1. Bogolyubov Institute for Theoretical Physics
Particle Number Fluctuations in Statistical Models: Effects of Quantum Statistics, Global Conservation Laws, Resonance Decays
Viktor Begun
Bogolyubov Institute for Theoretical Physics
Kiev, Ukraine
Viktor Begun

2. Multiplicity fluctuations
Scaled variance
- Event-by-event
- Statistical averaging:
GCE, CE, MCE
Non-relativistic case
(discussed in textbooks):
Relativistic Gas:
!!!
Viktor Begun

3. The beginning: CE vs GCE
grand canonical limit
They are different!
V.B., M.Gaździcki,
M.Gorenstein, O.Zozulya,
Phys. Rev. C70, (2004)
canonical limit
Viktor Begun

4. Some recent publications on fluctuations in GCE, CE and MCE:
G. Torrieri, nucl-th/
V.B., and M. I. Gorenstein, Phys. Rev. C 73, (2006)
G. Torrieri, S. Jeon and J. Rafelski, AIP Conf. Proc. 828, 55 (2006)
G. Torrieri, S. Jeon and J. Rafelski, nucl-th/
G. Torrieri, S. Jeon and J. Rafelski, Rom. Rep. Phys. 58, 031 (2006)
M. Gazdzicki, J. Phys. Conf. Ser. 27, 154 (2005)
F. Becattini, A. Keranen, L. Ferroni and T. Gabbriellini, Phys. Rev. C 72, (2005)
V.B., M. I. Gorenstein, A. P. Kostyuk and O. S. Zozulya, J.Phys.G32, (2006)
G. Torrieri, S. Jeon and J. Rafelski, nucl-th/
J. Cleymans, K. Redlich and L. Turko, J. Phys. G 31, 1421 (2005)
J. Cleymans, K. Redlich and L. Turko, Phys. Rev. C 71, (2005)
V.B., M. I. Gorenstein and O. S. Zozulya, Phys. Rev. C 72, (2005)
A. Z. Mekjian, Nucl. Phys. A 761, 132 (2005)
A. Keranen, F. Becattini, V.B., M. I. Gorenstein, O.S. Zozulya, J. Phys. G, (2005)
V.B., M. I. Gorenstein, A. P. Kostyuk and O. S. Zozulya, Phys. Rev. C 71, (2005)
F. Becattini and L. Ferroni, Eur. Phys. J. C 38, 225 (2004)
Viktor Begun

5. GCE, quantum statistics
Viktor Begun

6. CE microcorrelator M.Stephanov, K.Rajagopal,
E.Shuryak, Phys. Rev. D (1999)
V.B., M.Gorenstein, A.P. Kostyuk O.Zozulya,
Phys. Rev. C71, (2005)
V.B., M.Gorenstein, M.Hauer,
V.Konchakovski, O.Zozulya, nucl-th\
Viktor Begun

7. Scaled variance & Global Q, B, S conservation
Viktor Begun

8. Saddle point expansion
F.Becattini, Z.Phys., C69, (1996); F.Becattini, U.Heinz, ibid. 76, (1997);
F.Becattini, A.Keranen, L.Ferroni, T.Gabrielini, Phys.Rev., C72 (2006).
Viktor Begun

9. Effect of Resonance Decays
- the type of resonance:
- the number of resonances of a sort
- the particle specie that we are interested in
- a decay channel
the final number of particles that we are interested in
- the final probability to decay via -th channel
Viktor Begun

10. Generating function S.Jeon, V.Koch, Phys.Rev.Lett. 83 (1999)
V.B., M.Gorenstein, M.Hauer,
V.Konchakovski, O.Zozulya,
nucl-th\
Generating function
S.Jeon, V.Koch,
Phys.Rev.Lett. 83 (1999)
Viktor Begun

11. An example: one resonance type, two decay channels
V.B., M.Gorenstein, M.Hauer,
V.Konchakovski, O.Zozulya,
nucl-th\
the generating function
Viktor Begun

12. GCE CE S.Jeon, V.Koch, Phys.Rev.Lett. 83 (1999)
V.B., M.Gorenstein, M.Hauer,
V.Konchakovski, O.Zozulya,
nucl-th\ CE
Viktor Begun

13. Line of the chemical freeze-out
<E>/<N> = 1GeV → T(μB)
S = 0 → μS
Q/B = 0.4 → μQ
THERMUS — S. Wheaton, J. Cleymans, J. Phys. G (2005)
J. Cleymans and K. Redlich, Phys. Rev. Lett. 81, 5284 (1998)
J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, Phys. Rev. C 73, (2006)
F. Becattini, J. Manninen, M. Ga´zdzicki, Phys. Rev. C 73, (2006)
Viktor Begun

14. The prediction of CE hadron gas model
< 1 ?!
Small acceptance
Resonance decays
V.B., M.Gorenstein, M.Hauer,
V.Konchakovski, O.Zozulya,
nucl-th\
Viktor Begun

15. < 1 !!!
See talk of B. Lungwitz
Viktor Begun

16. Summary Scaled variances are different in different ensembles,
even in thermodynamic limit:
( ωCE(Q=0) = 1/2, ωGMCE(m=0) = 1/4, ωMCE(m=0,Q=0) = 1/8 )
The analytical formulae for the
resonance decay contribution in CE has been found
The prediction for the energy dependence of the scaled variances in the most central Pb+Pb collisions has been done
A comparison of the statistical model for hadron-resonance gas in CE with NA49 data was discussed
GCE is not valid for scaled variances,
even for “small” part of the system!
Viktor Begun

17.
Viktor Begun

18. Experiment
V.Konchakovski, S.Haussler,
M.Gorenstein, E.Bratkovskaya,
M.Bleicher, H.Stoecker, Nov 2005, Phys.Rev., (2006)
The volume fluctuations has the dominant effect
< 1 ? !!!
Viktor Begun

19. Scaled variance Independent source model
H.Heiselberg, Phys.Rep., (2001)
Viktor Begun