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Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Viscosity of hadron gas

Maneesha Sushama Pradeep

School of physical sciencesProject guide-Dr.Bedangadas Mohanty

NISER

20th March 2013

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Overview

1 Viscosity-A General Introduction

2 Motivation

3 System of hadron gases

4ηs

5 Computational Results

6 Comparison with known fluids

7 Summary

8 What more?

9 References

10 Acknowledgments

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Viscosity

• Bulk phenomenon that characterises the resistance offeredto the free flow of a fluid due to the interaction betweenthe various layers of itself

• Inverse of fluidity

•Fx = ηA

dvxdz

In principle η a 3× 3 tensor.

• Newtonian Fluids:η is a constant;eg:Water

• Non-Newtonian fluids:Stress is NOT linearly proportionalto strain rate.eg:paint,cement

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Viscosity

• Bulk phenomenon that characterises the resistance offeredto the free flow of a fluid due to the interaction betweenthe various layers of itself

• Inverse of fluidity

•Fx = ηA

dvxdz

In principle η a 3× 3 tensor.

• Newtonian Fluids:η is a constant;eg:Water

• Non-Newtonian fluids:Stress is NOT linearly proportionalto strain rate.eg:paint,cement

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Figure: Pitch

Figure: Laminar Flow of fluidbetween two plates

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Measuring viscosity-First year lab experiment

• A spherically uniform body was dropped• Terminal velocity assumed• Stokes’Law,F = 6πηrv where η is the viscosity, r is the

radius of the sphere, v its velocity and F is the viscousforce acting on it.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Temperature dependence of viscosity in liquids andgases

Liquid viscosity:

• In a liquid, every molecule has to overcome the forces dueto its neighbours to move

• Inversely proportional to mobility

• Probability that a molecule has the required energy at a

given temperature ∝ e−EaRT

• Mobility ∝ e−EaRT

• η = AeEaRT

• lnη = lnA+ EaRT

• Liquid viscosity decreases with increase in temperature

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Gas viscosity

• η = 13nl < |p| >,η-viscosity,l-Mean free

path,< |p| >-expectation value of momentum,n-numberdensity

• l = 1√2σ

,σ-Effective crossection for collisions

• < |p| >=√

8mkBTπ ,m-mass of gas molecule,T-temperature

of the system,kB-Boltzmann’s constant

• η = 23

√mkBTπσ2

• Viscosity of gas increases with increase in temperature

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Gas viscosity

• η = 13nl < |p| >,η-viscosity,l-Mean free

path,< |p| >-expectation value of momentum,n-numberdensity

• l = 1√2σ

,σ-Effective crossection for collisions

• < |p| >=√

8mkBTπ ,m-mass of gas molecule,T-temperature

of the system,kB-Boltzmann’s constant

• η = 23

√mkBTπσ2

• Viscosity of gas increases with increase in temperature

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Why should we study viscosity of hadron gases

• Elliptic flow in high enery heavy ion collisions-signature offluid like behaviour of particles formed?

• Initial anisotropy of the particles produced in positionspace leading to an anisotropy in the momentumspace-indicating strongly interacting medium?

Figure: Non-central collision seen in the transverse plane: the overlaparea, where particles are produced, is not a circle

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

My regime of study

• Two distinct phases:Quark Gluon plasma and Hadrons

• Two stages:Chemical freeze out and thermal freeze out

• My regime:Hadronic stage,only elastic collisions,betweenchemical freeze out and thermal freeze out

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

My regime of study

• Two distinct phases:Quark Gluon plasma and Hadrons

• Two stages:Chemical freeze out and thermal freeze out

• My regime:Hadronic stage,only elastic collisions,betweenchemical freeze out and thermal freeze out

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Figure: εT 4V s

T−TcTc

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A system of ideal hadron gases

• Baryons and mesons, composite relativistic particles

• Only interaction through elastic collisions which areinstantaneous

• Particles are point-like, volume occupied much lesscompared to distance between them

Figure: Proton

Figure: Pion

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A system of ideal hadron gases

• Baryons and mesons, composite relativistic particles

• Only interaction through elastic collisions which areinstantaneous

• Particles are point-like, volume occupied much lesscompared to distance between them

Figure: Proton

Figure: Pion

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A system of ideal hadron gases

• Baryons and mesons, composite relativistic particles

• Only interaction through elastic collisions which areinstantaneous

• Particles are point-like, volume occupied much lesscompared to distance between them

Figure: Proton

Figure: Pion

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A System of Volume excluded Vanderwaal gases

Particles have a finite radius, two particles cannot occupy thesame volume.

Figure: Excluded Volume Vanderwaal Gas

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Viscosity of a system of relativistic gases

• η ∝ nl < |p| >

• < |p| >=∫∞0 p3n(p)dp∫∞0 p2n(p)dp

where n(p) is the average number of

particles having momentum p

• For a system of identical particles,

η =5

64√π

√mT

r2

K 52(mT )

K2(mT )

where Kν is the modified Bessel function of order ν.

• For a system of x different hadrons(assuming same radius),

η =5

64√π

√T

r2Σxi=0

√mi

K 52(miT )

K2(miT )

where mi is the mass of the ith hadron.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Why ηs(Ratio of viscosity to entropy density)?

• Analogous to kinematic viscosity

• Entropy density is a factor times number density

• Dimensionless quantity

• This quantity facillitates a better comparison of theviscosities of two isentropic samples.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Finding entropy density of a system of volumeexcluded relativistic hadrons at a temperature T

• Aid of statistical mechanics

• Which ensemble to choose?

• Grand canonical pressure ensemble-Pressure, temperatureand chemical potential are the conserved quantities

• µi = biµB + siµS + qiµQ, µB >> µQ, µS ,µQ ≈ µS ≈ 0

• Determining the appropriate statistics

• Calculate entropy density using the commonly usedexpressions in Statistical Mechanics

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Finding entropy density of a system of volumeexcluded relativistic hadrons at a temperature T

• Aid of statistical mechanics

• Which ensemble to choose?

• Grand canonical pressure ensemble-Pressure, temperatureand chemical potential are the conserved quantities

• µi = biµB + siµS + qiµQ, µB >> µQ, µS ,µQ ≈ µS ≈ 0

• Determining the appropriate statistics

• Calculate entropy density using the commonly usedexpressions in Statistical Mechanics

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Statistical Analysis

• Grand Canonical Pressure Partition function for a generalsystem:

D(z, p, T ) =

∫ ∞0

dV e−V pZ(V, z, T )

• Grand canonical pressure partition function for a system ofvolume excluded gases:

D(z, s, T ) = Σ∞N=0

φN

N !

∫ ∞vN

dV e−(s(V−vN))(V − vN)N

, where φ is the ideal gas number density and v is theexcluded volume per molecule.

• Pressure is given by the transcedental equation:

p = e−vpT Tφ

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Statistical Analysis

• Grand Canonical Pressure Partition function for a generalsystem:

D(z, p, T ) =

∫ ∞0

dV e−V pZ(V, z, T )

• Grand canonical pressure partition function for a system ofvolume excluded gases:

D(z, s, T ) = Σ∞N=0

φN

N !

∫ ∞vN

dV e−(s(V−vN))(V − vN)N

, where φ is the ideal gas number density and v is theexcluded volume per molecule.

• Pressure is given by the transcedental equation:

p = e−vpT Tφ

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

• Number density:

ni =e−

vpT φi

1 + ve−vpT φ

,where φ is ideal gas number density

• Entropy density:

s =x

1 + vx(1 + vx+

T

φ

∂φ

∂T)

, where x = e−vpT Tφ

• s = An,where n is the total number density of the systemand A = 1 + vx+ T

φ∂φ∂T .

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Relating with experimental data

Furthermore, the viscosity of the medium consisting of theparticles produced at RHIC, SPS, AGS and SIS energies can befound out as follows:

• The experiment gives the particle densities as the observedquantity.

• These values are then fit into the expressions for particledensities obtained using statistical analysis and thetemperature and chemical potential of the systemproduced at those energies can be computed.

• Viscosity is calculated accordingly.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computation

• For a system of pions,

η =5

64√π

√mT

r2

K 52(mT )

K2(mT )

where Kν is the modified Bessel function of order ν,m isthe mass of pion, T is the temperature of the system.

• s = An,where n is the total number density of the systemand A = 1 + vx+ T

φ∂φ∂T (φ is the ideal gas number density).

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Viscosity as a function of temperature for varying radii

0

100

200

300

400

500

60 80 100 120 140 160 180

n(M

eV/fm

2 )

T(Mev)

Viscosity as a function of temperature for varying radii

r=0.5 fmr=0.3 fmr=0.4 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Viscosity as a function of temperature for varying radii

0

100

200

300

400

500

60 80 100 120 140 160 180

n(M

eV/fm

2 )

T(Mev)

Viscosity as a function of temperature for varying radii

r=0.5 fmr=0.3 fmr=0.4 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Entropy Density as a function of temperature for varying radii

0

0.2

0.4

0.6

0.8

1

60 80 100 120 140 160 180

Ent

ropy

Den

sity

(1/fm

3 )

T(Mev)

Entropy Density as a function of temperature for varying radii

r=0.5 fmr=0.3 fmr=0.4 fmr=0.1 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Entropy Density as a function of temperature for varying radii

0

0.2

0.4

0.6

0.8

1

60 80 100 120 140 160 180

Ent

ropy

Den

sity

(1/fm

3 )

T(Mev)

Entropy Density as a function of temperature for varying radii

r=0.5 fmr=0.3 fmr=0.4 fmr=0.1 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Ratio of Viscosity Vs Entropy Density as a function oftemperature for varying radii

0

5

10

15

20

25

30

35

40

60 80 100 120 140 160 180

n/s

T(Mev)

Ratio of viscosity to entropy density for varying radii

r=0.1 fmr=0.5 fmr=0.3 fmr=0.4 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Ratio of Viscosity Vs Entropy Density as a function oftemperature for varying radii

0

5

10

15

20

25

30

35

40

60 80 100 120 140 160 180

n/s

T(Mev)

Ratio of viscosity to entropy density for varying radii

r=0.1 fmr=0.5 fmr=0.3 fmr=0.4 fm

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Ratio of Viscosity Vs Entropy Density as a function of radiusfor varying beam energies

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

n/s

Radius(fm)

Viscosity to entropy density Vs radius for varying energies

SPSAGSSIS

RHIC

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Computational results for a system of pion gases

Figure: Ratio of Viscosity Vs Entropy Density as a function of radiusfor varying beam energies

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

n/s

Radius(fm)

Viscosity to entropy density Vs radius for varying energies

SPSAGSSIS

RHIC

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Table: Temperature and chemical potential of systems of differentbeam energies

Beam energy(GeV) T(MeV) µBSIS 2.32 64.3 800.8AGS 4.86 116.5 562.2SPS 17.3 154.4 228.6

RHIC 200 161.1 23.5

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A comparison between viscosity measurements ofHe, N2, Water and Pion gas with r = 0.5fm

Figure: ηs Vs T−Tc

Tcfor different fluids

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1

n/s

(T-Tc)/Tc

Comparing viscoity to entropy density ratios of fluids

WaterNitrogen

Hadron gasHelium

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

A comparison between viscosity measurements ofHe, N2, Water and Pion gas with r = 0.5fm

Figure: ηs Vs T−Tc

Tcfor different fluids

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1

n/s

(T-Tc)/Tc

Comparing viscoity to entropy density ratios of fluids

WaterNitrogen

Hadron gasHelium

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Summary

• The viscosity of the pion gas system was found to increasewith rise in temperature and decrease with increase inradius.

• As the temperature increases entropy density decreases.

• The ratio of viscosity to entropy density of pion was foundto decrease with temperature.

• The ratio of viscosity to entropy density was found to beof the same order as Helium, Nitrogen and Water for piongas with radius 0.5 fm at high temperatures.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

What more?

I am currently working on computationally determing theviscosity and η

s of a system consisting of all the discovered andproposed hadrons which have mass less than 2 GeV. Since theexperiments have shown close to a perfect liquid like behaviour,the η

s expected for the system is considerably low.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

References I

• Physical Review C 77,024911(2008),Viscosity in

the excluded volume hadron gas

model,M.I.Gorenstein,M.Hauer and O.N.Moroz

• The pressure ensemble as a tool for describing

hadron-quark phase transition,

R.Hagedorn,RE.Th-3392,CERN

• New Journal of Physics 13(2011)

055008(18pp),Elliptic Flow:A brief

review,Raimond Snellings

• Heat and Thermodynamics,Richard H Dittman

Mark W Zeemansky,TMH Publications

• Statistical Mechanics,R.K.Pathria Paule

d.Beale,Elsevier Publications

• Introductory Nuclear Physics,Krane,Wiley India

Edition

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

Acknowledgments

I express my sincere gratitute towards my guide for the valuableguidance and support he has been delivering me since thebegining of my project. I am also thankful to Dr. Victor, Dr.Chitrasen and Nasim for clearing my doubts and queries at theappropriate time. A special thanks to my friends for the fruitfuldiscussions I had with them.

Viscosity ofhadron gas

ManeeshaSushamaPradeep

Viscosity-AGeneralIntroduction

Motivation

System ofhadron gases

ηs

ComputationalResults

Comparisonwith knownfluids

Summary

What more?

References

Acknowledgments

THANK YOU