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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‘‘
2
•• 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
3
Models of heavyModels of heavy--ion collisionsion collisions
transporttransport
hydro hydro
thermal+expansion thermal+expansion
thermal modelthermal model
finalfinalinitialinitial
4
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)
7
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)
8
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)
15
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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Dynamical transport modelDynamical transport model
)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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Dynamical transport modelDynamical transport model
(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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Dynamical transport modelDynamical transport model
(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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Dynamical transport modelDynamical transport model
(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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Dynamical transport modelDynamical transport model
(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
2
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