Statistical Fluctuations in Different Ensembles Begun Victor Bogolyubov Institute for Theoretical...
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Statistical Fluctuations in Different Ensembles
Begun VictorBogolyubov Institute for Theoretical Physics
Ukraine
Victor Begun 2
1. Surprising success of statistical models for particle production in A+A collisions.
2. New data on fluctuations are coming.
3. Particle number fluctuations in different statistical ensembles were not studied up to now !?
4. We have found that they are different even in the thermodynamic limit!
Motivation
Victor Begun 3
Contents1. GCE,2. CE: Q=0, Q≠0,3. MCE neutral,4. MCE Q=0,5. Quantum statistics effects,6. Acceptance.
Victor Begun 4
1. V.V. Begun, M. Gaździcki, M.I. Gorenstein, O.S.Zozulya, "Particle Number Fluctuations in a Canonical Ensemble", Phys. Rev. C70 (2004) 034901; nucl-th/0404056, Apr 2004.
2. V.V.Begun, M.I.Gorenstein, A.P.Kostyuk, O.S.Zozulya, "Particle Number Fluctuations in the Microcanonical Ensemble", nucl-th/0410044, Phys. Rev. C, in print.
3. V.V. Begun, M.I. Gorenstein, O.S. Zozulya, "Fluctuations in the Canonical Ensemble", nucl-th/0411003.
4. A. Keranen, F. Becattini, V.V. Begun, M.I. Gorenstein, O.S. Zozulya, "Particle Number Fluctuations in Statistical Model with Exact Charge Conservation Laws“, nucl-th/0411116
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Microcanonical Canonical Grand canonical
Q V, E, QV,T, Qμ V,T,
Qμ Q TE
Statistical ensembles
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where j numerates the species,
is a single particle partition function,
,
μ/T)exp(λ j
jz
Tm
K Tm2
V
pd ]Tεexp[-)2(
V
j2
2j2
j
3p3
j
g
gjz
j
jzz
22p mpε j
V → ∞ z → ∞
Victor Begun 7
Partition function in GCE
/cosh 2 exp ) λ λ exp(
...!N)(λ
!N)(λ
...!N)(λ
!N)(λ
...... )T,(V,Z
NN
1
N11
1
N11
0N ,N 0N ,Ng.c.e.
11
11
Tjj
j
j
j
jjj
jj
zzz
zzzz
jjj
jj
Victor Begun 8
0N ,N 0N ,N
c.e.11
...... )QT,(V,Zjj
...!N)(λ
!N)(λ
...!N)(λ
!N)(λ
NN
1
N11
1
N11
11
j
j
j
jjj
jj zzzz
Partition function in CE
QNNNN 11 ............ jj
)(2IQ zK. Redlich, L. Turko, Z. Phys. C 5 (1980) 541; J. Rafelski, M. Danos, Phys. Lett. B 97 (1980) 279.
Victor Begun 9
The microcanonical partition function (m=0)
1N30N
NN
1k
(k)
(1)3(N)3N
3N
27 2
35
34
2Ex V)(E,W V) W(E,
N!1)!(3N
xE1 )|p|δ(E
pd...pd2π
VN!1 V)(E,W
xF ;,,;
,
g
. πVE xwhere 2
3g
V.V.Begun et al., PRC in print, nucl-th/0410044
Victor Begun 10
CE/GCE ratio
.,)(2I)(2I
a μ/T),exp(aaNQ
1Qc.e.g.c.e.
zz
z jj
M.Gorenstein, M.Gaździcki, W.Greiner, PLB 483 (2000) 60.
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. ...N 2304
491 NN , ... N 81 1 NN 2m.c.e.m.c.e.
MCE/GCE ratio (m=0)g.
c.e. g.
c.e.
g.c.e.g.c.e.
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Fluctuations
Variance:
Scaled variance:
N = N, N+, N- , Nch=N++N- ,
22 NN V(N)
N
NN ω22
1 ω ωωω chg.c.e.g.c.e.g.c.e.g.c.e.
Victor Begun 13
CE fluctuations
V.V. Begun, M. Gaździcki, M.I. Gorenstein, O.S.Zozulya, Phys. Rev. C70 (2004) 034901; nucl-th/0404056.
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CE fluctuations, Q>0
z2Qy
V.V. Begun, M.I. Gorenstein, O.S. Zozulya, nucl-th/0411003
Victor Begun 15
...N 1152
491 81ω , ...
N 811
41ω 2m.c.e.m.c.e.
MCE fluctuations (m=0)
V.V.Begun et al., PRC in print, nucl-th/0410044
g.c.e. g.c.e.
Victor Begun 16
Quantum statistics effects
p
pppg.c.e. ),(nw})({nW
, γ Tμ mp exp
1 n22
p
)γ1(
pp2
p2
p2p nn n)(n υ
. Δnq δ Δnε δ )(nw0)Q},({nW
, Δnq δ )(nw})({n W
p αp,
αp
α
αp,
αp
αpppm.c.e.
p αp,
αp
αpppc.e.
M.Stephanov, K.Rajagopal, E.Shuryak, Phys.Rev. D60 (1999) 114028
α =±1
Victor Begun 17
GCE CE
0.456,0)(Qω 0.912,ω
0.5,0)(Qω ω
0.684,0)(Qω 1.368,ω
Fc.e.
Fg.c.e.
c.e.g.c.e.
Bc.e.
Bg.c.e.
1/21,
V.V. Begun, M.I. Gorenstein, O.S. Zozulya, nucl-th/0411003
0μ
m=0:
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MCE
. 0.099 0)(Qω , 0.198 ω
, 0.125 0)(Qω , 0.25 ω
, 0.268 0)(Qω , 0.535 ω
Fm.c.e.
Fm.c.e.
m.c.e.m.c.e.
Bm.c.e.
Bm.c.e.
1/81/4
V.V.Begun et al., PRC in print, nucl-th/0410044
Blac
k-bo
dy ra
diat
ion
m=0:
Victor Begun 19
Limited kinematical acceptance
ω ω 1q acc
q)(1 ωq ωacc
1 ω 0q acc
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1. Particle number fluctuations calculated in the CE and MCE
2. + - ch
3.
4. In the limit
5. Large acceptance is required.
GCEneutr
al1 1
Q=0 1
Summary
for the first time! ≠ ≠ ≠
Conservation laws reduce fluctuations!
V→∞ CE MCE 1/4 1/2 1/8
Victor Begun 21
Examples when exact conservation laws are required
• collisions,• Strangeness production in A+A
collisions at low energies,• Antiproton production in peripheral
A+A collisions,• Charm and charmonium production.
ee , pp , pp
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Gaździcki, NA49, QM 2004:
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0 5 10 15 200
0.1
0.3
0.2
N
m.c.e. Q=0 m.c.e. c.e. Q=0 g.c.e.
P
Victor Begun 24
A system with two conserved charges (p,n,π-gas)
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Definitions of Fluctuations
1. Standard textbooks in statistical physics:a) Non-relativistic case: Nc.e.,Nm.c.e.=const,
b)
2. Scaled variance:
N
ΔN ω2
. 1 ω ω chg.c.e.g.c.e.
.N , 0 N1~
NΔN δ
2
Victor Begun 26
Conclusions1, ω ω ω ω ch
g.c.e.g.c.e.g.c.e.g.c.e. 1. 2. 3. Q=0, = 1/2 ,
4. Q>0, + < - ,
5. E=const, = 1/4 ,
6. E=const, Q=0 ± = 1/8 ,7. Large acceptance is required.
0.912, ω 1.368,ω Fg.c.e.
Bg.c.e.