Lecture 12 Models for heavy-ion collisions: thermal models...

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Lecture 12Lecture 12Models for heavyModels for heavy--ion collisions: ion collisions:

thermal models, thermal models, hydrodynamical models, hydrodynamical models,

transport modelstransport models

SS2011SS2011: : ‚‚Introduction to Nuclear and Particle Physics, Part 2Introduction to Nuclear and Particle Physics, Part 2‘‘

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•• Statistical models:basic assumption: the system is described by a (grand) canonical ensemble of non-interacting 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]

Basic models for heavyBasic models for heavy--ion collisionsion collisions

Microscopic transport models provide a unique dynamical description of nonequilibrium effects in heavy-ion collisions

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Models of heavyModels of heavy--ion collisionsion collisions

transporttransport

hydro hydro

thermal+expansion thermal+expansion

thermal modelthermal model

finalfinalinitialinitial

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Thermal models: ensemblesThermal models: ensembles

In equilibrium statistical mechanics one normally encounters three type of ensembles:

microcanonical ensemble: used to describe an isolated system with fixed E, N, V

canonical ensemble: used to describe a system in contact with a heat bath (presumed to represent an infinite reservoir) with fixed T, N, V

grand canonical ensemble: used to describe a system in contact with a heat bath, with

which it can exchange particles: with fixed T, V, μ

In a relativistic quantum system, where particles can be created and destroyed, one

computes observables conventionally in the grand canonical ensemble

Here: E – energy of the system, N – particle number in the system,

V – volume of the system, T – temperature of the system, μ – chemical potential

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with - Hamiltonian of the system, set of conserved number operators and

Thermal model: partition functionThermal model: partition function

The grand canonical partition function reads:

All other standard thermodynamic properties can be determined from the partition function

Pressure:

Entropy:

Particle number:

The 1st law of thermodynymicsconservation of energy E:

(1)

(2)

T1

=β

,...),,T,V(ZZ 21 μμ=

H iN

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Thermal model: Thermal model: nonnon--interacting gasinteracting gas

(3)

Consider a non-interacting gas of bosons or fermions of the same type.The many-body eigenstate Ψ is specified by the single-particle levels εi and the set of occupation numbers ni where i runs over all basis states:

∑∑ ==i

iii

i nE,nN εΨΨ

∏ ∑∑ ∏∑⎭⎬⎫

⎩⎨⎧

==∑= −−−−−−

i n

n)(

,...n,n i

n)(

,...n,n

n)(ii

21

ii

21

i ii eee),V,T(Z μεβμεβμεβμ

Summing over all states Ψ gives

(4)

(5)

For the fermions:the occupation numbers n can only be 0 or 1 the partition function is

∏ −−+=i

n)(F )e1(Z ii μεβ

For the bosons:any occupation number n is allowed the partition function is

∏∏ ∑ −−

∞

=

−−

−==

i)(

i 0n

n)(B i

ii

e11)e(Z μεβ

μεβ (6)

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Thermal model: Thermal model: nonnon--interacting gasinteracting gas

Replacing the sum over the eigenstates by an integral

the partition function for either fermions(„+“) or bosons („-“) becomes

∫∑ −→ pd)2(V 33i

hπ

1))p((3

3

)e1ln()2(

pdgVZln ±−−±= ∫ μεβ

πh(7)

We have added in (7) g – a degeneracy factor to account for degenerate single-particle levels.

1))p((3

3

)e1ln()2(

pdgTV

ZlnTP ±−−±=∂

∂= ∫ μεβ

πhPressure:

Particle number:1e

1)2(

pdgVZlnTN ))p((3

3

±=

∂∂

= −−∫ μεβπμ h

Energy:1e

)p()2(

pdgVNTSPVE ))p((3

3

±=++−= −−∫ μεβ

επ

μh

(8)

(9)

(10)

22 mp)p( +=εThe dispersion relation for single-particle energy:(11)

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Thermal model: Thermal model: thermal fitthermal fit

1e1

)2(pdg

VN

))p((3

3i

i ii ±== −−∫ μεβπ

ρh

Particle density for the hadrons of type i

Thermal fit of the experimental data –

from particle abundances (Ni) or particle ratios (in order to exclude the volume V) one can findthe Lagrange parameters T, μ

i – particle species

Good description of the hadron abundances by the thermal hadron gas model The hadron abundances are in rough agreement with a thermally equilibrated system !

Partial equilibration is approximately reached in central heavy-ion collisions at high energies!

(12)

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Thermal model: Thermal model: thermal fitsthermal fits

The phase diagram of QCD

From the thermal fits of the particle ratios

chemical freeze-out line T(μ)

phase boundary – phase transition from hadronic matter to a quark-gluon-plasma

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Ideal hydrodynamicsIdeal hydrodynamics

Basic ideas:assumption of local thermal and chemical equilibriumconservation laws + equation of state

oo For an isotropic system in local equilibrium, in the local rest frame (LR) of the fluid,the energy momentum tensor and conserved currents

are given by a few, local, macroscopic variables: i.e. energy density ε(x), pressure p(x),charge densities ni(x)

o In an arbitrary frame where the fluid could be moving

where where is the four-velocity of the fluid

(13)

(14)

)1uu( =μμ

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Ideal hydrodynamicsIdeal hydrodynamics

uμ(x) is fixed by energy-momentum and charge conservation

! Missing ingredient: equation of state

(15)

(16)

The set of partial differential equations (15) can then be solved for giveninitial conditions and freeze-out conditions

4 + I equations for 5 + I unknowns

(in heavy-ion physics, typically we only follow baryon charge nB, if at all),1I ≤

{ })n,u,p,( iμε

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Ideal hydrodynamics: initial conditionsIdeal hydrodynamics: initial conditions

Initial conditions :

(for each individual cell of the fluid)

Typical ingredients:

- thermalization time τ0

- shape of initial entropy (or energy) density profile

- maximum entropy (or energy) density s0 (or ε0)

- baryon density to entropy ratio nB/s (assumed to be constant)

- initial radial flow is typically set to zero

Parameters τ0 , s0 (or ε0) nB/s and also freezeout parameters Tf0 (εf0), are fitted to data for central collisions. Noncentral collisions can then be predicted.

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Ideal hydrodynamics: EoSIdeal hydrodynamics: EoS

Bag-model EoS - equation of state Lattice QCD EoS - equation of state

jump in effective degrees of freedom –from hadronic to partonic - at a critical temperature TC= 170-190 MeV

In the hadronic phase - a resonance gas equation of state is taken, while

in the plasma phase p = (ε-4B)/3,

where B = (0.23 GeV )4 is the bag constant

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Ideal hydrodynamics: freezeIdeal hydrodynamics: freeze--outout

Cooper-Frye freezeout:

sudden transition from the fluid to a gas on a 3D hypersurface

(typically Tf0 = const or εf0 = const), where f is the phase-space density of a local

equilibrium gas boosted with velocity u(x):

where dσμ is the hypersurface normal to x

Application:

evolve hydro until the end of the phase transition (for the EoS with a QGP)

then do Cooper-Frye - assume a noninteracting hadron gas

decay unstable resonances (Tf0 (εf0) and fit to data)

(17)

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Ideal hydrodynamicsIdeal hydrodynamicsSuccess of ideal hydro - spectra

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Ideal hydrodynamics: collective flowIdeal hydrodynamics: collective flow

v2 versus centrality

Ideal hydro (open boxes) reproduces the integrated chargeparticle v2 data (black dots) for impact parameters b < 7 fm. In peripheral collisions it overshoots v2 - the smaller and more dilute the system is, the more difficult for it to thermalize.

Success of ideal hydro – v2

The system behaves like aThe system behaves like a strongly interacting strongly interacting liquidliquid (of low viscosity). (of low viscosity).

The system is likely to beThe system is likely to be partonicpartonic but not but not ‘plasma‘plasma--like’ (weakly interacting).like’ (weakly interacting).

v2 versus pT

In the bulk (low In the bulk (low ppTT) : hydrodynamics works !) : hydrodynamics works !(full hydro or (full hydro or blastwave parametrizationblastwave parametrization) )

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Dynamical transport modelsDynamical transport models

Dynamics of heavy-ion collisions – a many-body problem!

General Schrödinger equation:

( ) ( )t,r,...,r,rHt,r,...,r,rt

i N21N21rrrrrr

h ΨΨ =∂∂ (18)

many-body wave-function many-body Hamiltonian

Approximation - time dependent Hartree-Fock (TDHF) theory:

( ) ( )t,rAt,r,...,r,rN

1N21 αα

α

ψΨrrrr ∏

=

⇒

• many-body wave-function antisymmetrized product of single particle wave functions

∑∑≠

−+⇒N

N21 )t,rr(V21)t,r(T)t,r,...r,r(H

βαβααβ

ααα

rrrrrr

(19)

(20)

• many-body Hamiltonian kinetic energy + potential with two-body interactions:

non-local local

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Dynamical transport modelDynamical transport model

TDHF equation for single-particle state α :

)t,r()t,r(h)t,r(t

irrr

h αα ψψ =∂∂

where h(r,t) is the single-particle Hartree-Fock Hamiltonian:

)t,r()t,rr(V)t,r(rd)r(T)t,r(hocc

*3 ′′−′′+= ∑∫rrrrrr

ββ

β ψψ

(21)

(22)

*in (22) we neglected the exchange (Fock) term (to reduce complexity in notation)

2r

2

m2)r(T ∇−=

rhrKinetic energy:

Introduce the single particle density matrix:

sum over all occupied states

)t,r()t,r()t,r,r( *

occ

rrrrββ

β

ψψρ ′≡′ ∑(23)

Thus, (22) can be written as

)t,r,r()t,rr(Vrd)r(T)t,r(hocc

3 ′′′−′+= ∑∫rrrrrr

ρβ

(24)

local potential

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)t,r()t,r(h)t,r()t,r(t

i)t,r( ** rrrrh

rαααα ψψψψ ′=

∂∂′

:)21()t,r(* ⋅′rαψ

Multiply (1) by:

:)t,r(|)21( rfor ′⋅′+ r

rαψ

)t,r()t,r()t,r(h)t,r()t,r(t

i ** rrrrrh αααα ψψψψ ′=⎥⎦

⎤⎢⎣⎡ ′∂∂

−

(25)

(26)

( ) :)26()25(∑ −α

[ ] )t,r,r()t,r(h)t,r(h)t,r,r(t

i ′′−=′∂∂ rrrrrr

h ρρ

)t,r()t,r()t,r,r( *

occ

rrrrββ

β

ψψρ ′≡′ ∑

)t,r(U)r(T

)t,r,r()t,rr(Vrd)r(T)t,r(hocc

3

rr

rrrrrr

+=

′′′−′+= ∑∫ ρβ

(27)

kinetic term + potential (local) term

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(28)

Rewrite (27) using x instead of r

0)t,x,x()t,x(Um2

)t,x(Um2

i)t,x,x(t

2x

2x =′⎥⎦

⎤⎢⎣⎡ ′−∇−+∇+′

∂∂

′rrrrhrrh

h

rrρρ

Instead of considering the denisity , let‘s find the equation of motion of its Fourier transform, i.e. Wigner transform density:

)t,x,x( ′rrρ

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛−= ∫ t,

2sr,

2srspiexpsd)t,p,r(f 3

rr

rrrr

h

rrρ (29)

New variables:

xxs,2

xxr ′−=′+

=rrr

rrr

xx:old ′rr

Density in coordinate space:

)t,p,r(frr

is the single particle phase-space distribution function

∫= )t,p,r(fpd)2(

1)t,r( 33

rr

h

r

πρ (30)

Density in momentum space: ∫= )t,p,r(frd)t,p(g 3 rrr (31)

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(32)

Make Wigner transformation of eq.(28)

0t,2sr,

2sr)t,

2sr(U)t,

2sr(Uspiexpsdi

t,2sr,

2srspiexpsd

m2i

t,2sr,

2sr

tspiexpsd

3

2

2sr

2

2sr

32

3

=⎟⎠⎞

⎜⎝⎛ −+⎥⎦

⎤⎢⎣⎡ −−+⎟

⎠⎞

⎜⎝⎛−+

⎟⎠⎞

⎜⎝⎛ −+⎥

⎦

⎤⎢⎣

⎡∇−∇⎟

⎠⎞

⎜⎝⎛−+

⎟⎠⎞

⎜⎝⎛ −+

∂∂

⎟⎠⎞

⎜⎝⎛−

∫

∫

∫

−+

rr

rr

rr

rrrr

hh

rr

rrrrrr

hh

h

rr

rrrr

h

rr

rr

ρ

ρ

ρ

sr2

2sr

2

2sr

2 rrrr

rr

rrrr∇⋅∇=∇−∇

−+Use that

(33)

Consider

)t,r(Us

|Us212|Us

21

!n1|Us

21

!n1

)t,2sr(U)t,

2sr(U

r

odd0s

n

r0s

n

r0n

0s

n

r0n

rrr

rrrrrr

rr

rr

r

rrr

∇⋅≈

⎟⎠⎞

⎜⎝⎛ ∇⋅=⎟

⎠⎞

⎜⎝⎛ ∇⋅−−⎟

⎠⎞

⎜⎝⎛ ∇⋅=

−−+

∑∑∑ ==

∞

==

∞

=

Make Taylor expansion around r; s 0

terms even in n cancel

Classical limit: keep only the first term n=1

(34)

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(35)

From (32) and (33),(34) obtain

0t,2sr,

2sr)t,r(Usspiexpsdi

t,2sr,

2sr2spiexpsd

m2i)t,p,r(f

t

r3

sr3

2

=⎟⎠⎞

⎜⎝⎛ −+∇⋅⎟

⎠⎞

⎜⎝⎛−+

⎟⎠⎞

⎜⎝⎛ −+∇⋅∇⎟

⎠⎞

⎜⎝⎛−+

∂∂

∫

∫r

rr

rrrvrr

hh

rr

rrrrrr

hh

hrr

r

rr

ρ

ρ

0)t,p,r(f)t,r(U)t,p,r(fmp)t,p,r(f

t prr =∇∇−∇+∂∂ rrrrrrrrrrr

rrr

Vlasov equation - free propagation of particles in the self-generated HF mean-field potential:

(36)

Eq.(36) is entirely classical (lowest order in s expansion).Here U is a self-consistent potential associated with f phase-space distribution:

)t,p,r(f)t,rr(pVdrd)2(

1)t,r(U 333

occ

rrrr

h

r ′′−′= ∑∫βπ

(37)

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(38)

Eq.(36) is equivalent to:

0)t,p,r(fprt

0)t,p,r(fdtd

pr =⎥⎦⎤

⎢⎣⎡ ∇+∇+∂∂

==rrr

&rr&rrr

rr

Classical equations of motion:

)t,r(Udtpdp

mp

dtrdr

rrvr

&r

rr&r

r∇−==

==(39)

Note: the quantum physics only plays a role in the initial conditions for f: the initial f must respect the Pauli principle, however, the identity of particles plays no role beyond the initial conditions

)t(r:trajectoty r

1

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Lecture 12 Models for heavy-ion collisions: thermal models...

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Transcript of Lecture 12 Models for heavy-ion collisions: thermal models...