Multiplicity Fluctuations in Statistical Models

Transcription

1 Multiplicity Fluctuations in Statistical Models Michael Hauer Outline Introduction Particle Anti Particle Gas Hadron Gas Comparison with Data Summary 1

2 Statistical Ensembles Grand Canonical Canonical Microcanonical V,T, V,T, Q V, E,Q E, Q, P =0 E, P =0 P=0 Thermodynamic Limit : N,V, N V =const 2

3 Why has there been so little interest so far? Experiment Experimentally this means event by event analysis of data Very precise determination of collision centrality Good understanding of acceptance and resolution 3

4 Why has there been so little interest so far? Thermal Model Statistical ensembles are equivalent under the thermodynamic limit Grand canonical ensemble often sufficient Canonical effects only become important when only few particles are produced Quantum statistics effects are generally small (10% for pions) But: all this changes when one is interested in multiplicity fluctuations 4

5 Grand Canonical Boltzmann Pion Gas Single particle partition function z = gv 2 2 dp p 2 e E T = gv 2 2 m2 T K 2 m T System partition function Z GCE T = exp z e z e T Expectation values N = 1 Z GCE N 2 = 1 Z GCE Z GCE ± =1 2 Z GCE ± =1 = z e T 2 = z e T T z e Scaled variance = N 2 N 2 N = 1 Poisson! 5

6 Canonical Boltzmann Pion Gas System partition function Z Q = d 2 e iq exp z e i z e i ± =1 =I Q 2z Expectation values N = 1 Z Q Z Q ± =1 = I Q 1 2z I Q 2z z N 2 = 1 Z Q 2 Z Q ± =1 = I Q 2 2z I Q 2z z 2 I Q 1 2z I Q 2z z Scaled variance = N 2 N 2 N V 1 2 Not 1! ( for a neutral system ) V.Begun, M.Gorenstein, M.Gazdzicki, O.Zozulya, Phys.Rev. C 70 (2004)

7 Canonical Ensemble Finite Volume Pion gas (Boltzmann) Microcanonical Ensemble massless gas V.Begun, M.Gorenstein, A.Kostyuk, O. Zozulya Phys.Rev. C 71 (2005) V.Begun, M.Gorenstein, O.Zozulya, Phys.Rev. C 72 (2005) Very fast convergence to asymptotic values for a neutral sytem! 7

8 Thermodynamic Limit Pion Gas Gas Q=0=0 = Quantum effects can be quite large, even in a neutral system! V.Begun, M.Gorenstein, A.Kostyuk, O.Zozulya 8 nucl th/

9 Hadron Resonance Gas 9

10 Hadron Resonance Gas Canonical Partition Function =[ Z Q j 3 j=1 d j] j j 2 e iq Z GC { l } N n k = 1 Z Q j k k n Z Q j Z GC { l }=exp [ l z l l ] 3 l =exp i j=1 q l j j z l l = g V l p 2 dp ln 1±exp [ E l T ] l±1 q l j = b l, s l, q l No practical analytical solution is known Only in Boltzmann approximation is an analytical reduction of integrals possible heavily oscilating integrand makes numerical evaluation expensive A.Keränen, F.Becattini, Phys.Rev. C 65 (2002)

11 Central Limit Theorem Expansion the state we are interested in : P E, Q, P, N = number of all states with E, Q, P, and N number of all states with normalization : P E, Q, P = number of all states with E, Q, P number of all states and finally our distribution : P E, Q, P N = P E, Q, P, N P E, Q, P 11

12 Central Limit Theorem Expansion Z Q j 3 j=1 [ d j 2 j] e i Q j exp[ V n=0 n j 1 j n n! j1 j n] Cumulant tensor n j 1 j n = i n n j1 jn = l z l V Generating function 3 dim Gaussian with first correction term O V 1 /2 Z Q j Z GCE 2 V 2/3 det exp [ j j 2 ] 1O V 1/2 where 2 1/2,and j = Q k k V 1 1 j k V 1/2 Find physical fugacities from Z Q Q = 0 1 B, 1 S, 1 Q V B, S, Q 12

13 Central Limit Theorem Expansion Canonical Normalization 4 dim Gaussian P V 1 j P Q j, N k = Z V j 1 Z 1 GC 2 V 3/2 det = Z Q j, N k Z GC 1 2 V 4/2 det exp [ j j ] 2 1 dim Gaussian P V 1 j N k = Z V 1 j, N k Z V j 1 det 2 V 1/2 det exp [ j j ] 2 With variance D 2 = And scaled variance k = V det 2 det 2 det 2 = det 2 det 2 k k det 2 1 Finite volume corrections can be done by Gram Charlier Expansion 13

14 Particle Decay Central Limit Theorem Expansion z l = g l p 2 dpln j[ 2 El l 2 1±e i ql e j C l,k nk=0 n k l, qe i n k k]±1 n k l,q C l, k = c k =n k k q n k 1 q c n c k k n k c k l C l, k n k =0 n k l, q=1 q Detection Probability c k multiplicity of particle k in this channel l c k branching ratio of resonance l into c_k particles k n k number of detected particles n k j, q corrected branching ratio 14

15 Chemical Freeze Out Line constant average energy per partilce E N ~1GeV Pb Pb B B S S S = 0 Q Q Q = 0.4 B S exp 1.23 T B J.Cleymans, K.Redlich Phys.Rev.Lett. 81 (1998) F.Becattini, J.Manninen Phys.Rev. C73 (2006) J.Cleymans, H.Oeschler, K.Redlich, S.Wheaton Phys.Rev. C73 (2006) B S NN 1.27GeV 1 S NN /4.3GeV A.Andronic, P.Braun Munzinger, J.Stachel Phys.Rev. A772 (2006)

16 Scaled Variance V.Begun, M.Gorenstein, M.H., V.Konchakovski, O.Zozulya Phys.Rev. C74 (2006) V.Begun, M.Gazdzicki, M.Gorenstein, M.H., V.Konchakovski, B.Lungwitz nucl th/

17 Hadron Resonance Gas (again) 17

18 Acceptance Scaling Uncorrelated detection P acc n, N = q n 1 q N n N! n! N n! Distribution of detected particles P n = N =n P 4 N P acc n, N with scaled variance acc = 1 q q 4 This would be exact only in (Boltzmann) GCE and CE. Valid for any type of distribution. Transfroms Poissonians into Poissonians. Certainly only a approximation, if particles are correlated in momentum space. 18

19 Comparison with NA49 Data negatively charged hadrons positively charged hadrons V.Begun, M.Gazdzicki, M.Gorenstein, M.H., V.Konchakovski, B.Lungwitz nucl th/

20 Comparison with NA49 Data 20

21 Summary and Outlook Data is well described by our MCE model! However : There should be energy fluctuations! How good is acceptance scaling in MCE? How would flow change our results? What about chemical non equillibrium? Phase transition? What about clusters?... 21

22 Finite Volume Hadron Gas T 160 MeV S =1.0 V =90fm 3 r 2.8fm CE E=20GeV E =0.4GeV r 2.3 fm MCE {B, S,Q}={0,0,0} {B, S,Q}={2,0,2} F.Becattini, L.Ferroni, T.Gabbriellini, Phys.Rev. C 65 (2002)

23 Finite Volume Corrections Gram Charlier Expansion P N j = z N j j N j! Q N Z j Q j j excl Z Q Primordial Boltzmann T =0.160GeV S =1.0 B, S, Q = 0,0,0 r=2.0 fm approx exact Norm N

24 Comparison of Methods Comparison is half hearted, since different particle tables were used! Microscopic correlator method and Central limit theorem expansion agree on the dot. (same table!) F.Becattini, L.Ferroni, T.Gabbriellini, Phys.Rev. C 72 (2005)

25 Techniques Microscopic correlaor (GCE and CE) 1 n p,k = exp [ E k k ] k p 3 gv v p, k = n p, k = n p, k n p, k = n p, k 1 k n p, k 2 n p,k n q,l =v p,k p,q k,l GCE correlator 2 v p, k 2 q v k q l q,l 2 2 p, k v p, k q k k k =q k Bosons Fermions Boltzmann E k = p 2 m k 2 In a canonical ensemble the variation needs to vanish Q= p,k q k n p,k =0 N k 2 = p,q,l N k = p n p, k n q,l n p,k k N k 2 N k V.Begun, M.Gorenstein, O.Zozulya, Phys.Rev. C 72 (2005)

26 Particle Decay Microscopic Correlator G R r b r R i i n i,r N R b r R n i, r Branching ration of channel r of resonance R multiplicity if species i in channel r N i k R = [ i ]k G i n i R average multplicity of i from decay of R N R average primordial density of R N i N j = N i N j R [ N R 2 n i R n j R N R n i n j R ] GCE N i N j = N i N j R R N i N R n j R R [ N R 2 n i R n j R N R n i n j R ] CE N j N R n i R R R ' N R N R ' n i R n j R ' V.Begun, M.Gorenstein, M.H., V.Konchakovski, O.Zozulya Phys.Rev. C74 (2006)

27 Pseudo intensive quantities Reservoir, CE or MCE GCE GCE is defined as a small subsystem of a large reservoir N D 2 = N 2 N 2 k=1 D 2 = N 2 M M = k=1 l=1 N k 2 N k 2 N k N l Variance is an extensive quantity, but not additive Correlators vanish only in GC = N 2 N 2 N Hence scaled variance is not intensive, since different in different ensembles, but pseudo intensive F.Becattini, L.Ferroni, T.Gabbriellini, Phys.Rev. C 72 (2005)