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Landau Hydrodynamics Cheuk-Yin Wong Oak Ridge National Laboratory

Dubna July 14, 2008

• Introduction• Landau hydrodynamics -- predictions on total N -- prediction on differential dN/dy -- space-time dynamics• Modification of Landau’s dN/dy• Comparison with experiment• Conclusions

C.Y. Wong, Dubna Lecture notes (to be posted on preprint archive).

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Why study Landau hydrodynamics?

• It gives a good description of experimental data• It gives a simple description of the space-time dynamics

of the dense hot matter produced in heavy-ion collisions• Dense hot matter evolution is needed in many problems• It is very simple

• L.D. Landau, Izv. Akad. Nauk SSSR, 17, 51 (1953)

• Belenkij and L.D.Landau,Usp.Fiz.Nauk. 56, 309 (1955)

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Recent revived interest in Landau hydrodynamics

• BRAHM dN/dy data agree with Landau hydrodynamics

Murray, J. Phys. G30, S667 (2004)

factorn contractio Lorentz

theof logarithm theis L

2 ln

2exp

2

1 24/1

p

NN

m

sL

Ly

LKs

dydN

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Landau hydrodynamics give the correct multiplicity in AA collisions

Steinberg, arxiv:nucl-ex/0702019

Ly

LKs

dydN

2exp

2

1 24/1

Landau AA

Landau pp

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Landau hydrodynamics exhibits limiting fragmentation

byyy

yL

y

LdydN

'

'2

'exp

1~

2

Steinberg, arxiv:nucl-ex/0702019

AA Collisions

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Unanwsered questions:

• Is Landau’s formula for y or η ?• Landau rapidity distribution is actually

The Gaussian Landau distribution is only an approximation

• Does the original Landau distribution agree with data?

224/1 exp yLAKsdydN

norm

We need to answer these questions in Landau hydrodynamics

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Landau Hydrodynamics • Local thermal equilibrium • Particle number is proportional to entropy• Entropy is conserved in Landau hydrodynamics • First stage, independent

(i) 1-D longitudinal expansion

(ii) transverse expansion• Second stage of rapidity freeze-out

when the transverse displacement is equal

to the transverse dimension of the system

x(tm) = a ; tm=rapidity freeze-out time

.2 ,exp 224/1 KyLAKsdydN

norm

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Consrvation of Entropy in Landau hydrodynamics

• Mean-free path is small, low viscosity• The only means to destroy entropy conser

vation is shock wave. • Landau hydridyncmics is applied after the

completion of the compression stage. It deals only with the expansion of the dense matter after shock-wave compression

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Landau hydrodynamics predictions on total Nch

GeV.in for 2 )2/(

particles ofnumber Total

34

34

entropy Total

34

densityopy Entr

34

densityEnergy

; 3

4 volumeInitial

22/energy totalInitial

entropy. initial thefrom particles ofnumber total the

getcan that wemeansentropy ofon Conservati

number particle toalproportion is Entropy

).conserving(entropy isentropic is flow icalHydrodynam

2/1

2/1

2/1

30

4/33

0

4/33

04/3

30

30

NNNNpartch

NN

NN

NN

NN

NN

p

NN

NNNN

sKsKNN

AsN

As

ArrscV

rscc

rsVE

m

sArV

AsAsE

NS

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Landau prediction on Nch/(Npart/2) agrees with experiment

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1-D longitudinal expansion

)/ln(

of logarithm Introduce

sinh cosh Introduce

3/

state ofequation simple use We

)(

0

0

equations icHydrodynam

10

1101

0100

ty

t

zttztt

yuyu

p

pguupTzT

tT

zT

tT

13

1-D longitudinal expansion solution

conditions initial of choices limited (ii)

part edge not the fluid, ofpart bulk for theonly (i)

:gesdisadvanatMinor

simplicity :Advantage

on.substitutidirect by solution eapproximatan is that thisprovecan We

2/)(

)(3

4exp

solution eapproximat Simple

0)(

2

0)(

2

becomes equations icHydrodynam

0

2

2

yyy

yyyy

tte

te

ty

y

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Solution of 1-D longitudinal expansion

15yat

uaut

tx

attx

uu

ttx

tv

ttv

tx

xxp

tv

uu

gpguupT

vuuvuuuupT

xT

tT

xx

x

xx

2

2

00

2

200

22

00

222222

02002002

0202

cosh44)(

3

1)(2

3

4

isequation icHydrodynam

)(2 ;

2

1)(

isnt displaceme Transverse3

1

3

4

becomesequation icHydrodynam

)(

;3

4)(

0 equation icHydrodynam

Transverse expansion during the first stage

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Landau condition for rapidity freeze-out

Angle (and rapidity ) will not change after the transverse displacement is equal to the transverse dimension.

diameter tranversecosh4

)(2

2

aya

ttx

Rapidity freeze-out occurs at different time t for different rapidity (and z).

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Rapidity distribution

exp :ondistributirapidity Final

2exp

ondistributientropy Final

2ln

cosh )(exp

)(at on distributirapdity theneed We

.cosh

)(exp

cosh/ cosh/

cosh/sinh

at timeon distributiRapidity

22

22

))((

4/30

4/30

4/3

20

0

yydydN

SN

dyyyydS

yyya

y

ydy

tyyyycdS

ytt

ydy

tyyyycdS

c

ydytydytudS

yytzdzudS

t

b

bb

b

ytt

m

m

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Comparison with experiment

Modified distribution gives better agreement than Landau distribution

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Comparison with experiment

Modified distribution gives better agreement than Landau distribution

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Conclusions• Landau’s prediction on Nch agrees with data

• The rapidity distribution in Landau hydrodynamics should be modified. The modified rapidity distribution gives better agreement with experimental data than the Landau distribution

• The quantitative agreement of Landau hydrodynamics supports its use in other problems of heavy ion collisions, such as J/psi suppression, jet quenching, and ridge jet-medium interaction,….