Systematic Approach to Knudsen Number and Viscosity Extraction

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Systematic Approach to Knudsen Number and Viscosity Extraction. Jamie Nagle (University of Colorado, Boulder) Peter Steinberg (Brookhaven National Laboratory) Bill Zajc (Columbia University) arXiv:0908.3684. How to Quantify h /s?.

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Jamie Nagle (University of Colorado, Boulder)

Peter Steinberg (Brookhaven National Laboratory)

Bill Zajc (Columbia University)

arXiv:0908.3684

How to Quantify h/s?

Need 3-d relativistic viscous hydrodynamics to compare to bulk medium flow.

h/s ~ 0

h/s = 1/4p

h/s = 2 x 1/4p

h/s = 3 x 1/4p

In fact this was a major theory milestone for the field.

This example is part of a very promising

“work in progress” in the field.

Alternative Approach (Boltzmann Style)

Knudsen

Number

R = size when flow develops

ST = transverse overlap area

dN/dy = number of partons

e = eccentricity

Postulate relation of Knudsen number to flow and then relate to h/s.

First, we reproduced the work of

Drescher, Dumitru, Gombeaud, Ollitrault (arXiv:arXiv:0704.3553v2)

Zero viscosity limit determined from fit

Deviation (less flow) due to finite viscosity

Drescher et al. with Glauber initial conditions h/s = 2.4 x 1/4p

and Color Glass Condensate initial conditions h/s = 1.4 x 1/4p

However, there is a mistake in the CGC case from using two different speeds of sound, that algebraically cancel.

It should be h/s =1.9 x 1/4p

Issue #1

Statement that this form obeys the correct limits for K0 and K∞, but not derived from fundamentals.

Challenge, are there other forms that obey these limits and give equally good fits, but different h/s?

Yes, an infinite set based on

One standard deviation range

h/s = 0.34 - 2.55 (x 1/4p)

Including well below the bound

Issue #2

What if data indicates saturation of v2/e due to quantum bound, not approach to ideal hydro limit? Can we tell the difference?

Introduce modification to mean free path that goes to limiting value of deBroglie wavelength.

Increases extracted h/s value by ~ +1/4p and provides an equally good fit.

Of course this is a crude approximation to the breakdown of the Boltzmann picture.

Compare to previous

value w/o bound

h/s x 4p = 2.59 ± 0.53

x=0.0

x=0.13

x=1.00

Issue #3

Very sensitive to initial geometry and centrality dependence.

Monte Carlo Glauber with

two-component contribution for spatial density.

Mostly < 30% changes in

Eccentricity versus centrality.

Results in >120% change in extracted h/s because of curvature sensitivity.

Issue #4,5,6,…

Assumed data systematics are point-to-point uncorrelated. Most likely they are quite the opposite.

Nearly double all uncertainties.

There are significant uncertainties in the parameters characterizing the medium and these values vary as a function of space and time, one needs to specify the precise space and time interval for averaging, and that averaging of intermediate quantities is consistent with determining an average h/s.

Anonymous quote from response to our paper:

“Why don't you just say that this approach is useless for quantitative work?”

Initial Condition Side Lesson #1

In this particular viscous hydrodynamic constraint, the two initial conditions considered were:

I. “Glauber”

In this paper x=1.00

(positions of binary collisions

only) and optical model

(i.e. no fluctuations)

II. “CGC”

fKLN model (from Drescher+Nara)

(i.e. no fluctuations)

PHOBOS MCGlauber

x=0.13 (with rp fluctuations)

Large, centrality dependent e changes

with correct

inclusion of fluctuations.

Drescher MC-KLN

(with rp fluctuations)

Drescher f-KLN

(no rp fluctuations)

PHOBOS MCGlauber

x=0.13 (no rp fluctuations)

Ratio MC Glauber

with rp fluctuations / without

Ratio MC-KLN (fluctuations)/

fKLN (no fluctuations)

Initial Condition Side Lesson #2

arXiv:0908.2617v2

Utilizes ~ 60% difference in eccentricities of CGC and Glauber to “exclude the possibility … [of] Glauber initial conditions.”

PHOBOS MCGlauber

x=0.13 (with rp fluctuations)

Fluctuations dramatically reduce the e

differences between

CGC and Glauber.

Drescher MC-KLN

(with rp fluctuations)

Drescher f-KLN

(no rp fluctuations)

PHOBOS MCGlauber

x=0.13 (no rp fluctuations)

Ratio fKLN/Glauber

both with no rp fluctuations

Ratio MC-KLN/Glauber

both with rp fluctuations

Public viscous hydrodynamic code from Paul Romatchke

Modifications and running (KZ Mendoza and Mike McCumber).

Project to understand how these fluctuating initial conditions fully impact our physics conclusions.

Average over 1M Events

One Single Event

PHOBOS MCGlauber

x=0.13 (with rp fluctuations)

Drescher MCGlauber

x=0.13 (with rp fluctuations)

Drescher MCGlauber

x=0.13 (no rp fluctuations)

PHOBOS MCGlauber

x=0.13 (no rp fluctuations)

Ratio Drescher/PHOBOS MCGlauber

with no rp fluctuations

Ratio Drescher/PHOBOS MCGlauber

with rp fluctuations