Heavy-ion collisions at high baryon densities: …nfqcd2018/Slide/Steinheimer.pdfHeavy-ion collisions at high baryon densities: Modelling the QCD phase transition Jan Steinheimer 06.06.2018

Heavy-ion collisions at high baryon densities:Modelling the QCD phase transition

Jan Steinheimer

06.06.2018

Jan Steinheimer (FIAS) 06.06.2018 1 / 34

Phase Transitions

Important features of thethermodynamic properties ofmaterials can be depicted in a socalled Phase Diagram.

1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 1 0 1 1 0 20

4 0

8 0

1 2 0

1 6 0

2 0 0

Temp

eratur

e [Me

V]

B a r y o n d e n s i t y [ f m - 3 ]

N u c l e a r L i q u i d - G a sC o e x i s t e n c e

Q u a r k - H a d r o nC o e x i s t e n c e ?

H a d r o n R e s o n a n c e G a s

Q u a r k G l u o n P l a s m a

S u p e r c o n d u c t i n gQ u a r k s ?

D e n s e N u c l e a r M a t t e r ?

E x o t i c N u c l e a r M a t t e r ?

Q u a r k i o n i c M a t t e r ?

Different Types of PhaseTransitions

Crossover: Smooth transitionwhere the first derivatives of thefree energy are alwayscontinuous. Butter.

1st order: Phase transitionwhere the first derivatives of thefree energy show a discontinuityand the latent heat is finite.

2nd order: Derivatives of thefree energy are also continuous,but higher order derivativesdiverge. Critical behavior.


Motivation

This talk: Only the phase transition! Nothing about the CeP.

How do we find the first order phase transition?

A worthwhile discussion: What gives a bigger effect, phase transitionor CeP. Maybe at the end.

If it is a phase transition

Do we have the models that can tell us what to look for?E.g. fluid-dynamics.


Motivation







Motivation







Phase separation

Nucleation vs. spinodal decomposition

Nucleation: Thermal fluctuations serve as seeds for bubble formation.(e.g. ice in water). SLOW!

Spinodal decomposition: System is quenched below separationtemperature. Instabilities occur (e.g. hot oil + water). FAST!


Importance of models

Why dynamical models?

Finite number of particles and volume

Conservation laws

Finite Lifetime

Can separate out different physics effects

...


Important ingredients

1 A dynamic model: In this case fluid dynamics.


Solving Fluid-Dynamics

The equations for relativistic fluid-dynamics are solved numerically on agrid of cell size ∆x = 0.2 fm:

∂µTµν = 0 and ∂µN

µ = 0

Tµν is the relativistic energy momentum tensor and Nµ the baryonfour-current. In ideal fluid-dynamics these can be written as:

Tµν = (ε+ p)uµuν − pgµν and N µ = nuµ

To close this system of equations one needs the equation of state, of theform p = p(ε, n), as an additional input.




2 A ”proper” equation of state.


The Maxwell Construction

Problem

Most model calculations, including a ’phase transition’, are based on aMaxwell constructed EoS!

Van der Waals equation isotherm.

”In thermodynamic equilibrium,a necessary condition for stability isthat pressure P does not increase withvolume V.”” The Maxwell construction is a way ofcorrecting this deficiency.”

In a dynamical scenario, locally thesystem is not necessarily stable.

Phase separation occurs.

11http://www.wikipedia.comJan Steinheimer (FIAS) 06.06.2018 9 / 34

The Maxwell Construction

Problem

Most model calculations, including a ’phase transition’, are based on aMaxwell constructed EoS!

Van der Waals equation isotherm.

”In thermodynamic equilibrium,a necessary condition for stability isthat pressure P does not increase withvolume V.”” The Maxwell construction is a way ofcorrecting this deficiency.”

In a dynamical scenario, locally thesystem is not necessarily stable.

Phase separation occurs.1

1http://www.wikipedia.comJan Steinheimer (FIAS) 06.06.2018 9 / 34

Non-Equilibrium Phase Transition

”Spinodal decomposition is essentially a mechanism for the rapid unmixingof a mixture of liquids or solids from one thermodynamic phase, to formtwo coexisting phases.”

I takes place for example when one quenches a mixture of two substancesrapidly below the demixing temperature. Then, the two substancesseparate locally, giving rise to the complicated structures which can beseen on the leftmost picture.



Equilibrium Phase Transition(Maxwell construction)

As the system dilutes, the phasesare always well separated

Phase II

Den

sity

Length


Phase separation is a dynamicalprocess.





Phase II

Den

sity

Length

Phase I

Interface







Phase I

Den

sity

Length







Phase II

Den

sity

Length

Phase I

Interface



Den

sity

Length





Phase II

Den

sity

Length

Phase I

Interface



Den

sity

Length





Phase II

Den

sity

Length

Phase I

Interface



Den

sity

Length


A better example: A Model for the Liquid-Gas P.T.

Take a mean field model: Walecka or hadronic σ-ω model.

σ serves as the order parameter.

Calculate the Grand Canonical potential Ω/V for fixed T and µ asfunction of σ

0.4 0.5 0.6 0.7 0.8 0.9 1.0-3

-2

-1

0

1

2

3

Local Maxima Local Minima

[MeV] =

CO= 922 B = 916 A = 938

/V [M

eV/fm

3 ]

/ 0

At T = 0 and µ = µCO, Ω/Vhas two, equally deep, minima.

Within the spinodal regionalways two minima!

Each corresponding to ameta-stable phase.

Maxima: unstable!


A Model for the Liquid-Gas P.T.

P = −Ω/V as a function of densityat T = 0

0.00 0.05 0.10 0.15 0.20

-1

0

1

2

Unstable Stable

P [M

eV/fm

3 ]

[MeV] =

CO= 922 B = 916 A = 938

/ 0

c2s < 0

Mechanically Unstable

Curve through the points givesthe nuclear EoS.

Mechanically unstable regiondefined by imaginary speed ofsound

∂P∂ε

∣∣T=0

= c2s < 0


The Phase Diagram in Density

A More convenient representation of the phase diagram is in T and ρ.

0.00 0.04 0.08 0.12 0.160

5

10

15

20

Metastable

Nucle

atio

n

Coexistence Line Spinodal B Spinodal A

T [M

eV]

/ 0

Unstable

Spino

dal d

ecom

posit

ion

Nucle

atio

n

Metastable and Unstableregions visible.

Note: Order ofSpinodals is reversed

Supercooled andOverheated phases.


Constructing an Effective EoS: Another way

If we don’t have an effective model that describes both phases

Obtain the free energy density fT (ρ) = εT (ρ)− TsT (ρ) by a splinebetween a Gas of int. nucleons+pions and a QGP.

∂2ρfT (ρ) > 0: (Meta-)Stable∂2ρfT (ρ) < 0: Unstable

Fluid evolution: T 6= const.but S/A = const.

c2s

∣∣T< 0: Isothermal spinodal

c2s

∣∣S/A

< 0: Isentropic spinodal

0 2 4 6 8 10 12

Compression ρ/ρs

-500

0

500

1000

1500

2000

2500

MeV

/fm

3

nucleons

quarks

fT=0

(ρ)

Maxwell

T=0

free energ

y densit

y

pressure

Alternatively: Do a Maxwell construction.


Constructing an Effective EoS: Another way

If we don’t have an effective model that describes both phases

Obtain the free energy density fT (ρ) = εT (ρ)− TsT (ρ) by a splinebetween a Gas of int. nucleons+pions and a QGP.

∂2ρfT (ρ) > 0: (Meta-)Stable∂2ρfT (ρ) < 0: Unstable

Fluid evolution: T 6= const.but S/A = const.

c2s

∣∣T< 0: Isothermal spinodal

c2s

∣∣S/A

< 0: Isentropic spinodal

Alternatively: Do a Maxwell construction.




2 A ”proper” equation of state.

3 Long range interactions. Lead to a surface energy.


The Gradient Term

A proper description of spinodal decomposition requires that finite-rangeeffects be incorporated.

0 2 4 6 8 10 124

6

8

10

Phase II

No Surface Tension With Surface Tension

B/

S

r [fm]

Phase I

We rewrite the local pressure as

p(r) = p0(ε(r), ρ(r))−a2 εsρ2s

ρ(r)∇2ρ(r)

The gradient term will cause a diffuseinterface to develop when matter of twocoexisting phases are brought intophysical contact.


The dispersion relation in Numerical Fluid-Dynamics

Calculation in a box with periodic boundaries

Ideal fluid dynamics: ω2 = c2sk

2

+ Gradient term: ω2 = c2sk

2 + a2 (εs/h)(ρ/ρs)2

k4

+ Shear and bulk viscosity: ω2 = c2sk

2 + a2 (εs/h)(ρ/ρs)2

k4 − ζ ωh0k2

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

=cs kNumerical:

a=0 a=0.033

[fm]

k [fm-1]

cs>1


The dispersion relation in Numerical Fluid-Dynamics


Ideal fluid dynamics: ω2 = c2sk

2

+ Gradient term: ω2 = c2sk

2 + a2 (εs/h)(ρ/ρs)2

k4

+ Shear and bulk viscosity: ω2 = c2sk

2 + a2 (εs/h)(ρ/ρs)2

k4 − ζ ωh0k2

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

=cs kNumerical:

a=0 a=0.033

[fm]

k [fm-1]

cs>1

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0 cs= /k

Numerical: a=0 a=0.033

d/d

k (~

c2 s)

k [fm-1]


Growth Rates in the unstable phase


The amplitude of a density undulation should grow exponentially withinthe unstable region

A(t) = At=0(eγkt + e−γkt)

20 40 60 80 100 120 140 160 1802

4

6

8

10

12

14

16

18

2020 40 60 80 100 120 140 160 180

2

4

6

8

10

12

14

16

18

20

Numerical viscosity cuts offlarge wave number growth

The gradient term modifies thegrowth rates:

γ2k = |vs|2k2 − a2(εs/h)(ρ/ρs)

2k4

Depending on a certain wavenumbers are favored






0.0 0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4 k=0.8 k=1.6 k=2.4 k=3.2 k=4.0

Am

plitu

de A

(t)

time t [fm]




2k4







0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

x=0.2 fm,a [fm]:

0.0 0.01 0.033 0.05

a=0.0; x=0.1 fm:k and k scaled by 1/2

a=0.033; x=0.1 fm

Gro

wth

rate

k [

fm-1]

wave number k [fm-1]




2k4



Show Animation I

Initialize Random noise in the unstable region and let it evolve.


Show Animation I

Initialize Random noise in the unstable region and let it evolve.Show movies.


Show Animation I

Initialize Random noise in the unstable region and let it evolve.Show movies.

0 10 20 30 40 50 60 70 800

5

10

15

20

Time t [fm/c]

Dis

tanc

e r [

fm]

0.92

0.96

1.00

1.04

1.08Spatial density correlation C(r)(a)


Initial State And Setup

We apply the UrQMD transport model for the initial, non-equilibrium, partof the collision.

When contracted nuclei havepassed through each other,

energy-, momentum- and baryondensities are mapped onto thecomputational grid.

-10 -5 0 5 10

-10

-5

0

5

10

X-Axis [fm]

Y-A

xis

[fm]

0

2

4

6

8

10

in units of s

t-t0= 0.00 fm


Evolution in Fluid-Dynamics

EoS with unstable phase:

EoS with Maxwell construction:




EoS with Maxwell construction:-5 -4 -3 -2 -1 0 1 2 3 4 5

-5-4-3-2-1012345

t-t0= 2.50 fm

Y-A

xis

[fm]

X-Axis [fm]

unstable - Maxw

in units of s

-6

-4

-2

0

2

4

6

Notable difference observed.Unstable phase leads to clustering ofbaryonnumber!




EoS with Maxwell construction:

0 2 4 6 8 100

5

10

15

20

Time t [fm/c]

Dis

tanc

e r [

fm]

0

20

40

60

70

Spatial density correlation C(r)(k)

0 2 4 6 8 100

5

10

15

20

Time t [fm/c]

Dis

tanc

e r [

fm]

0

20

40

60

70

Spatial density correlation C(r)(l)


Moments of the Baryon Density

Let’s be more quantitative

Define Moments of the net baryon density distribution:

〈ρN 〉 ≡ 1

A

∫ρ(r)Nρ(r) d3r

0 1 2 3 4 5 6

Elapsed time t - t0 (fm/c)

100

101

102

103

Density m

om

ents

<ρN

>/<

ρ>N

N = 7

N = 6

N = 5

N = 4

N = 3

N = 2

Elab

= 3 A GeV

As a function of time

0

5

10

15std EoS

Maxw EoS

0 1 2 3 4 5 6Beam energy E

lab (A GeV)

0

1

2

3

Ma

xim

um

en

ha

nce

me

nt

of

<ρN

>/<

ρ>N

std EoS

Maxw EoS

N = 5

N = 7

As a function of beam energy


How to make particles from densities

Typically done by use of the Cooper-Frye equation

EdN

d3p=

∫σf(x, p)pµdσµ (1)

Just calculating the integral will conserve all relevant quantumnumbers but lead to non-integer particle numbers.

To do resonance decays, final state rescattering and compare withdata on an event-by-event basis one usually does a Monte-Carlosampling of this equation.

One may even conserve some or all conserved quantities.

Important

Since the volume elements (cells) are usually small ∆x < 1fm, theparticle number in a cell is also Np << 1 and so one assumes it is Poissondistributed!




EdN

d3p=





Important





EdN

d3p=





Important





EdN

d3p=





Important





EdN

d3p=





Important



WHAT we calculate and WHEN?

Cumulants of net baryon number

K1 = M = 〈N〉K2 = σ2 =

⟨(δN)2

⟩K3 = Sσ3/2 =

⟨(δN)3

⟩K4 = κσ4 =

⟨(δN)4

⟩− 3

⟨(δN)2

⟩2

In coordinate space.

At a fixed time.


WHAT we calculate and WHEN?

Cumulants of net baryon number

K1 = M = 〈N〉K2 = σ2 =

⟨(δN)2

⟩K3 = Sσ3/2 =

⟨(δN)3

⟩K4 = κσ4 =

⟨(δN)4

⟩− 3

⟨(δN)2

⟩2

In coordinate space.

At a fixed time.

0 1 2 3 4 5 60.50

0.75

1.00

1.25

1.50

Enha

ncem

ent o

f 2 /M

Elapsed time t-t 0 [fm/c]

Cooper-Frye time


Effect on the cumulants - numerical results

Comparison with numerical results

Take the hydro results with thespinodal clumping.

Calculate the baryon numberfluctuations in a spatial volumedirectly

Use the C-F equation andsample baryons, conservingbaryon number globally.

K2/K1

0 1 2 3 4 5 60.00

0.25

0.50

0.75

1.00

Hydro only

Hydro + C-F: QB=1 QB=1/5 QB=1/20

Multinomial: QB=1 QB=1/5 QB=1/20

2 /Mzmax [fm]







K2/K1

0 1 2 3 4 5 60.00

0.25

0.50

0.75

1.00

Hydro only



2 /Mzmax [fm]







K2/K1

0 1 2 3 4 5 60.00

0.25

0.50

0.75

1.00

Hydro only



2 /M

zmax [fm]







Analytic and numerical resultsagree well.

K3/K2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-2

-1

0

1 Hydro only



Szmax [fm]








K3/K2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-2

-1

0

1 Hydro only



Szmax [fm]








K3/K2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-2

-1

0

1 Hydro only



S

zmax [fm]


Effect on the cumulants - analytic - multinomial

In case you want to see the equations:

KB,CF,multi1 = 〈B〉 = KB

1

KB,CF,multi2 = KB

2 +QB

(KB

1 −KB

12

+K2

Btot

)

KB,CF,multi3 = KB

3 + 3QB

(KB

2 −2KB

1 KB2 +KB

3

Btot

)+Q2

B

(KB

1 − 3KB

12

+KB2

Btot

+2KB

13

+ 3KB1 K

B2 +KB

3

B2tot

)(2)


How important are nuclear potentials?

Nuclear interaction do havean influence on the baryonnumber susceptibilities.

Expected effects: increase ofall cumulant ratios!

Including an additionaltransition (even crossover)can make the whole scenariomuch more complicated.

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 002 04 06 08 0

1 0 01 2 01 4 01 6 01 8 0

( d )

V D W - H R G

- 1

2

0

2

0

T (Me

V)µ B ( M e V )

χB4 / χB

2

- 1 0

01

2

1 0h e a v y - i o n - c o l l i s i o n s

V. Vovchenko, M. I. Gorenstein and H. Stoecker, Phys. Rev. Lett.118, no. 18, 182301 (2017)






0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 002 04 06 08 0

1 0 01 2 01 4 01 6 01 8 0

( d )

V D W - H R G

- 1

2

0

2

0

T (Me

V)

µ B ( M e V )

χB4 / χB

2

- 1 0

01

2

1 0h e a v y - i o n - c o l l i s i o n s

V. Vovchenko, M. I. Gorenstein and H. Stoecker, Phys. Rev. Lett.118, no. 18, 182301 (2017)

0 200 400 600 800 1000 12000

25

50

75

100

125

150

175

B [MeV]

T [M

eV]

-2

1

4

B4/ B

2

I: TFO= 165 II: TFO= 120 Crossover 1st Order

A. Mukherjee, JS and S. Schramm, Phys. Rev. C 96, no. 2, 025205(2017)Jan Steinheimer (FIAS) 06.06.2018 31 / 34





0 200 400 600 800 1000 12000

25

50

75

100

125

150

175

B [MeV]

T [M

eV]

-2

1

4

B4/ B

2

I: TFO= 165 II: TFO= 120 Crossover 1st Order

A. Mukherjee, JS and S. Schramm, Phys. Rev. C 96, no. 2, 025205(2017)

0

0.5

1

1.5

2

2.5

3

10 100

Susceptibility Ratios

√sNN (GeV)

χ3B/χ2

B for Tlim=165 MeV

χ4B/χ2

B for Tlim=165 MeV

A. Mukherjee, JS and S. Schramm, Phys. Rev. C 96, no. 2, 025205(2017)


A (qualitative) study of nuclear effects

Use a microscopic transport model (UrQMD)

Nuclear forces can be included via mean fieldapproach.

Has been used to describe particle spectra andcollective flow at low beam energies(Elab < 10A GeV).

We will employ the model for central collisionsof Au+Au as measured by the HADESexperiment at GSI (Elab = 1.23A GeV)

They have enough statistics to actuallymeasure baryon number fluctuations.

Simulation results

The simulations show a trend similar to thegrand canonical predictions.

However, effects of finite size and baryonnumber conservation have a significant impact.

Also: measuring only protons and not baryonsis a problem!

0.3

0.6

0.9

1.2

1.5

-0.5

0.0

0.5

1.0

1.5

0.0 0.4 0.8 1.2 1.6 2.0-3

-2

-1

0

1

2

10% most central eventsbaryons protons w/o potentials with potentials Binomial

K2/M

K3/K

2K4/K

2

y

JS, Y. Wang, A. Mukherjee, Y. Ye, C. Guo, Q. Li andH. Stoecker, arXiv:1804.08936 [nucl-th].








Simulation results

The simulations show a trend similar to thegrand canonical predictions.



0.3

0.6

0.9

1.2

1.5

-0.5

0.0

0.5

1.0

1.5

0.0 0.4 0.8 1.2 1.6 2.0-3

-2

-1

0

1

2


K2/M

K3/K

2K4/K

2

y









Simulation resultsThe simulations show a trend similar to thegrand canonical predictions.



0.3

0.6

0.9

1.2

1.5

-0.5

0.0

0.5

1.0

1.5

0.0 0.4 0.8 1.2 1.6 2.0-3

-2

-1

0

1

2


K2/M

K3/K

2K4/K

2

y



The Critical Point

0.4 0.5 0.6 0.7 0.8 0.9 1.0-3

-2

-1

0

1

2

3

Local Maxima Local Minima

CO= 922 B = 916 A = 938 Critical Point

/V [M

eV/fm

3 ]

/ 0

0.00 0.05 0.10 0.15 0.20

-1

0

1

2

Unstable Stable

P [M

eV/fm

3 ]

CO= 922 B = 916 A = 938 Critical Point

/ 0

c2s < 0

Mechanically Unstable

Fluctuation Amplitude

Phase Transition:

Density domains created dynamically inside theunstable and metastable region → Driven bypressure gradient: Timescales similar to expansiontimes.

Amplitude of fluctuations favor certain values.

Thermal fluctuations add but are not necessary

Critical Point:

Density domains created only at the critical point→ In equilibrium, the critical point is reached onlyby precise tuning.

Amplitudes of fluctuation are not driven by pressuregradient: Timescales?

Fluctuation on all possible length scales

No creation of fluctuations by the EoS. Driven bythermal fluctuations and long range interactions(e.g. gradient term).

requires cs = 0 for some time.


Summary

I have described a way to include effects of instabilities due to a firstorder phase transition in fluid dynamics.

The associated instabilities may cause significant amplification ofinitial density irregularities.

The present study demonstrates that the phase structure does affectthe character of the density evolution.

Hopefully, these results will stimulate efforts to develop analysistechniques for extracting the related observables from experimentaldata.

So far we are still trying to identify observables sensitive on theenhanced fluctuations.

(Average) Radial flow, nuclei production and angular correlationsshow no large signal of the unstable dynamics of the phase transition.


Summary

I have described a way to include effects of instabilities due to a firstorder phase transition in fluid dynamics.

The associated instabilities may cause significant amplification ofinitial density irregularities.

The present study demonstrates that the phase structure does affectthe character of the density evolution.

Hopefully, these results will stimulate efforts to develop analysistechniques for extracting the related observables from experimentaldata.

So far we are still trying to identify observables sensitive on theenhanced fluctuations.

(Average) Radial flow, nuclei production and angular correlationsshow no large signal of the unstable dynamics of the phase transition.


Cluster Distribution

Let’s be more quantitative

Define Clusters: A coherent chunk of baryon number with density aboveρ > 7 ρS

0 5 10 15 20 2510-4

10-3

10-2

10-1

100

Unstable EoS Maxwell EoS Exponential Fit Unstable - Maxwell EoS

Pro

babi

lity

Baryon number in cluster

Probability distributionfor cluster size isexponential (at timet− t0 ≈ 3 fm)

Probability to find alarge cluster drasticallyenhanced with unstablephase!


Radial Flow

0 1 2 3 4 5 60.15

0.20

0.25

0.30

0.35

0.40

0.45 With unstable phase With Maxwell constr.

u R

time - t0 [fm]

Average durationof fluid-phase

The average radial velocity hardly showsa difference.

Nuclei Production

Density fluctuations might lead to an enhanced production of(composite) nuclei:

EAdNA

d3PA= BA

(Ep

dNp

d3Pp

)Z (En

dNn

d3Pn

)N∣∣Pp=Pn=PA/A

.


Nuclei Production


EAdNA

d3PA= BA

(Ep

dNp

d3Pp

)Z (En

dNn

d3Pn

)N∣∣Pp=Pn=PA/A

.

Also the energy density is enhanced

The increased excitation again counters the effect of the higherbaryon density

⇒ No observed enhancement


Nuclei Production


EAdNA

d3PA= BA

(Ep

dNp

d3Pp

)Z (En

dNn

d3Pn

)N∣∣Pp=Pn=PA/A

.

Also the energy density is enhanced

The increased excitation again counters the effect of the higherbaryon density

⇒ No observed enhancement


Angular Correlations

Initial state fluctuation can be observed in final state angular correlations.

VN∆(pa⊥, pb⊥) =

⟨cos(n(φa − φb))

⟩= vn(pa⊥)×vn(pb⊥)+p⊥cons.+dec.+?a

Can this be true also for fluctuations at the phase transition?

aK. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 708, 249 (2012)

No enhancement in the angularcorrelations is observed!

A comparison with themicroscopic model shows theimportance of momentumconservation.









0 1 2 3 4 5 6 7 8 9 10 11-0.02

0.00

0.02

0.04

0.06 Baryons Unstable EoS Maxwell EoS

Charged particles Unstable EoS Maxwell EoS

Vn

[10-2

]

n

Elab= 3.0 A GeVb = 0.0 fm











0 1 2 3 4 5 6 7 8 9 10 11-0.14-0.12-0.10-0.08-0.06-0.04-0.020.000.020.040.06

Baryons Unstable EoS Maxwell EoS std UrQMD

Charged particles Unstable EoS Maxwell EoS std UrQMD

Vn

[10-2

]

n

Elab= 3.0 A GeVb = 0.0 fm




Differences to the Liquid-Gas P.T.

0 50 100 150 200

Temperature T (MeV)

0

100

200

300

400

500

Pre

ssu

re p

X(T

) (

Me

V/f

m3)

T =

0

T =

156.4

MeV

120

90

150

603

0

x100

Phase crossing

Phase diagramm in pressure andtemperature reveals that theyare of different type.

While the L.-G.- Phasetransition is driven by thedensity, with no change ofdegrees of freedom

The Hadron-Quark transition isdriven by the entropy, withsignificant change of degrees offreedom

Still, both can be a ’true’ firstorder transition including alatent heat and a ’mixed phaseregion’.


Differences to the Liquid-Gas P.T.

The nuclear Liquid-Gas phasetransition

0− 1 times nuclear groundstate densityρ0 ≈ 0.15− 0.16 fm−3

Nuclear ground state stablew.r.t. vacuum

Several experimentalconstraints, e.g. bindingenergy, compressibility.

Droplets/Fragments can beobserved in detector

The hadron-quark phase transition

2− 10? times nuclear ground statedensity

Quark droplets/domains/clustersstable in dense hadronic mediumbut unstable w.r.t. vacuum

Almost no theoretical constraints.Lattice QCD not conclusive (yet?).

How to observe evidence fordroplets/fragments in detector?


How Important is the specific EoS?

Use the PQM model, a constituentquark σ model with Polyakov Looppotential.

Unstable

Pres

sure

[MeV

/fm3 ]

Density [ 0]

Use an EoS constructed from ahadronic nucleon-pion model plus abag model.

Unstable

Density [ 0]

HQ EoS T = 100 MeV

Stable Quark matter at T = 0? Is this physical?


How Important is the specific EoS?

Use the PQM model, a constituentquark σ model with Polyakov Looppotential.

Use an EoS constructed from ahadronic nucleon-pion model plus abag model.

Stable Quark matter at T = 0? Is this physical?


Differences in the EoS

There are significant differences in the EoS along the transition curve,depending on the nature of the transition:

0 30 60 90 120 150 1800

100

200

300

400 Hadron-Quark EoS PQM model PQM + PQM model x 50 PQM + x 50 Nuclear L.-G. x 50

Pres

sure

[MeV

/fm3 ]

Temperature [MeV]

Phase crossing

PQM x 50

Nuclear L.-G. x 50

90 120 150 1800

50

100

150

200

Hadron-Quark EoS PQM model

Lattice : 2

2 + 4

2 + 4 + 6 with c = 1 2 + 4 + 6 with c = -1

Pre

ssur

e [M

eV/fm

3 ]Temperature [MeV]

Phase crossing

Is the PQM just a bad model for the Liquid gas transition(at finite density)?


Droplet Formation

Initialize a symmtric system + fluctuations and let it evolve:

0.0 0.50 1.0 1.5 2.0 2.5

t = 0.0 fm (a)

(b)

2 fm

b [ ]

t = 160.0 fm

0 1 2 3 4 5 6 7

t = 2.0 fm (a)

(b)

2 fm

b [ ]

t = 10.0 fm

The PQM model produces almost stable dropletts of quark matter.Physical???


Fluctuations Compared

Initialize a symmtric system + fluctuations and let it evolve:

1 10 100 20010-3

10-2

10-1

100

2x100

HQ EoS Unstable Maxwell

PQM EoS Unstable Maxwell

<(t)

5 >/<

(t=0)

5 >

Time [fm/c]0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4 NA=8/NA=2

Unstable Maxwell

NA=5/NA=2

Unstable Maxwell

Yie

ld ra

tioTime [fm/c]

The dropletts in the PQM model have a long lifetime, one can even seeoscillation. Average 5th power of density is enhanced in both cases,however translation into cluster creation is small.


Effects of Viscosity

In actual systems there is some degree of physical dissipation which givesrise to both viscosity and heat conduction.

To leading order, the viscosity reduces the growth rate by

≈ 12 [4

3η + ζ]k2/h (3)

where η and ζ are the shear and bulk viscosity coefficients, respectively.

A heat conductivity generally increases the growth rate because the speedof sound will be closer to the iso-thermal vT instead of the isentropic,increasing the size of the unstable region (and also decreasing the squaredspeed of sound).


Changing the Surface Tension

Parameter Dependencies

We change the value of the surface tension by varying a from 0.01 to 0.05.

0 1 2 3 4 5 6101

102

103

Elapsed time t - t0 [fm]

7 >/<

>7

a = 0.01 fm a = 0.033 fm a = 0.05 fm

0 5 10 15 20 2510-4

10-3

10-2

10-1

100

a = 0.05 fm a = 0.033 fm a = 0.01 fm

Pro

babi

lity

Baryon Number in Cluster

Quantitative results depend on choice of a.Cluster formation is stronger for small values of a, as expected.


Changing the Initial Fluctuations

Parameter Dependencies

We change the value of the initial fluctuation widthfrom σini = 0.5 to 1.0 fm.

0 1 2 3 4 5 610

100

1000

Elapsed time t - t0 [fm]

7 >/<

>7

INI= 0.5 INI= 0.7 INI= 1.0

0 5 10 15 20 251E-4

1E-3

0.01

0.1

1

INI= 0.5 INI= 0.7 INI= 1.0

Pro

babi

lity

Baryon Number in Cluster

Strongest clustering observed for intermediate width!


Changing the Initial State

Model Dependencies?

For the initial state we can apply a version of the UrQMD model thatincludes nuclear interactions. a

aQ. -f. Li, Z. -x. Li, S. Soff, M. Bleicher and H. Stoecker, J. Phys. G 32, 407(2006)

The densities achieved in theUrQMD+Potentials calculations areconsiderably smaller than in thosewithout (close to the geometricaloverlap values).This is mainly due to the repulsiveinteraction.

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

N/P W/P N=5 N=7

N>/

<>N

MA

X

Elab [A GeV]


Documents

Heavy-ion collisions at high baryon densities: …nfqcd2018/Slide/Steinheimer.pdfHeavy-ion collisions at high baryon densities: Modelling the QCD phase transition Jan Steinheimer 06.06.2018