北沢正清（阪大理）重イオン衝突実験と非ガウスゆらぎ MK, Asakawa, Ono, in preparation 松本研究会, 2013/Jun./22.

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北沢正清（阪大理）重イオン衝突実験と非ガウスゆらぎ MK, Asakawa, Ono, in preparation 松本研究会, 2013/Jun./22

北沢の生い立ち北沢の生い立ち飯田市実家大町市誕生～ 1.5 歳南箕輪村 1.5 歳～小１信州新町小２～小６長野市中、高茨城県へ大学

Beam-Energy Scan Beam-Energy Scan 0 Color SC  T Hadrons

Beam-Energy Scan Beam-Energy Scan 0 Color SC  T Hadrons high low beam energy STAR 2012

Fluctuations Fluctuations Observables in equilibrium are fluctuating. P(N) V N N

Fluctuations Fluctuations P(N) V N N  Skewness:  Variance:  Kurtosis: Observables in equilibrium are fluctuating.

Event-by-Event HIC Event-by-Event HIC Fluctuations can be measured by e-by-e analysis in experiments. Detector V

Event-by-Event HIC Event-by-Event HIC Fluctuations can be measured by e-by-e analysis in experiments. Detector STAR, PRL105 (2010)

Fluctuations Fluctuations  Fluctuations reflect properties of matter.  Enhancement near the critical point Stephanov,Rajagopal,Shuryak(’98); Hatta,Stephanov(’02); Stephanov(’09);…  Ratios between cumulants of conserved charges Asakawa,Heintz,Muller(’00); Jeon, Koch(’00); Ejiri,Karsch,Redlich(’06)  Signs of higher order cumulants Asakawa,Ejiri,MK(’09); Friman,et al.(’11); Stephanov(’11)

Fluctuations Fluctuations Free Boltzmann  Poisson

Fluctuations Fluctuations Free Boltzmann  Poisson RBC-Bielefeld ’09

Central Limit Theorem Central Limit Theorem Large system  Gaussian Higher-order cumulants suppressed as system volume becomes larger? CLT

Central Limit Theorem Central Limit Theorem :Gaussian CLT  Cumulants are nonzero.  Their experimental measurements are difficult.  In a large system,

Proton # STAR-BES Proton # STAR-BES No characteristic signals on phase transition to QGP nor QCD CP STAR,QM2012

Charge STAR-BES Charge STAR-BES STAR, QM2012 No characteristic signals on phase transition to QGP nor QCD CP

Charge LHC Charge LHC ALICE, PRL110,152301(2013) D-measure D ~ 3-4 Hadronic D ~ 1 Quark is not equilibrated at freeze-out at LHC energy! LHC: significant suppression from hadronic value

 ALICE  ALICE ALICE PRL 2013 t z  rapidity acceptane

Dissipation of a Conserved Charge Dissipation of a Conserved Charge

Time Evolution in HIC Time Evolution in HIC Quark-Gluon Plasma Hadronization Freezeout

Time Evolution in HIC Time Evolution in HIC Pre-Equilibrium Quark-Gluon Plasma Hadronization Freezeout

Time Evolution in HIC Time Evolution in HIC Pre-Equilibrium Quark-Gluon Plasma Hadronization Freezeout

 ALICE  ALICE ALICE PRL 2013 t z  Higher-order cumulants as thermometer? HotQCD,2012 rapidity acceptane  dependences of fluctuation observables encode history of the hot medium!

Conserved Charges : Theoretical Advantage Conserved Charges : Theoretical Advantage  Definite definition for operators -as a Noether current -calculable on any theory ex: on the lattice

Conserved Charges : Theoretical Advantage Conserved Charges : Theoretical Advantage  Definite definition for operators  Simple thermodynamic relations -as a Noether current -calculable on any theory ex: on the lattice -Intuitive interpretation for the behaviors of cumulants ex:

Fluctuations of Conserved Charges Fluctuations of Conserved Charges t z yy  Under Bjorken expansion  Variation of a conserved charge in  is slow, since it is achieved only through diffusion. NQNQ  Primordial values can survive until freezeout. The wider , more earlier fluctuation. Asakawa, Heintz, Muller, 2000 Jeon, Koch, 2000 Shuryak, Stephanov, 2001

LHC ? LHC ? should have different  dependence.

@ LHC LHC ? Left (suppression) How does behave as a function of  ? Right (hadronic) or

Hydrodynamic Fluctuations Hydrodynamic Fluctuations Diffusion equation Stochastic diffusion equation Landau, Lifshitz, Statistical Mechaniqs II Kapusta, Muller, Stephanov, 2012

Hydrodynamic Fluctuations Hydrodynamic Fluctuations Diffusion equation Stochastic diffusion equation Landau, Lifshitz, Statistical Mechaniqs II Kapusta, Muller, Stephanov, 2012 Conservation LawFick’s Law

Fluctuation-Dissipation Relation Fluctuation-Dissipation Relation Stochastic force  Local correlation (hydrodynamics)  Equilibrium fluc.

 Dependence  Dependence  Initial condition:  Translational invariance initial equilibrium initial Shuryak, Stephanov, 2001

 Dependence  Dependence  Initial condition:  Translational invariance initial equilibrium initial Shuryak, Stephanov, 2001

Non-Gaussian Stochastic Force ?? Non-Gaussian Stochastic Force ??  Local correlation (hydrodynamics)  Equilibrium fluc. Stochastif Force : 3rd order

Caution! Caution! diverge in long wavelength   No a priori extension of FD relation to higher orders

Caution! Caution!  Theorem cf) Gardiner, “Stochastic Methods”  HydrodynamicsLocal equilibrium with many particles Gaussian due to central limit theorem Markov process ＋ continuous variable  Gaussian random force diverge in long wavelength   No a priori extension of FD relation to higher orders

Three “NON”s Three “NON”s  Non-Gaussian  Non-critical  Non-equilibrium Physics of non-Gaussianity in heavy-ion collisions is a particular problem. Non-Gaussianitiy is irrelevant in large systems Fluctuations are not equilibrated fluctuations observed so far do not show critical enhancement

Diffusion Master Equation Diffusion Master Equation Divide spatial coordinate into discrete cells probability

Diffusion Master Equation Diffusion Master Equation Divide spatial coordinate into discrete cells Master Equation for P(n) x-hat: lattice-QCD notation probability Solve the DME exactly, and take a  0 limit No approx., ex. van Kampen’s system size expansion

Solution of DME Solution of DME Appropriate continuum limit with  a 2 =D initial 1st Deterministic part  diffusion equation at long wave length (1/a<<k)

Solution of DME Solution of DME Deterministic part  diffusion equation at long wave length (1/a<<k) Appropriate continuum limit with  a 2 =D initial 1st 2nd Consistent with stochastic diffusion eq. (for sufficiently smooth initial condition)

Total Charge in  Total Charge in  01x FT of step func.

Net Charge Number Net Charge Number Prepare 2 species of (non-interacting) particles Let us investigate at freezeout time t

Initial Condition at Hadronization Initial Condition at Hadronization  Boost invariance / infinitely long system  Local equilibration / local correlation  Initial fluctuations suppression owing to local charge conservation strongly dependent on hadronization mechanism

 Dependence at Freezeout  Dependence at Freezeout Initial fluctuations: 2nd 4th

@ LHC boost invariant system tiny fluctuations of CC at hadronization short correlaition in hadronic stage Assumptions Suppression of 4 th -order cumulant is anticipated at LHC energy! Suppression of 4 th -order cumulant is anticipated at LHC energy!

 Dependence at STAR  Dependence at STAR STAR, QM2012 decreases as  becomes larger at RHIC.

Chemical Reaction 1 Chemical Reaction 1 Cumulants with fixed initial condition Master eq.: x: # of X a: # of A (fixed) initial equilibrium

Chemical Reaction 2 Chemical Reaction 2 2nd 3rd 4th Higher-order cumulants grow slower.

 Dependence at Freezeout  Dependence at Freezeout Initial fluctuations: 2nd 4th

Summary Summary Plenty of physics in  dependences of various cumulants Diagnozing dynamics of HIC  history of hot medium  mechanism of hadronization  diffusion constant Physical meanings of fluctuation obs. in experiments.

Summary Summary Plenty of physics in  dependences of various cumulants Diagnozing dynamics of HIC  history of hot medium  mechanism of hadronization  diffusion constant Search of QCD Phase Structure Physical meanings of fluctuation obs. in experiments.

Open Questions & Future Work Open Questions & Future Work  Why the primordial fluctuations are observed only at the LHC, and not the RHIC ?  Extract more information on each stage of fireballs using fluctuations  Model refinement  Including the effects of nonzero correlation length / relaxation time global charge conservation  Non Poissonian system  interaction of particles

Evolution of Fluctuations Evolution of Fluctuations Fluctuation in initial state Time evolution in the QGP approach to HRG by diffusion volume fluctuation experimental effects particle missID, etc.