Transcript Do small systems equilibrate chemically? Ingrid Kraus TU Darmstadt

Do small systems equilibrate chemically?
Ingrid Kraus
TU Darmstadt
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
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Statistical Ensembles
• Micro-canonical
– closed system
E, V, N
– E, V, N fix
• Canonical
– heat bath
– T, V, N fix
T, Vb, Nb
T, V, N
Laplace
transformation
SE
Z grand can   N (e
m /T N
) Zcan
SN
T, Vb, Nb
• Grand-canonical
– open system
– heat bath and particle reservoir
T, V, m
– T, V, m fix
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Partition function and its derivations
• Partition function of a grand canonical ensemble ln Z (T ,V , m )
• Energy density

Entropy density
T  (T ln Z ) 
 mn
V
T
s
• Particle number density
n
1  (T ln Z )
V
m
Pressure
P
• Grand-canonical partition function
1  (T ln Z )
V
T
 (T ln Z )
V


ln Z (T ,V , m )  i ln Zi (T ,V , m )
– i: species in the system
– Mesons m < 1.5 GeV, Baryons m < 2 GeV
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Partition function and model parameters
• Partition function for species i with degenaracy factor gi

V  gi
ln Zi (T ,V , m ) 

2 
2 0

• with
 
N
i m  Ei

p 2 dp ln1  e T e T

 
N i m  N Bi m B  N Si m S  N Qi mQ



Ei 
pi2  mi2
– (+) for fermions, (-) for bosons
• Model parameters
– T and mB
mS constrained by strangeness neutrality
– V cancels in ratios
mQ constrained by charge of nuclei
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Comparison to Experimental Data
A.Andonic, P. Braun-Munzinger, J. Stachel, nucl-th/0511071
– Accurancy in T, mB: few MeV
– Different data selected for fits
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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T - mB – systematics, extrapolation to LHC
Chemical decoupling conditions
extracted from SIS up to RHIC
hep-ph/0511094
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
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Predictions for Pb+Pb
• Reliable for stable
particles
• Benchmark for
resonances
• Errors:
T = 170 +/- 5 MeV
μB = 1 +- 41 MeV
All calculations with THERMUS hep-ph/0407174
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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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 W /  and/or W / K
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Resonance Decays
•
Hadron Resonance gas
Ni  Ni
thermal
  j  j i N j
thermal
•
W no resonance contribution
•
X
–
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
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Canonical suppression
• Grand canonical ensemble
T, Vb, Nb
– large systems, large number of produced hadrons
T, V, m
• 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
nicanonical  nigrand canonical 
I S ( x)
I 0 ( x)
T, Vb, Nb
T, V, N
• 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
Ingrid Kraus, TU Darmstadt
nicanonical  ni (V )  ni ( R)
Hot Quarks 2006, Sardinia, May 16, 2006
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Canonical suppression
– Stronger suppression for
multi-strange hadrons
– Suppression depends on
strangeness content, not
difference
(expected from gS)
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Suppression by undersatured phase-space
– Stronger suppression for
multi-strange hadrons
– Suppression depends on
difference of strangeness
content
(power of gS)
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Suppression in small systems
SPS √(sNN) = 17 AGeV
• Suppressed strangeness
production beyond canonical
suppression
– addressed by canonical treatment
and undersaturation factor gS
– new: equilibrated clusters
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Modification of the model
• Statistical Model approach: T and μB
– Volume for yields → radius R used here
• Deviations: strangeness undersaturation factor gS
– Fit parameter
• Alternative: small clusters (RC) in fireball (R): RC ≤ R
– Chemical equilibrium in subvolumes: canonical suppression
– RC free parameter
R
RC
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Fit Example
• All Fits were performed with
THERMUS
hep-ph/0407174
• Fits with gS / RC give better
description of data
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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System size and energy dependence of T and mB
•
T independent of
– System size
– Data selection
– Energy
Ingrid Kraus, TU Darmstadt
• μB smaller at RHIC
Hot Quarks 2006, Sardinia, May 16, 2006
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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)
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
18
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
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
19
Canonical Suppression
• Particle ratios saturate
at RC ≈ 2 - 3 fm
– no precise determination
for small strangeness
suppression
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
20
Summary
• Grand canonical ensemble
– successful description of Au+Au,
Pb+Pb data
• Canonical ensemble
– volume dependend suppression
– extrapolations allow for predictions
– stronger suppression modeled with
smaller, thermally equilibrated clusters
– determination of thermal
parameters with few particle ratios
– successful description of p+p, C+C,
Si+Si data
– proper treatment of resonances is
mandatory
– strangeness production in small
systems reproduced with equilibrated
subvolumes
• Outlook
– strangeness production in p+p at LHC
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Going into formulas

V  gi
ln Zi (T ,V , m ) 

2 
2 0

 
N
i m  Ei

p 2 dp ln1  e T e T




• performing the momentum integration
 V
ln Zi (T ,V , m ) 
  k
N
m
 T  gimi2  (1) k 1  Ti 
e
2
2


2
k 1 k



 km 
K2  i 
 T 
– (+) for bosons, (-) for fermions
– mi: mass of hadron i
• Particle number density
1  (T ln Zi ) T

ni (T , m ) 

V
m
Ingrid Kraus, TU Darmstadt
  k
N
m
 gi  mi2  (1) k 1  Ti 
e
2


k
2
k 1



Hot Quarks 2006, Sardinia, May 16, 2006
 km 
K2  i 
 T 
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Density and Ratios

• Approx. modified Bessel function
2
2
2
 
Ni m
e T
  
3 / 2 ( N1  N 2 ) m
m1 
 e T
n1 g1 
  
n2 g 2  m2 
• Particle ratio
• Antiparticle/Particle ratio
Ingrid Kraus, TU Darmstadt
ni 
 gi  (T  mi )3 / 2
 
2N1m
n1
e T
n1
Hot Quarks 2006, Sardinia, May 16, 2006
e
e


mi
T
m1  m2
T
2 N B, 1 m B 2 N S , 1 mS
e
T
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System size dependence of T and mB
•
μB decreases at mid-rapidity in small systems ….
• …. as expected from increasing antibaryon / baryon ratio
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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System size dependence of the cluster size
Same trend as K / 
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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More SPS and RHIC 200 GeV Data
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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Model setting with gS
•
gS
– sensitive on data sample
– increase with size
– increase with energy
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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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
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
28
Prediction for p+p
• significant increase of
ratios at RC ≈ 1.5 fm
• K /  and W / X
behave differently
– multistrange hadrons
suffer stronger
suppression
• RC will be determined
with ALICE data
Ingrid Kraus, TU Darmstadt
Hot Quarks 2006, Sardinia, May 16, 2006
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