Second-order Relativistic Hydrodynamic Equations Compatible with

Boltzmann Equation and Critical Opalescence around the QCD Critical Point

Quark Matter 2009 Knoxville, March 30 --- April 4, 2009

Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co.)

Critical Opalescence around the QCD Critical Pointand

Second-order Relativistic Hydrodynamic

Equations Compatible with Boltzmann Equation

Quark Matter 2009 Knoxville, March 30 --- April 4, 2009

Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co.)

ContentsI Density fluctuations around QCD critical point with rel.

dissipative hydrodynamics; new possible signal for identifying the QCD critical point*

II Derivation of Israel-Stewart type hydrodynamic equations on the basis of the (dynamical) renormalization group method** ;

only brief exposition of the result

* Yuki Minami and T.K., in preparation,** Kyosuke Tsumura and T.K., in preparation

The same universality class; Z2

Density fluctuation is the soft mode of QCD critical point!The sigma mode is a slaving mode of the density.

H. Fujii, PRD 67 (03) 094018;H. Fujii and M. Ohtani, Phys.Rev.D70(2004)Dam. T. Son and M. A. Stephanov, PRD70 (’04) 056001

T

P

SolidLiq.

Classical Liq.-Gas

c.f. The coupling of the density fluctuation with the scalar mode was discussed in, T.K. Phys. Lett. B271 (1991), 395

Critical point

Triple.P

gas

Spectral function of density fluctuations The density fluctuation depends on the transport as well as

thermodynamic quantities which show an anomalous behavior around the critical point.

Especially, the existence of the density-temperature coupling. For non-relativistic case with use of Navier-Stokes eq. L.D. Landau and G.Placzek(1934), L. P. Kadanoff and P.C. Martin(1963), R. D. Mountain, Rev. Mod. Phys. 38 (1966), 38 H.E. Stanley, `Intro. To Phase transitions and critical phenomena’ (Clarendon, 1971)

We apply for the first time relativistic hydrodynamic equations to analyze the spectral properties of density fluctuations, and examine possible critical phenomena.

Missing in the previous analyses.

Relativistic Hydrodynamics

0u 0u

dissipative terms

(1) Energy-frame

(2) Particle frame ; Eckart(1940), unstable Tsumura-Kunihiro-Ohnishi, Phys.Lett.B646(2007)

(3) Israel-Stewart

0u u

0

In the rest frame of the fluid,

Inserting them into , and taking the linear approx.

etc

Linear approximation around the thermal equilibrium;

Rel. effects

・ Linearized Landau equation (Lin. Hydro in the energy frame);

Solving as an initial value problem using Laplace transformation, we obtain, in terms of the initial correlation.

with

Spectral function of density fluctuations in the Landau frame

In the long-wave length limit, k→0

/p nc c t

Long. Dynamical ：： specific heatratio

： sound velocity

thermal expansion rate ：

Rel. effectsRel. effects appear only in the width of the peaks.

rate of isothermal exp.

Notice:

As approaching the critical point, the ratio of specific heats diverges!

P

The strength of the sound modes vanishes out at the critical point.

enthalpy

sound modesthermal mode

Eq. of State of idealMassless particles

[MeV]

[1/fm]

[1/fm]

[MeV]

In the energy(Landau) frame, relativistic effects appear only in the peakheightand width of the Rayleigh peak.

Rel. Non-rel.

Rayleigh peak

Brillouin peak Brillouin peak

sound mode

thermal mode

Spectral function from I-S eq.

[1/fm]

IS

Non-rel. (N-S)

No contribution in the long-wave length limit k→0.

For

Conversely speaking, the first-order hydro. Equations have no problem to

describe the hydrodynamic modes with long wave length, as it should.

[1/fm]

Particle frame; the new equation

Rel. case (TKO)

Non-rel. (Navier-Stokes)

Rel. effects appear in the Brillouin peaks (sound mode) but not inThe Rayleigh peak.

K. Tsumura, T.K. K. Ohnishi; Phys. Lett. B646 (2007) 134TKO=

Critical behavior of the density-fluctuation spectral functions

/p nc c t

BSo, the divergence of and the viscocities therein can not be observed, unfortunately.

In the vicinity of CP, only the Rayleigh peak stay out, while the sound modes (Brillouin peaks) die out.

Critical opalescence

c.f. 2 ( ) / 3z a

Spectral function of density fluctuation at CP

( ) /c cT T T

( ) /c ct T T T

The sound mode (Brillouin) disappearsOnly an enhanced thermal mode remains.

Suggesting interesting criticalphenomena related to sound mode. Eg. Disappearance of Mach cone at CP! A hint for detecting CP!

Spectral function at CP

0.4

0. 1t

The soft mode around QCD CP is thermally induced density fluctuations,but not the usual sound mode.

Needs explicit examination.

Cf. STAR, arXiv:0805/0622, to be published in PRL

R. B. Neufeld, B. Muller, and J. Ruppert,arXiv:0802.2254gT gL

(z -

ut)

x (

rx)

mD2 T

x gz (

rx)

mD2 T

Why at all do sound modes die out at the Critical Point ?

The correlation length

0t

The wave length of sound mode

s

The hydrodynamic regimel

as t 0

s However, around the critical point

So the hydrodynamic sound modes can not be developed around CP!

s

<<

<<

II An RG derivation of 2nd order hydrodynamic equations

K.Tsumura and T.K. , preliminary results was presented at JPS meeting Sep. 23, 2008, in preparation

Relativistic Boltzmann equation

Conservation law of the particle number and the energy-momentum

H-theorem.

The collision invariants, the system is local equilibrium

Maxwell distribution (N.R.)Juettner distribution (Rel.)

Geometrical image of reductionof dynamics

nR

t X

M

dim M m n

dim X n

( )ts

O dim ms

Invariant and attractive manifold

( )d

dt

XF X

( )d

dtsG s

M={ ( )}X X X s

( , )fX r p ; distribution function in the phase space (infinite dimensions)

{ , , }u T ns ; the hydrodinamic quantities (5 dimensions), conserved quantities.

eg.

RG/E equation

Slow dynamics (Hydro dynamics + relaxation equations ）

Energy frame

Particle frame0

Relaxation equations (very long)

(I)

(II)

T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179

S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236

Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006),

The viscocities are frame-independent, in accordance with Lin. Res. Theory.

However, the relaxation times and legths are frame-dependent.

The form is totally different from the previous ones like I-S’s,And contains many additional terms.

contains a zero mode of the linearizedcollision operator. 2p p m

Conformal non-inv.gives the ambiguity.

0

0u u

Summary• Density fluctuations is analyzed using rel. hydro.• The second-order terms in IS theory do not affect the (long-

wave length) hydrodynamic modes.• As a dynamical critical phenomena, Brillouin peaks due to

sound modes disappear. The disappearance of the sound mode suggests the suppresion or even disappearance of Mach cone, which can be a signal of the created matter hitting CP. Need further explicit calculation for confirmation,

• The (dynamical) RG method is applied to derive generic second-order hydrodynamic equations.

• There are so many terms in the relaxation terms which are absent in the previous works, especially due to the conformal non-invariance, which gives rise to an ambiguity in the separation in the first order and the second order terms (matching condition).

• A practical use of I-S level equations, however, may be problematic.

© University of CambridgeDoITPoMS, Department of Materials Science and Metallurgy, University of CambridgeInformation provided by doitpoms@msm.cam.ac.uk.

Critical Opalescence is a general phenomenon for the matter with 1st order transition.

Experiment at lab.

HBT?

http://www.msm.cam.ac.uk/doitpoms/tlplib/solid-solutions/videos/laser1.mov

Back Ups

for

for

Definitions of critical exponents

Israel-Stewart eq. in particle frame

： relaxation times

tends to coincide with the Eckart equation.

→ diverges; unstable

Eckart eq.(1st order);

I-S eq.

→ stable.

Stability of I-S eq.

→ unstable! even with finite rel. time.

Rel. time of thermalconductivity

W.A.Hiscock and L.Lindblom Phys.Rev.D35(1987)

STAR dataAway side shape modification

STAR

2.5 < pT

trig< 4 GeV/c

1< pT

assoc < 2.5 GeV/c

Technique: Measure 2- and 3- particle correlations on the away-side triggered by “high” pT hadron in central coll’s. Cone-shaped emission should show up in 3-particle correlations as signal on both sides of backward direction.

Central Au+Au 0-12% (STAR)

(1-2)/2

B. Muller@NFQCD08

The Mach cone

(z - u

t)

Energy density

Momentum density

x (

rx)

mD2 T

x gz (

rx)

mD2 T

gT gL

Unscreened source with min/max cutoff

B. Muller@NFQCD08

R. B. Neufeld, B. Muller, and J. Ruppert,arXiv:0802.2254

References on the RG/E method:• T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179• T.K.,Jpn. J. Ind. Appl. Math. 14 (’97), 51• T.K.,Phys. Rev. D57 (’98),R2035• T.K. and J. Matsukidaira, Phys. Rev. E57 (’98), 4817• S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236• Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24• T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006),

8089 (hep-th/0512108)• K. Tsumura, K. Ohnishi and T.K., Phys. Lett. B646 (2007),

134

L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(’95),376; Phys. Rev. E54 (’96),376.

C.f.