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

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Contents1. GCE,2. CE: Q=0, Q≠0,3. MCE neutral,4. MCE Q=0,5. Quantum statistics effects,6. Acceptance.

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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 → ∞

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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

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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.

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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

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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.

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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

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...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.

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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

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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:

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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

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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

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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

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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.