Rapidity Dependence of Transverse Momentum Correlations from Fluctuating Hydrodynamics

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

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

WWND 2013 Rajendra Pokharel 2/4/13

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

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 !

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

WWND 2013 Rajendra Pokharel 2/4/13

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

Results

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First order vs second order

Results

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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.

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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Thank You !

Contact: rajpol@wayne.edu, rajpol@hotmail.com

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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

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