nuclear and particle physics 4b physics of the quark gluon...

25/05/2016 Alberica Toia 1

Nuclear and Particle Physics 4bPhysics of the

Quark Gluon PlasmaAlberica Toia

Goethe University Frankfurt GSI Helmholtzzentrum für Schwerionenforschung

Lectures and Exercise Summer Semester 2016


Organization● Language: English● Lecture:

● Wednesday 13:00-15:00 ● Phys 01.402

● Marks / examination→ only if required / desired● Seminar presentation → schein ● Oral Exam → grade

● Office hours: tbd on demand


Info: Email and Website● E-Mail:

[email protected]

● Website: https://web-docs.gsi.de/~alberica/lectures/KT4_SS16.html

mailto:[email protected]


Hydrodynamics and collective phenomena

● Momentum distributions

● pT and mT, mT scaling and breaking● Radial flow

● Hydrodynamics and the evolution of the system

● Directed flow● Other collective motion

● Elliptic Flow

● Methods● Experimental Results


Basic models for heavy-ion collisions● Statistical models:

basic assumption: the system is described by a (grand) canonical ensemble of noninteracting fermions and bosons in thermal and chemical equilibrium

[ -: no dynamics]

● (Ideal-) hydrodynamical models:basic assumption: conservation laws + equation of state; assumption of local thermal and chemical equilibrium

[ -: - simplified dynamics]

● Transport models:based on transport theory of relativistic quantum many-body systems -off-shell Kadanoff-Baym equations for the Green-functions S<h(x,p) in phase-space representation. Actual solutions: Monte Carlo simulations with a large number of test-particles

[+: full dynamics | -: very complicated]

● Microscopic transport models provide a unique dynamical description of non equilibrium effects in heavy-ion collisions


Basic models for heavy-ion collisions


pt- Distributions

● High pT (>>1 GeV/c):● Particle production

mechanisms are hard

● The dN/pTdpT distributions depart from the exponential trend and follow a power-law

transverse momentum (pT) distributions of particles produced in the collisions allow to extract important information on the system created in the collision

● Low pT (<≈ 1-2 GeV/c ): ● particle production mechanisms soft

● The dN/pTdpT distributions have a decreasing exponential trend (Boltzmann) pratically indipendent of energy s

Energy-independentAt low pT→ <pt>~ 300-400 MeV


pT and mT● From the definition of transverse mass:

● Therefore:

● The pT spectra are commonly expressed in terms of transverse mass mT

● mT is the energy of the particle in the transverse plane

or mT-m ● which is the kinetic energy in the transerse plane (ET

KIN)

TTTTT

T

T

TT

TT

T dppdmmm

p

pm

ppm

dp

d

dp

dm

22

22

TTTT dpp

dN

dmm

dN


Spectra in pT and in mT

slope

T

slope

T

T

pm

T

m

TTTT

eedmm

dN

dpp

dN22


mT scaling in pp● The transverse mass distributions

(dN/mTdmT) for low momentum particles have an exponential trend

● The dN/mTdmT spectrum in pp collisions is identical for all hadrons (mT scaling)● The value of the Tslope coefficient is

≈ 167 MeV for all particles

slope

T

slope

T

T

m

TT

T

m

TT

emdm

dNe

dmm

dN

Interpretations: the spectra are thermal Boltzmann spectra and Tslope represents the temperature when the emission of particles occur, i.e. the temperature of the system at the thermal freeze-out (Tfo)


● The transverse mass mT cannot have values smaller than the mass m of the particle

● The transverse kinetic energy mT-m starts from zero

Spectra in mT and ETKIN


Spectra in pT for different Tslope

● Increasing Tslope:

● The slope of the pT

spectrum decreases● The spectrum becomes

more “hard”● The mean mT and mean pT

increases


Breaking of mT scaling in AA (1)

● The slope of the spectra decreases (i.e. Tslope increases) for increasing mass of the particle● Heavier particles are “shifted”

towards higher values of pT


Breaking of mT scaling in AA (2)

● The mean pT increases with the mass of the particele● Conseguence of the increase of Tslope with the mass of the particle

● For each particle <pT> increaes with centrality

200 GeV130 GeV130 GeV200 GeV


Breaking of mT scaling in AA (3)● Tslope depends linearly on

the mass of the particle● Interpretation: on top of

thermal excitation there is a collective motion of all particles in the transverse plane with velocity v so that:

● The apparent temperature is larger than the original temperature by a blue shift

2

2

1 mvTT foslope

This collective expansion in the transverse plane is called radial flow


Flow in heavy ion collisions● Flow = collective motion of particles on top of thermal excitation

● The collective motion is caused by the high pressure generated when compressing of heating nuclear matter

● The flow velocity of a fluid element of the system is given by the sum of the velocity of all particles inside the system itself

● The collective flow is a correlation between the velocity v of a fluid element and its position in the space-time

x

yv

v


Blast-wave model● Thermal models cannot describe the differential spectra – e.g. pT or

rapidity spectra of hadrons

● One needs to account for the collective flow!

● The simplest way: Blast-wave model(P..J.. Siiemens and J..O.. Rasmussen,, Phys.. Lett.. 42 (1979) 880)

● add a collective velocity which is common for all hadrons!

● all particle spectra are described by a universal formula with common thermal freeze-out parameters:

● a temperature T of the fireball and a radial-flow velocity b

Here Ei, pi are the total energy and momentum of the considered particle I while Ai are normalization factors


Radial Flow at SPS

● Radial flow breaks the “mT -scaling” at low pT

● A fit to the identified particle spectra allows to separare the thermal excitation component from the collective motion

● In central collisions at top SPS energy (s=17 GeV):

● Tfo ≈ 120 MeV

● = 0.5

x

y

BLAST WAVE


Radial Flow at RHIC

x

y

AuAu s =200GeV

● Radial flow breaks the “mT -scaling” at low pT

● A fit to the identified particle spectra allows to separare the thermal excitation component from the collective motion

● In central AuAu collisions at top RHIC energy (s=200 GeV):

● Tfo ≈ 110 MeV

● = 0.7

BLAST WAVE


Radial flow at LHC (1)

● Spectra of pions, Kaons and protons

● Simultaneous fit with blast-wave model to extract Tfo and

BLAST WAVE


Radial flow at LHC (2)

● Spectra at LHC harder than those at RHIC● Radial flow stronger at LHC

•Blast-wave fit parameters

•Centrality

•STAR pp √s=200 GeV

From the fit to the spectra one can see that the radial flow velocity is 10% larger in central collisions at LHC than in central collisions at RHIC


Chemical and Thermal Freeze-out

Thermal Freeze out

- Stop elastic interactions

- Dynamics of particle (“momentum spectra”) frozen

Tfo (RHIC/LHC) ~ 100-130 MeV

Chemical Freeze-out

- Stop inelastic interactions

- Particle abundance (“chemical composition”) forzen

Tch (RHIC/LHC) ~ 170 MeV


Dynamic evolution of the system

● Fit to pT spectra allow to derive the temperature T and the radial expansion velocity at thermal freeze-out and indicate that: ● the fireball created in a collision of ions goes under thermal

freeze-out at a temperature of 100-130 MeV ● at freeze-out there is a rapid radial collective expansion, with a

velocity of the order of 0.5-0.7 the speed of light

● NOTE: the values of Tfo and are results of a fit to spectra and it is not a priori guaranteed that the values have a physical meaning● To understand if the values of freeze-out temperature and radial

flow velocity have a physical meaning, one needs to verify that they are reproduced by theoretical models based on the dynamic evolution of the system HYDRODYNAMICS


Hydrodynamics


Hydrodynamics

● As thermodynamics, hydrodynamics tries to explain a system using macroscopic variables (temperature, pressure … ) connected to microscopic variables

● Microscopic parameters of the fluid:● Mean free path between two collisions ()● Mean thermal excitation velocity of the particles (vTHERM)

● Macroscopic parameters of the fluid:● Dimension of the system (L)● Velocity of the fluid (vFLUID)● Pressure (p)● Density of the fluid ()● Sound Velocity in the fluid: cS = dp/d● Viscosity: ~ vTHERM


Characteristics of the fluid (1)● Knudsen Number

● Kn > 1 : The particles of the fluid do not interact – “Free-streaming” – “Ballistic limit” PERFECT GAS

● Kn << 1 : The particles of the fluid interact strongly– Liquid

● The Knudsen number is connected to the reach of thermal equilibrium● A strongly interacting system (mean free path small

compared to the dimension L of the system → Kn<<1) reaches thermal equilibrium faster

LKn

Mean free path

Dimension of the system

Can we usehydrodynamics?


Characteristics of the fluid (2)● Mach Number

● Ma < 1 : subsonic regime

● Ma > 1 : supersonic regime

● The Mach number is connected to the compressibility of the fluid● A fluid incompressible (/ ≈ 0) and stationary (so that it

Bernoulli equation holds) has a Mach number Ma ≈ 0 :

S

FLUID

c

vMa

Speed of sound

Speed of sound in the fluid

222 2

1111Mav

cp

dp

d

S

Is it compressibile?


Characteristics of the fluid (3)● Reynolds number

● Re >> 1 : Ideal fluid (not viscous)● Re <≈ 1 : viscous fluid

● For the viscosity we have the relation cS from where one derives

● For a compressible fluid (Ma>≈1): if the fluid is thermalized (Kn<<1) then it is ideal (Re>>1)

● Viscosity means a departure from equilibrium

FLUIDvL

Re Viscosity

Velocity of the fluid

Kn

Ma

c

vLRe

S

FLUID

Viscous or ideal?

≥≈1 → compressible

<<1 → thermalized>>1 → ideal


Hydrodynamics in heavy ion collisions

● After the collision a dense gas of particles is created● At certain point equ the mean free path becomes smaller than

the dimension of the system L● Kn<<1 we can use hydrodynamics for an ideal liquid

● The fluid expands, the density decreases and therefore the mean free path increases as well as the dimension of the system

● At certai point fo the mean free path is of same order of magnitued as the dimension of the system L● Kn>≈1 wee can no longer assume ideal liquid

● This point is called Thermal (or Kinetic) Freeze-out and it is characterized by the freeze-out temperature Tfo


Hydrodynamic Equations ● Hydrodynamic equations are the energy-momentum

conservation laws● In case of heavy ion collisions they have to be written for a fluid

● In non stationary motion (velocity in one point not constant in time)

● Compressible (velocity of the fluid >> speed of sound in the fluid)● Relativistic (collective velocity in the order of 0.5c)● Ideal, ie non viscous

– This last assumption is needed to simplify the problem but since some years there are hydrodynamical model which include viscosity


● This kind of fluid is described by:Non-Relativistic ● Euler equations (conservation of momentum)

Continuity Equation (conservation of mass)

Relativistic

● mass density is not a good degree of freedom: does not account for kinetic energy (large for motions close to c).

● Replace by the total energy density .Replace u by Lorentz four-vector u.→ Ideal energy momentum tensor T is built frompressure, energy density, flow velocity, and the metric tensor.

Conservation of energy-momentum

with

Continuity equation:conservation of baryonic number

with

→ 5 partial derivative differential equations with 6 unknowns (, p , nB and the 3 components of velocity)

0 T

pguupT )(


32

Equation of state● To solve the system of 5 equations for conservation of energy,

momentum and baryon number we need an additional relation● Therefore we use an equation of state for nuclear matter which

relates the pressure and the energy density of the system

First order phase transitionFor T<Tc: equation of state of non interacting hadron

gas Speed of sound: cS

2 = dp/d≈ 0.15

For T>Tc: equation of state of non interacting quark-

gluon gas with zero mass and bag-pressure B (=3p+4B)

Speed of sound: cS2 = dp/d= 1/3


Initial Conditions● In the early phases of the fireball evolution the system is not in equilibrium,

therefore hydrodynamics cannot be applied● The hydrodynamical evolution is started at a time equ starting from the state

of the system (= spatial distributions of energy and entropy) at time equ

● Modeling of initial conditions can be done with:● Monte Carlo code that describe partonic cascades (UrQMD, AMPT)● Derive energy and entropy density from participant and collision density

calculated with Glauber modelC

olli

sio

n de

nsi

ty

Par

ticip

ant

de

nsity


Thermal Freeze-out

● The hydrodynamical evolution stops when the mean free path of particles becomes of the order of the dimention of the system and therefore the system is no longer able to maintain a thermal equilibrium

● The end of the hydrodynamical evolution is normally descibed with the Cooper-Frye prescription● We postulate a sudden transition of all particles inside a fluid

element from the thermal equilibrium (mean free path = zero) to a free expansion (mean free path ∞)

● The energy density at freeze-out is one of the parameters of hydrodynamical models optimized to reproduce experimental data


After-burner: late hadronic stage● Combining hydrodynamic evolution with microscopic hadronic

transport models.

● The alternative being to just take the thermal spectra and compute resonance decays.

● Use of a hadron cascade like UrQMD in hadron gas: large dissipation and freeze-out naturally included

● Less extreme transition than going from hydro right to free streaming

TSW

switching temperature

from hydro to hadronic cascade(155-165 MeV)


Summary: from Hydrodynamics to Particle Spectra

Equation of State

1)

2)


Summary: from Hydrodynamics to Particle Spectra

Temperature Contours3) Hydro evolution

4)Particle spectra


Hydrodynamics and radial flow (1)

● The free parameters of hydrodynamics are fixed to reproduce the pT spectra of pions and antiprotons in central collisions

● Once the parameters are fixed for pions and protons in central collisions, the pT distributions in other centralities and for other hadrons are calculated without other parameters.

Pion and antiproton specra in central events fix the parameters of the model

All other spectra are calculated without other free parameters


Hydrodynamics and radial flow (2)

● The parameters in the hydrodynamical evolution initial time , temperature T, energy (or entropy s) density depend on the collision energy

● The time to equilibrate the system decreases with increasing s


Hydrodynamics and radial flow at LHC

● Hydro Prediction based on the extrapolation of RHIC parameters

● OK for pions and kaons, for protons neither the shape nor the yield are reproduced

Hydrodynamics for QGP + hadronic cascade (= microscopic model) for phases after hadronization OK for pions and kaons, for

protons the shape is reproduced


Other collective motion


Anisotropic transverse flow● In heavy ion collisions the impact

parameter generates a preferential direction in the transverse plane● The reaction plane is the plane

defined by the impact parameter and the beam direction

x

y

RP

The anisotropic transverse flow is a correlation between the azimuthal angle [=tan-1 (py/px)] of produced particles and the impact parameter (i.e. the reaction plane)

An anisotropic flow is generated if the moments of the particles in final state depend not only the local conditions in their production point but also on the global geometry of the event anisotropic flow is a non ambiguous signature of a collective behavior


Reaction Plane● The anisotropic transverse flow is therefore a correlation

between the direction (= momentum) of produced particles and the impact parameter of the collision ● The plane defined by the impact parameter and the beam

direction is called reaction plane● The azimuthal angle of the impact parameter vector in the

transverse plane is indicated with RP

x

y

RP

Reaction plane


Anisotropic transverse flow● Correlation between the velocity of the produced

particles and the impact parameter● In collisions with b≠0 (non central) a fireball is created

with a geometric anisotropy● The overlap region has an elliptic shape

• macroscopically: •The pressure gradients (and therefore the forces that push the particles) in the transverse plane are anisotropic (= depend on ) The pressure gradient is larger in the x,z plane

(along the impact parameter) than along y

•The flow velocity depend on •The azimuthal distribution of detected particles will be anisotropic


Anisotropic transverse flow

microscopically: The interactions between produced

particles (if sufficiently strong) can convert this initial geometric anisotropy in an anisotropy in the momentum distribution of particles that can be measured

● Correlation between the velocity of the produced particles and the impact parameter

● In collisions with b≠0 (non central) a fireball is created with a geometric anisotropy● The overlap region has an elliptic shape


Anisotropic transverse flow

Let's start from the azimuthal distribution of particles with respect to the reaction plane (- RP)

Using a Fourier serie decomposition :

The terms with sin do not appear because the distribution must be symmetric wrt RP

The coefficients of the different harmonics (v1, v2,…) describe the differences from an isotropic distribution

From the properties of Fourier series:

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

RPn nv cos


Coefficient v1: Directed flow

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

Directed flow

RPv cos1If v1≠0 there is a difference between the number of particles going parallel (0°) and anti-parallel (180°) to the impact parameter

directed flow therefore represent a preferential direction of particle emission


Coefficient v1: Directed flow

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

Directed flow

RPv cos1

cos21 1v

directed flow represent a translation of the particle source in the transverse planeIn the transverse plane


Directed flow


CCoefficiente v2: Elliptic flow

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

Elliptic flow

RPv 2cos2If v2≠0 there is a difference between the number of particles directed parallel (0° and 180°) and perpendicular (90° and 270°) to the impact parameter

This is the expectation from the difference between the pressure gradients parallel and ortogonal to the impact parameter


Coefficiente v2: Elliptic flow

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

Elliptic flow

RPv 2cos2

OUT OF PLANE

OUT OF PLANEIn transverse plane

The elliptic flow represents a deformation (≈ elliptical) of the distribution of the particles in the transverse plane

2cos21 2v


In plane vs. out of plane

....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

RPv 2cos2

v2 > 0 flow in-plane

v2 < 0 flow out-of-plane

2cos21 2vElliptic flow


....2cos2)cos(212)( 21

0 RPRP

RP

vvN

d

dN

Higher order harmonics

Third harmonic: v3

For collisions of identical nuclei it must be v3 = 0 (and so all the odd harmonics) for symmetry reasons wrt y

If v3≠0 it arises purely due to fluctuations

However uncorrelated to the reaction plane

Fourth harmonic: v4

For large v2 it must be ≠ 0 tor reproduce the geometry of the overlap region.

In case of ideal fluid v4=0.5 v22

3cos31 3v 4cos41 4v


Importance of elliptic flow


Elliptic flow - characteristics

The geometrical anisotropy which originates the elliptic flow decreases with the evolution of the system → self-quenching Even in case of free expansion (non interacting system) the eccentricity of the

fireball decreases with increasing dimensions of the system

The pressure gradients which originate the elliptic flow are stronger in the first instants after the collision

The elliptic flow is therefore very sensitive to the equation of state (i.e. sound velocity) of the system just after the collision (earlytimes)


Sensitivity to Equation of State

The geometrical anisotropy (X = elliptical deformation of the fireball) decreases with timeThe anisotropy of the moments (P, which is what you measure): Develops fast in the first instants after the collision ( < 2-3 fm/c), when

the system is in the QGP stateEffect of the equation of state of QGP which has a high speed of sound cS

(“hard equation of state”) Stays constant during the phase transition (2 < < 5 fm/c) which is

typically first order in hydro modelsEffect of “softening” of the equation of state during phase transition (cS = 0 )

Increases sligthly in the hadron gas phase ( < 5 fm/c)In this phase the speed of sound is lower (cS

2 ≈ 0.15 )


Sensitivity to Initial Conditions


Sensitivity to viscosity


Elliptic flow: Experimental Results


Event Plane method


Two particle correlations


Non-flow effects


v2 vs. centrality at RHIC (1)The observed elliptic flow depends on:

Eccentricity of the overlap region Decreases with increasing centrality

Amount of interactions suffered byt the particles Increases with increasing density of particles (and therefore with centrality)

Central collisions: eccentricity ≈ 0 distribution ≈ isotropic (v2 ≈ 0)

Semi-peripheral collisions:large eccentricity e re-interactionsv2 large

Very peripheral collisions: large eccentricity, few re-interactions v2 small


v2 vs. centrality at RHIC (2)

Hydrodynamic limit

STAR

PHOBOS

Hydrodynamic limit

STAR

PHOBOS

RQMD

s=130 GeV

● The v2 values measured are well described by ideal (i.e. viscosity = 0 ) hydrodynamics for central and semi-central collisions using parameters extracted from pT spectra

● Models (e.g. RQMD) based on hadronic cascade do not reproduce the observed elliptic flow which therefore seems originating from a partonic (= deconfined) phase


v2 vs. centrality at RHIC (3)

Hydrodynamic limit

STAR

PHOBOS

Hydrodynamic limit

STAR

PHOBOS

RQMD

s=130 GeV

● Interpretation:● In semi-central collisions there is a fast thermalization (equ≈0.6–1

fm/c) and the system created is an ideal fluid● For more peripheral collisions (smaller and less interacting fireball) the

thermalization is incomplete and/or slower● Note that hydro limit is that of a perfect fluid, the effect of viscosity is

to reduce the elliptic flow


v2 vs. pT at RHIC

● At low pT ideal hydrodynamics reproduces the data● At high pT the data depart from ideal trend

● Natural explanation: high pT particles escape fast from the fireball without suffering enough re-scattering and thermalizing, therefore hydrodynamics is not applicable


v2 vs. pT for identified particles

● Hydrodynamics can reproduce also the v2 dependence on particles mass at low pT


Quark number scaling

Ideal hydro:v2 scales with KE

T


Elliptic flow from RHIC to LHC● From the first 20-30% central PbPb collisions at s=2.76 TeV● v2 increases by 30% wrt values measured at RHIC

● More than foreseen by ideal hydro calculations● in agreement with models which include viscous corrections

In-plane v2 (>0) at relativistic energies (AGS and above) driven by pressure gradients (collective hydrodynamics)

Out-of-plane v2 (<0) for low √s, due to

absorption by spectator nucleons

In-plane v2 (>0) for very low √s: projectile and target form a rotating system

low energies: vspectators

< vfireball

. → spectators prevent particles frombeing emitted in the reaction plane → therefore they are preferably emitted orthogonal to it (”squeeze-out”)→ v

2<0

very low energies: small amount of pressure is builtThe interaction zone of a non-central collision is rapidly rotating and has a large lifetime. When the fireball decays particles are emitted preferably in the reaction plane due to the centrifugal forces → v

2>0


Elliptic flow from RHIC to LHC

● v2 vs. pT does not change within uncertaintes between √sNN=200 GeV and 2.76 TeV● The 30% increase of elliptic

flwo pT id therefore due to the fact that <pT> is larger at LHC due to larger radial flow

● The difference between v2 vs. pT of pions, kaons and protons (mass splitting) is more pronounced at LHC than at RHIC● Also this is due to the larger radial

flow which pushes the protons to larger values of transverse momentum


v2 vs. hydrodynamics at LHC

● Hydro predictions describe well v2(pT) measured for and K in semi-peripheral (40%-50%) and semi-central (10%-20%) collisions

● Disagreement for anti-protons in centrality 10%-20%● Larger radial flow in data than in the hydro model● Important rescattering in hadronic phase: the models with hadronic

cascade after QGP hydro reproduces bettwer the protons


● Low pT/m

T: m

T scaling is broken

● Intermediate pT/m

T: approximate (within 20%) scaling with n

q

suggest that basic degrees of freedom are quarks, whilerelation to deconfined (QGP) phase is not straightforward

Quark number scaling v

2/n

qratio to pion v

2/n

q

ALICE PreliminaryALICE Preliminary


ConclusionsIn heavy ion collisions at relativistic energies (RHIC and LHC) we observed: A strong elliptic flow The hydrodynamical evolution of an ideal fluid reproduces the observed values

and the mass dependence of the elliptic flow using an equation of state with a phase transition from QGP to a hadron gas

The elliptic flow is one of the signatures used in 2005 to state that in AuAu collisions at RHIC a “Strongly interacting QGP” (sQGP) is created The fireball reaches fast thermal equilibrium (tequ ≈ 0.6-1 fm/c)

A “perfect liquid” behavior is observed

Mean free path << dimensions of the system and viscosity ≈ 0

Note:● There are signs of incomplete thermalization (“early viscosity”) or presence of

dissipative effects in the hadronic pahse (“late viscosity”)

● The viscosity is certainly “exceptionally low”, but its estimate is affected by uncertainties since due to the initial conditions (pure Glauber vs. gluon saturation models) different viscosity values are needed in hydroto reproduce the data

● The study of higher order harmonics (v3, v4) and event-by-event v2 fluctuations provide useful information to define initial conditions and viscosity


Homework● Collective flow and viscosity in relativistic heavy-ion collisions, Ulrich W

Heinz and Raimond Snellings, Annu. Rev. Part. Sci. 63 (2013) 123-151, e-Print: arXiv:1301.2826

● Elliptic Flow: A Brief Review. Raimond Snellings New. J. Phys. 13 (2011) 055008, e-Print: arXiv:1102.3010

● Flow analysis with cumulants: Direct calculations, Ante Bilandzic, Raimond Snellings and Sergei Voloshin Phys. Rev. C83 (2011) 044913, e-Print: arXiv:1010.0233

● Event-plane flow analysis without nonflow effects, Ante Bilandzic, Naomi van der Kolk, Jean-Yves Ollitrault, and Raimond Snellings, Phys. Rev. C 83 (2011) 014909, e-Print: arXiv:0801.3915

● Collective phenomena in non-central nuclear collisions., S. A. Voloshin, A. M. Poskanzer, and R. Snellings, in Landolt-Boernstein, Relativistic Heavy Ion Physics Vol. 1/23 (Springer-Verlag, Berlin, 2010), e-Print: arXiv:0809.2949


Directed flow at FAIR energies

v1 of protons (slope at midrapidity)

● Sign changes of v2 about √s

NN ~ 3-4 GeV

● Slope of proton v1 at η ~ 0: sensitive to the phase transition / EoS

elliptic flow v2

FAIR energy regime is interesting for “flow phenomena” studies

● Can one see flow fluctuations at low energies (statistics hungry):extra constrains to disentangle physical mechanisms

Andronic arXiv:1210.8126 Steinheimer arxiv:1402.7236SIS-100 SIS-100


Continuity Equations

● Mass Conservation● The variation in time dt of the mass of fluid inside a volume V is:

● If there are no wells or sources, it has to equal the mass flow which enters/exits from the external surface of the volume V

where the – sign is due to the fact that dS is directed towards external and therefore for an outgoing flow the mass in the volume V decreases (dm/dt negative)

● Therefore:

dVdt

d

dt

dm

dVvSdvM )(divergenzadellateorema

0)( vdt

d


Euler Equations of motion (1)

● Pressure on a fluid element V=xyz:

● Pressure per unit volume:

Vdz

dpyxz

dz

dppyxpF

Vdy

dpzxy

dy

dppzxpF

Vdx

dpzyx

dx

dppzypF

zzz

yyy

xxx

)(

)(

)(

00

00

00

pf p

x

y

z

xdx

dppx 0

0xp

0yp

ydy

dppy 0


Euler Equations of motion (2)● If the only force which act on the fluid is gravitation, we can

write Newton law F=ma as

● where D/Dt represents the total derivative of velocity (which depends on t, x, y and z) wrt time:

Dt

vDgp

vvt

v

z

vv

y

vv

x

vv

t

v

dt

dz

z

v

dt

dy

y

v

dt

dx

x

v

t

v

Dt

vD

zyx

Pressuregravity

Derivative of velocity wrt time


Euler Equations of motion (3)● The Euler's equations are therefore :

● These are 3 non linear partial derivative equations which represent the momentum conservation

● In case of stationary and incompressible fluid the Euler equations reduce to Bernoulli

● In case of viscous fluid the equations are those (more complicated) of Navier-Stokes

gpvvt

v

1

)(

zz

zz

yz

xz

yy

zy

yy

xy

xx

zx

yx

xx

gz

p

z

vv

y

vv

x

vv

t

v

gy

p

z

vv

y

vv

x

vv

t

v

gx

p

z

vv

y

vv

x

vv

t

v

1

1

1


Relativistic Hydrodynamics

● In case of fluid in motion with relativistic velocity, the equations of energy/mass – momentum conservation are expressed in tensorial form as:

with

● Moreover there is a continuity equation to express the baryon number conservation :

with

● These are therefore 5 partial derivative differential equations with 6 unknowns (, p , nB and the 3 components of velocity)

0 T pguupT )(

0 Bj

unj B


Types of collective flow


• Radial flow = isotropic flow (i.e. indipendent on azimuthal angle ) in transverse plane

Due to the pressure difference between inside and outside the fireball Only kind of collective motion for b=0 Experimental Observablesi: pT (mT) spectra

Anisotropic transverse flow = flow velocity depends on azimuthal angle , tipical of collisions with b≠0 Due to pressure gradients generated by geometrical anisotropy of the

fireball Experimental Observables: azimuthal distributions of particles wrt

reaction plane, Fourier coefficients v1 , v2 , ….

Flows in nuclear collisions

x

y

x

y


Sensitivity to Equation of State


Pions vs. protons● Pions (light) are more sensitive to Tfo and

● Protons (and heavy hadrons) are more sensitive to the equation of state of the fluid● The datat clearly favor an equation of state with a partonic

phase, a hadronica phase and a phase transition

nuclear and particle physics 4b physics of the quark gluon...

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