Do small systems equilibrate chemically?

Do small systems equilibrate chemically?

Ingrid Kraus

TU Darmstadt

Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 2

Outline

• Introduction to the Statistical Model– Ensembles, partition function

• Grand canonical ensemble– Comparison to data

– Extrapolation and predictions for heavy-ion collisions at LHC

– Experimental observables for T and μB determination

– Relevance of resonances

• From Pb+Pb to p+p: system size and energy dependence– Canonical suppression

– Concept of equilibrated clusters

– Comparison to data

• Summary


• Micro-canonical– closed system

– E, V, N fix

• Canonical– heat bath

– T, V, N fix

• Grand-canonical– open system

– heat bath and particle reservoir

– T, V, fix

Statistical Ensembles

E, V, N

T, V, N

T, Vb, Nb

T, V,

T, Vb, Nb

E

Laplacetransformation

N

canN

NT

cangrand ZeZ )( /


• Partition function of a grand canonical ensemble

• Energy density Entropy density

• Particle number density Pressure

• Grand-canonical partition function– i: species in the system

– Mesons m < 1.5 GeV, Baryons m < 2 GeV

Partition function and its derivations

),,(ln VTZ

nT

ZT

V

T

)ln(T

ZT

Vs

)ln(1

)ln(1 ZT

Vn

V

ZTP

)ln(

i i VTZVTZ ),,(ln),,(ln


• Partition function for species i with degenaracy factor gi

• with– (+) for fermions, (-) for bosons

• Model parameters

– T and B S constrained by strangeness neutrality

– V cancels in ratios Q constrained by charge of nuclei

Partition function and model parameters

0

22

1ln2

),,(ln TE

TN

ii

ii

eedppgV

VTZ

QQiSSiBBii NNNN 22iii mpE


Comparison to Experimental Data

– Accurancy in T, B: few MeV

– Different data selected for fits

A.Andonic, P. Braun-Munzinger, J. Stachel, nucl-th/0511071


T - B – systematics, extrapolation to LHC

hep-ph/0511094Chemical decoupling conditions

extracted from SIS up to RHIC

Feature common behavior

On the freeze-out curve:

TLHC ≈ TRHIC ≈ 170 MeV

T ≤ TC ≈ 170 MeV

μB from parametrised freeze-out

curve:

μB (√(sNN) = 5.5TeV) = 1 MeV

Nucl. Phys. A 697 (2002) 902

Grand canonical ensemble

for Pb+Pb predictions


Predictions for Pb+Pb

• Reliable for stable particles

• Benchmark for resonances

• Errors:

T = 170 +/- 5 MeV

μB = 1 + 4 MeV

- 1

All calculations with THERMUS hep-ph/0407174


Extraction of thermal parameters from data

• determine μB from p/p

• sensitivity on T– increases with mass

difference

– decay contribution affect

lighter particles stronger

– increasing feed-down with

increasing T

– decay dilutes T dependence

• T from and/or

K

_


Resonance Decays

• Hadron Resonance gas

• no resonance contribution

•

– 50% from feed-down

– both exhibit same T dependence

• K decay exceeds thermal at LHC

• – thermal production ≈ constant

– resonance contribution dominant

• 75% of all from resonances

j

thermaljij

thermalii NNN


• Grand canonical ensemble– large systems, large number of produced hadrons

• Canonical ensemble– small systems / peripheral collisions, low energies

– suppressed phase-space for particles related to conserved charges

– density of particle i with strangeness S approxiamtely

• S: order of Bessel functions

• x: sum over strange hadrons, related to volume

– Volume enters as additional parameter V

– here: radius R of spherical volume V

Canonical suppression

)(

)(

0 xI

xInn Scanonicalgrandi

canonicali

)()( RnVnn iicanonicali

T, V, N

T, Vb, Nb

T, V,

T, Vb, Nb


Canonical suppression

– Stronger suppression for

multi-strange hadrons

– Suppression depends on

strangeness content, not

difference

(expected from S)


Suppression by undersatured phase-space

– Stronger suppression for

multi-strange hadrons

– Suppression depends on

difference of strangeness

content

(power of S)


Suppression in small systems

• Suppressed strangeness

production beyond canonical

suppression– addressed by canonical treatment

and undersaturation factor S

– new: equilibrated clusters

SPS √(sNN) = 17 AGeV


Modification of the model

• Statistical Model approach: T and μB

– Volume for yields → radius R used here

• Deviations: strangeness undersaturation factor S

– Fit parameter

• Alternative: small clusters (RC) in fireball (R): RC ≤ R

– Chemical equilibrium in subvolumes: canonical suppression

– RC free parameter

R

RC


Fit Example

• All Fits were performed with

THERMUS

hep-ph/0407174

• Fits with S / RC give better

description of data


System size and energy dependence of T and B

• T independent of– System size

– Data selection

– Energy • μB smaller at RHIC


System size and energy dependence of the cluster size

• Small clusters in all systems

• Small system size dependence

• p+p– energy dependence?

• Pb+Pb– depends on data selection

(multistrange hadrons needed)


System size and energy dependence of the cluster size

• A+A: clusters smaller than fireball

• RC not well defined for RC ≥ 2 fm because suppression vanishes


Canonical Suppression

• Particle ratios saturate

at RC ≈ 2 - 3 fm

– no precise determination

for small strangeness

suppression


• Grand canonical ensemble– successful description of Au+Au,

Pb+Pb data

– extrapolations allow for predictions

– determination of thermal

parameters with few particle ratios

– proper treatment of resonances is

mandatory

Summary

• Canonical ensemble– volume dependend suppression

– stronger suppression modeled with

smaller, thermally equilibrated clusters

– successful description of p+p, C+C,

Si+Si data

– strangeness production in small

systems reproduced with equilibrated

subvolumes

• Outlook– strangeness production in p+p at LHC


Going into formulas

• performing the momentum integration

– (+) for bosons, (-) for fermions

– mi: mass of hadron i

• Particle number density

T

kmKe

k

mgTVVTZ i

k

k

TNk

iii

i

122

1

2

2 )1(

2),,(ln

0

22

1ln2

),,(ln TE

TN

ii

ii

eedppgV

VTZ

T

kmKe

k

mgTZT

VTn i

k

TN

k

kiii

i

i

21

1

2

2 )1(

2

)ln(1),(


• Approx. modified Bessel function

• Particle ratio

• Antiparticle/Particle ratio

Density and Ratios

Tm

TNii

i

ii

eemTg

n

2

2/3

2

)(2

Tmm

TNN

eem

m

g

g

n

n 2121 )(2/3

2

1

2

1

2

1

T

NN

T

N SSBB

een

n 1,1,1

222

1

1


System size dependence of T and B

• μB decreases at mid-rapidity in small systems ….

• …. as expected from increasing antibaryon / baryon ratio


System size dependence of the cluster size

Same trend as K /


More SPS and RHIC 200 GeV Data


Model setting with S

• S

– sensitive on data sample

– increase with size

– increase with energy


Extrapolation to LHC

• does strangeness in

p+p at LHC behave

grand canonical ?

• multiplicity increases

with √(sNN)

– canonical and grand

canon. event classes?

plot from PPR Vol I


Prediction for p+p

• significant increase of

ratios at RC ≈ 1.5 fm

• K / and

behave differently– multistrange hadrons

suffer stronger

suppression

• RC will be determined

with ALICE data

Do small systems equilibrate chemically?

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