Statistical Models A. ) Chemical equilibration B Statistical Models A.) Chemical equilibration B.) Thermal equilibration C.) Hydrodynamics

Basic Idea of Statistical Hadronic Models Assume thermally (constant Tch) and chemically (constant ni) equilibrated system Given Tch and  's (+ system size), ni's can be calculated in a grand canonical ensemble Chemical freeze-out (yields & ratios) inelastic interactions cease particle abundances fixed (except maybe resonances) Thermal freeze-out (shapes of pT,mT spectra): elastic interactions cease particle dynamics fixed

Particle production: Statistical models do well We get a chemical freeze-out temperature and a baryochemical potential out of the fit

Ratios that constrain model parameters

Statistical Hadronic Models : Misconceptions Model says nothing about how system reaches chemical equilibrium Model says nothing about when system reaches chemical equilibrium Model makes no predictions of dynamical quantities Some models use a strangeness suppression factor, others not Model does not make assumptions about a partonic phase; However the model findings can complement other studies of the phase diagram (e.g. Lattice-QCD)

Thermalization in Elementary Collisions ? Seems to work rather well ?! Beccatini, Heinz, Z.Phys. C76 (1997) 269

Thermalization in Elementary Collisions ? Is a process which leads to multiparticle production thermal? Any mechanism for producing hadrons which evenly populates the free particle phase space will mimic a microcanonical ensemble. Relative probability to find a given number of particles is given by the ratio of the phase-space volumes Pn/Pn’ = fn(E)/fn’(E)  given by statistics only. Difference between MCE and CE vanishes as the size of the system N increases. This type of “thermal” behavior requires no rescattering and no interactions. The collisions simply serve as a mechanism to populate phase space without ever reaching thermal or chemical equilibrium In RHI we are looking for large collective effects.

Statistics  Thermodynamics p+p Ensemble of events constitutes a statistical ensemble T and µ are simply Lagrange multipliers “Phase Space Dominance” A+A We can talk about pressure T and µ are more than Lagrange multipliers

Are thermal models boring ? Good success with thermal models in e+e-, pp, and AA collisions. Thermal models generally make tell us nothing about QGP, but (e.g. PBM et al., nucl-th/0112051): Elementary particle collisions: canonical description, i.e. local quantum number conservation (e.g.strangeness) over small volume. Just Lagrange multipliers, not indicators of thermalization. Heavy ion collisions: grand-canonical description, i.e. percolation of strangeness over large volumes, most likely in deconfined phase if chemical freeze-out is close to phase boundary.

T systematics it looks like Hagedorn was right! [Satz: Nucl.Phys. A715 (2003) 3c] filled: AA open: elementary it looks like Hagedorn was right! if the resonance mass spectrum grows exponentially (and this seems to be the case), there is a maximum possible temperature for a system of hadrons indeed, we don’t seem to be able to get a system of hadrons with a temperature beyond Tmax ~ 170 MeV!

Does the thermal model always work ? Data – Fit (s) Ratio . Particle ratios well described by Tch = 16010 MeV, mB = 24 5 MeV Resonance ratios change from pp to Au+Au  Hadronic Re-scatterings!

Identified particle spectra : p, p, K-,+, p-,+, K0s and L

Identified Particle Spectra for Au-Au @ 200 GeV The spectral shape gives us: Kinetic freeze-out temperatures Transverse flow The stronger the flow the less appropriate are simple exponential fits: Hydrodynamic models (e.g. Heinz et al., Shuryak et al.) Hydro-like parameters (Blastwave) Blastwave parameterization e.g.: Ref. : E.Schnedermann et al, PRC48 (1993) 2462 Explains: spectra, flow & HBT BRAHMS: 10% central PHOBOS: 10% PHENIX: 5% STAR: 5%

“Thermal” Spectra Invariant spectrum of particles radiated by a thermal source: where: mT= (m2+pT2)½ transverse mass (Note: requires knowledge of mass) m = b mb + s ms grand canonical chem. potential T temperature of source Neglect quantum statistics (small effect) and integrating over rapidity gives: R. Hagedorn, Supplemento al Nuovo Cimento Vol. III, No.2 (1965) At mid-rapidity E = mT cosh y = mT and hence: “Boltzmann”

“Thermal” Spectra (flow aside) Describes many spectra well over several orders of magnitude with almost uniform slope 1/T usually fails at low-pT ( flow) most certainly will fail at high-pT ( power-law) N.B. Constituent quark and parton recombination models yield exponential spectra with partons following a pQCD power-law distribution. (Biro, Müller, hep-ph/0309052)  T is not related to actual “temperature” but reflects pQCD parameter p0 and n.

“Thermal” spectra and radial expansion (flow) Different spectral shapes for particles of differing mass  strong collective radial flow Spectral shape is determined by more than a simple T at a minimum T, bT explosive source T,b mT 1/mT dN/dmT light heavy mT 1/mT dN/dmT light heavy T purely thermal source

Thermal + Flow: “Traditional” Approach Assume common flow pattern and common temperature Tth 1. Fit Data  T 2. Plot T(m)  Tth, bT is the transverse expansion velocity. With respect to T use kinetic energy term ½ m b2 This yields a common thermal freezeout temperature and a common b.

Blastwave: a hydrodynamic inspired description of spectra Spectrum of longitudinal and transverse boosted thermal source: bs R Ref. : Schnedermann, Sollfrank & Heinz, PRC48 (1993) 2462 Static Freeze-out picture, No dynamical evolution to freezeout

The Blastwave Function Increasing T has similar effect on a spectrum as increasing bs Flow profile (n) matters at lower mT! Need high quality data down to low-mT

Heavy (strange ?) particles show deviations in basic thermal parametrizations STAR preliminary

Blastwave fits Source is assumed to be: In local thermal equilibrium Strongly boosted , K, p: Common thermal freeze-out at T~90 MeV and <>~0.60 c : Shows different thermal freeze-out behavior: Higher temperature Lower transverse flow Probe earlier stage of the collision, one at which transverse flow has already developed If created at an early partonic stage it must show significant elliptic flow (v2) Au+Au sNN=200 GeV STAR Preliminary  68.3% CL  95.5% CL  99.7% CL

Hydrodynamics in High-Density Scenarios Assumes local thermal equilibrium (zero mean-free-path limit) and solves equations of motion for fluid elements (not particles) Equations given by continuity, conservation laws, and Equation of State (EOS) EOS relates quantities like pressure, temperature, chemical potential, volume = direct access to underlying physics Kolb, Sollfrank & Heinz, hep-ph/0006129

Hydromodels can describe mT (pT) spectra Good agreement with hydrodynamic prediction at RHIC & SPS (2d only) RHIC: Tth~ 100 MeV,  bT  ~ 0.55 c

Blastwave vs. Hydrodynamics Tdec = 100 MeV Kolb and Rapp, PRC 67 (2003) 044903. Mike Lisa (QM04): Use it don’t abuse it ! Only use a static freeze-out parametrization when the dynamic model doesn’t work !!

Basics of hydrodynamics Hydrodynamic Equations Energy-momentum conservation Charge conservations (baryon, strangeness, etc…) For perfect fluids (neglecting viscosity), Need equation of state (EoS) P(e,nB) to close the system of eqs.  Hydro can be connected directly with lattice QCD Energy density Pressure 4-velocity Within ideal hydrodynamics, pressure gradient dP/dx is the driving force of collective flow.  Collective flow is believed to reflect information about EoS!  Phenomenon which connects 1st principle with experiment

Input for Hydrodynamic Simulations Tchemical Final stage: Hadronic interactions (cascade ?)  Need decoupling prescription Intermediate stage: Hydrodynamics can be applied if thermalization is achieved.  Need EoS (Lattice QCD ?) Initial stage: Pre-equilibrium, Color Glass Condensate ? Instead parametrization (a) for hydro simulations

Caveats of the different stages Initial stage Recently a lot of interest (Hirano et al., Heinz et al.) Presently parametrized through initial thermalization time t0, initial entropy density s0 and a parameter (pre-equilibrium ‘partonic wind’) QGP stage Which EoS ? Maxwell construct with hadronic stage ? Nobody uses Lattice QCD EoS. Why not ? Hadronic stage Do we treat it as a separate entity with its own EoS Hadronic cascade allows to describe data without an a

Interface 1: Initial Condition Need initial conditions (energy density, flow velocity,…) Initial time t0 ~ thermalization time Take initial distribution from other calculations Parametrize initial hydrodynamic field y y Hirano .(’02) x x x Energy density from NeXus. (Left) Average over 30 events (Right) Event-by-event basis e or s proportional to rpart, rcoll or arpart + brcoll

Main Ingredient: Equation of State One can test many kinds of EoS in hydrodynamics. EoS with chemical freezeout Typical EoS in hydro model H: resonance gas(RG) Q: QGP+RG p=e/3 Kolb and Heinz (’03) Hirano and Tsuda(’02) Latent heat PCE:partial chemical equiliblium CFO:chemical freeze out CE: chemical equilibrium

Interface 2: Hadronization Kolb, Sollfrank, Huovinen & Heinz; Hirano;… Hirano & Tsuda; Teaney; Kolb & Rapp Teaney, Lauret & Shuryak; Bass & Dumitru QGP phase Ideal hydrodynamics Tc Chemical Equilibrium EOS Partial Chemical Equilibrium EOS Tch Hadronic Cascade Hadron phase Tth Tth t

The Three Pillars of Experimental Tests to Hydrodynamics Identified Spectra Radial Flow in partonic and hadronic phase Identified Elliptic Flow (v2) Spatial to Momentum anisotropy, mostly in partonic phase HBT results Kinetic Freezeout Surface Lifetime of Source Conclusions from hydro Early local thermalization Viscosity, mean free path Coupling, Collectivity

π-, K-, p : reasonable agreement Best agreement for : Tdec= 100 MeV α = 0.02 fm-1 α ≠ 0 : importance of inital conditions Only at low pT (pT < 1.5 – 2 GeV/c) Failing at higher pT (> 2 GeV/c) expected: Less rescattering Central Data Tdec = 165 MeV Tdec = 100 MeV Thermalization validity limit P.F. Kolb and R. Rapp, Phys. Rev. C 67 (2003) 044903 α : initial (at τ0) transverse velocity : vT(r)=tanh(αr)

π-, K-, p : apparent disagreement? Predictions normalized to data Limited range of agreement Hydro starts failing at 62 GeV? different feed-down treatment in data and hydro? Different initial / final conditions than at 200 GeV ? Lower Tdec at 62 GeV ? Larger τ0 at 62 GeV ? Increasing τ0 gives much better agreement! Tdec = 100 MeV STAR preliminary data

PHENIX proton and pion spectra vs. hydro

Conclusions from spectra Central spectra well described either by including a pre-equilibrium transverse flow or by using a hadron cascade for the hadronic phase. Multistrange Baryons can be described with common decoupling temperature. Different result than blast wave fit. Blast wave fit is always better. Centrality dependence poorly described by hydro Energy dependence (62 to 200 GeV) indicates lower decoupling temperature and longer initial thermalization time at lower energy. System thermalizes slower and stays together longer.

Collective Radial Expansion From fits to p, K, p spectra: <r > increases continuously Tth saturates around AGS energy Strong collective radial expansion at RHIC high pressure high rescattering rate Thermalization likely Slightly model dependent here: Blastwave model