results
Relaxation time:τπ = 5-6, AMY, Phys. Rev. D79, 054011 (2009), 0811.0729τπ = 6.3, J. Hong, D. Teaney, and P. M. Chesler (2011), 1110.5292
STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467
R. Pokharel, S. Gavin, G. Moschelli in preparation
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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) 467STAR (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
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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
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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) 054915
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.
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Backups
Constant , first order vs second order
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Backups
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Backups
STAR: H. Agakishiev et al, Phys.Lett. B704 (2011) 467
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Backups
M. Sharma WWND 2011 presentation
WWND 2013 Rajendra Pokharel 2/4/13