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What is the meaning of the statistical model ? F.B. hep-ph OUTLINE oIntroduction oDiscussion: phase space dominance, triviality and Lagrange multipliers oBasic microcanonical formulation oFuture tests F. Becattini, Kielce workshop, October

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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...

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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 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

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Why? From: L. Mc Lerran, Lectures “The QGP and the CGC”, hep-ph 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

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Phase space dominance Discussed in detail in J. Hormuzdiar et al., Int. J. Mod. Phys. E (2003) 649, nucl-th VS 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

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IF for large mults Where is such that: 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!

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Example: quite restrictive: again, only one scale and factorization Phase space dominance is not trivial In principle, |M if | 2 may depend on as well as on 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 )

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Disagreement with “Triviality” arguments e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th , talk given at QM |M if | 2 may not depend just on N, also on specific particle content in the channel (through mass, isospin etc.) 2.In analyses of e.g. pp collisions overall multiplicities are not large enough to make fluctuations negligible |M if | 2 depends on N; N is large; small fluctuations of N |M if | 2 is unessential at high N and therefore the statistical model results are trivially recovered

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“Lagrange multiplier” or what is a temperature? See e.g. V. Koch, Nucl. Phys. A715 (2003) 108, nucl-th , talk given at QM2002; U. Heinz, hep-ph “Concepts in heavy ion physics” U. Heinz, hep-ph “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!

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Derive the statistical features within other models A. Bialas, Phys. Lett. B 466 (1999) 301 W. Florkowski, Acta Phys. Pol. B 35 (2004) 799 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

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Need to test exclusive channel rates Much more sensitive to the integration measure (V d 3 p vs d 3 p/2 ) because information is not integrated away How to probe a genuine statistical model ? 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

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Basic scheme Clusters: extended massive objects with internal charges Every multihadronic state within the cluster compatible with conservation laws is equally likely with conservation laws is equally likely

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The microcanonical ensemble and its partition function P i projector on the cluster’s initial state cluster’s initial state | h V > multihadronic state within the cluster A usual definition reads Can be generalized as What is the probability of an asymptotic free state | f > ? Define canonical:

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The cluster is described by the mixture All p f are positive definite as: In principle, projection P V should be made on localized field states: In all studies, relativistic quantum field effects are neglected: good approximation for V 1/3 > C (at most 1.4 fm) Note: P i P V P i P V P i P V Used in Eur.Phys. J. C 35 (2004) 243

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Full microcanonical ensemble Projection onto an irreducible state P 4-momentum J spin helicity helicity parity C-parity Qabelian charges I, I 3 isospin Decompose the projector: The projector P P,J, can be written (formally) as an integral over the extended Poincare’ group IO(1,3) ↑ The projector P P,J, can be written (formally) as an integral over the extended Poincare’ group IO(1,3) ↑ The projectors on 4-momentum, spin-helicity and parity factorize if P=(M,0) F.B., L. Ferroni, Eur. Phys. J. C 35 (2004) 243 F.B., L. Ferroni, hep-ph , Eur. Phys. J. C in print

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Other projectors: Integral projection technique already used in the canonical ensemble Integral projection technique already used in the canonical ensemble (Cerulus,Turko, Redlich,Cleymans, et al.) Restricted microcanonical ensemble: only four-momentum and abelian charges

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Rate of a multi-hadronic channel {N j }=(N 1,...,N K ) For non-identical particles: For identical particles: cluster decomposition Generalization of the expression in M. Chaichian, R. Hagedorn, Nucl. Phys. B92 (1975) 445 which holds only for large V partitions

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Comparison between C and C hadron multiplicities Q=0 cluster, M/V=0.4 GeV/fm 3 Mesons Baryons pp-like cluster, M/V=0.4 GeV/fm 3

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Comparison between C and C hadron multiplicity distributions Inequivalence between C and C in the thermodynamic limit Q=0 cluster, M/V=0.4 GeV/fm 3 pp-like cluster, M/V=0.4 GeV/fm 3

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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.

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The partition functions Microcanonical ensemble (ang mom and parity cons’n disregarded) Canonical ensemble Grand-canonical ensemble ensemble Q = (Q, B, S, C,...) = ( Q, B, S,...)

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The C and GC ensembles can be obtained from C by means of asymptotic expansions From C to C can be written as the following inverse Laplace transform (with some provisoes): with If V and M are large, a saddle-point expansion can be done

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At the leading order of the saddle-point expansion where is a four-vector such that is real and timelike. Set = u/T with u 2 = 1 is the four-temperature T is the temperature Z is the canonical partition function: Mean energy-momentum in the canonical ensemble = given initial total four-momentum

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From C to GC k are the fugacities = exp( k /T) k are the fugacities = exp( k /T) Z G = exp(F( , )) is the grand-canonical partition function where Mean charge in the GC ensemble= given initial charge

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Average multiplicities (Boltzmann statistics limit) Grand-canonical ensemble Canonical ensemble Microcanonical ensemble

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