Transcript STATISTICAL AND THERMAL EQUILIBRIUM MODELS
F. Becattini, Kielce workshop, October 15 2004
What is the meaning of the statistical model ?
F.B. hep-ph 0410403
OUTLINE
o o o o Introduction Discussion: phase space dominance, triviality and Lagrange multipliers Basic microcanonical formulation Future tests
The statistical model is successful in describing soft observables in Heavy Ion Collisions
M. Gazdzicki, M. Gorenstein F. B., A. Keranen, J. Manninen J. Cleymans, H. Satz P. Braun-Munzinger, J. Stachel, D. Magestro W. Broniowski, W. Florkowski J. Letessier, J. Rafelski K. Redlich, A. Tounsi A. Panagiotou, C. Ktorides Nu Xu, M . Kaneta And many more...
The statistical model is even more successful in describing relevant soft observables in elementary collisions
F.B. Z. Phys. C 69 (1996) 485 F. B., Proc. XXXIII Eln. Workshop Erice, hep-ph 9701275 F. B., U. Heinz, Z. Phys. C 76 (1997) 269.
F. B., G. Passaleva, Eur. Phys. J. C 23 (2001) 551 F.B., Nucl. Phys. A 702 (2002) 336 F.B., G. Passaleva, Eur. Phys. J. C 23 (2002) 551 Warning! Strangeness phase space is undersaturated
Why?
From: L. Mc Lerran, Lectures “The QGP and the CGC”, hep-ph 0311028 Three kind of answers (criticisms) The thermodynamical-like behaviour is only mimicked by the data. It should be rather called
“phase space dominance”
The Statistical Model results are somehow trivial due to large involved multiplicities The Statistical Model results can be obtained as a by -product of other models, at least in elementary collisions The temperature is not a real temperature
Phase space dominance
Ω
j
V
( 2
N
) 3
N
j
( 2
J j
1 )
N j N j
!
d
3
p
1
d
3
p N
4
P
i N
1
p i
Γ f
1 ( 2 ) 3
N
j
( 2
J j
1 )
N j N j
!
d
3 2
p
1 1
d
3 2
p N N M if
2 4
P
i N
1
p i
VS
Discussed in detail in nucl-th 0001044 J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, If
|M if | 2
has a very weak dependence on kinematical independent variables, e.g.
p i · p j
,we could somehow recover a pseudo thermal shape of the multiplicity and
p T
spectrum function
IF
M if
2
N
for large mults
n j
( 2
J j
( 2 1 ) ) 3
d
3
p
2 exp(
p
2
m
2
j
) Where is such that:
M
j
( 2
J j
( 2 1 ) ) 3
d
3 2
p
exp(
p
2
m
2
j
) 0 Conclusion: is not a temperature and inclusive particle multiplicities are not sensitive enough to the different integration measure to distinguish between a genuine thermal behaviour and this pseudo-thermal function (phase space dominance) However, there is a quantitative difference!
n j
( 2
J j
( 2 1 )
V
) 3
d
3
p
exp(
p
2
m
2
j
)
Phase space dominance is not trivial
In principle, as well as on
|M if | 2 I
1 , may depend on
I
2 , ,
I
1
I
2 ,
I
1
m
1 2
I
3 , ,
m
2 2 , , , 2
m
12 , 2
m
13 , Example:
M if
2 ( 3
M
)
N i N
1
f
(
m i
)
g
(
I i
) quite restrictive: again, only one scale and factorization
n j
( 2
J j
1 )( 3
M
) ( 2 ) 3
g
(
I j
)
f
(
m j
)
d
3 2
p
exp(
p
2
m
2
j
) The thermal-like behaviour can be easily distorted at ANY scale of Multiplicity (just take
g(I)=AI 2 +C
or
f(
m)=(
m) 5
)
Disagreement with “Triviality” arguments e.g.
V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002
|M if | 2 |M if | 2
depends on
N
;
N
is large; small fluctuations of is unessential at high
N
model results are trivially recovered
N
and therefore the statistical
1.
2.
|M if | etc.) 2
may not depend just on
N
,
also on specific particle content in the channel (through mass, isospin
In analyses of e.g. pp collisions overall multiplicities are not large enough to make fluctuations negligible
“Lagrange multiplier” or what is a temperature?
See e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th 0210070, talk given at QM2002; U. Heinz, hep ph 0407360 “Concepts in heavy ion physics” Seem to advocate the idea that the temperature determined with hadron abundances is not a “real temperature” , rather a “Lagrange multiplier constraining maximization of entropy” This is just a possible definition of temperature There might be different definitions in
small
systems (e.g.
1/T=
S/
E
, saddle point for microcanonical partition function, etc.) but ALL OF THEM converge to the same quantity in the thermodynamic limit A quantitative difference is needed: if you have volume, energy and statistical equilibrium, temperature is a temperature regardless of how the system got there!
Derive the statistical features within other models
A. Bialas, Phys. Lett. B 466 (1999) 301 W. Florkowski, Acta Phys. Pol. B 35 (2004) 799 Fluctuation of the string tension may lead to an exponential shape, e.g. of the p T spectrum
Occam razor argument
From: Delphi collaboration, CERN-PPE 96-120
How to probe a genuine statistical model ?
Need to test exclusive channel rates
BR
j BR
j Ω
Ω
Much more sensitive to the integration measure ( V d 3 p d 3 p/2 ) because information is not integrated away vs Data available at low energy ( s < 10 GeV) Need full microcanonical calculations see e.g. W. Blumel, P. Koch, U. Heinz, Z. Phys. C63 (1994) 637
Basic scheme
Clusters:
extended massive objects with internal charges
Every multihadronic state within the cluster compatible with conservation laws is equally likely
The microcanonical ensemble
and its partition function A usual definition reads Can be generalized as canonical:
h V h V Ω
states
4 (
P
P state
)
Ω
h V h V
P
i h V P i | h V
exp( H /
T
)
h V >
projector on the cluster’s initial state multihadronic state
within
the cluster What is the probability of an asymptotic free state
| f >
?
Define
p f
f
P
i
P
V
P
i f
with P
V
h V h V h V
f p f
tr ( P
i
P
V
P
i
) tr ( P
i
P
V
)
h V h V
P
i h V
Ω
All
p f
are positive definite as:
f
P
i
P
V
P
i f
h V f
P
i h V
2 The cluster is described by the mixture
W
h V
P
i h V h V
P
i
P
i
P
V
P
i
Note: P i P V P i P V P i P V Used in Eur.Phys. J. C 35 (2004) 243 In principle, projection
P V
states: should be made on localized field P
V
1
particle
V D
with
x
(
x
) In all studies, relativistic quantum field effects are neglected: good approximation for
V 1/3 >
l
C
(at most 1.4 fm)
F.B., L. Ferroni, Eur. Phys. J. C 35 (2004) 243 F.B., L. Ferroni, hep-ph 0407117, Eur. Phys. J. C in print
Full microcanonical ensemble
Decompose the projector: P
i
P
P
, Projection onto an irreducible state
P
4-momentum
J
, l , P P
I
,
I
3 P
Q
J
l
Q
spin helicity parity C-parity abelian charges
I, I 3
isospin The projector
P P,J,
l, can be written (formally) as an integral over the extended Poincare’ group IO(1,3) ↑ P
P
,
J
, l , 1 2 z Π, I dim
d
(
g z
)
D
(
g z
)
i i U
(
g z
)
The projectors on 4-momentum, spin-helicity and parity factorize if
P=(M,0)
P
P
,
J
, l , 1 (2 ) 4
d
4
x
e
iP
x
exp(
i
x
) ( 2
J
1 )
d
R
D J
( R ) l l *
U
( R ) I
U
( Π ) 2
Other projectors: P
I
,
I
3 (2
I
1)
dg D I
(
g
)
I
3 *
U I
3 (
g
) P P
Q
1 2 I C ( 2 1 )
M
d M
exp[
i
(
Q
) ] Integral projection technique already used in the canonical ensemble (Cerulus,Turko, Redlich,Cleymans, et al.) Restricted microcanonical ensemble: only four-momentum and abelian charges P
P
,
Q
4 (
P
) ,
Q
Rate of a multi-hadronic channel
{N
j
}=(N
1
,...,N
K
)
For
non-identical particles: Γ
Ω
j
N V
( 2 ) 3
N i N
1 ( 2
J i
1 )
d
3
p
1
d
3
p N
4
P
i N
1
p i
For identical particles: cluster decomposition
Ω
j
d
3
p
1
d
3
p N
4
P
P f
j
{ }
h n j
N j
H n N
j
1
j n j j
( 2
J h n j j h n j
1 )
H
!
j j l H
j
1
F n l j H j
j n N
j
1
h n j N j
j n N
j
1
n j h n j F n l
i l n l
1 1 ( 2 ) 3
V d
3
x
exp
i
x
p
i l
p
c l
(
i l
) Generalization of the expression in M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445 which holds only for large
V
partitions
Comparison between
C and C hadron multiplicities
Q
=0
M/V=0.4
GeV/fm 3 3 Mesons Baryons
Summary and Conclusions
Discussion on the statistical model Temperature, phase space dominance: only quantitative differences are differences.
More quantitative tests of the picture, e.g. on exclusive channels (at low energy) require full microcanonical calculations and Monte Carlo implementations (matching with parton shower) Microcanonical ensemble sampling algorithm for hadron system accomplished. Ongoing work to include ang. Mom., isospin etc.