LectureLecture 1212 ModelsModels forfor heavyheavy--ionion collisions:collisions: thermalthermal models,models, hydrodynamicalhydrodynamical models,models, transporttransport modelsmodels
SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ 1 BasicBasic modelsmodels forfor heavyheavy--ionion collisionscollisions
•• 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]
Î Microscopic transport models provide a unique dynamical description of nonequilibrium effects in heavy-ion collisions 2 ModelsModels ofof heavyheavy--ionion collisionscollisions
thermalthermal modelmodel
thermal+expansionthermal+expansion finalfinal initialinitial hydrohydro
transporttransport
3 ThermalThermal models:models: ensemblesensembles
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 4 ThermalThermal model:model: partitionpartition functionfunction
The grand canonical partition function reads: (1)
1 with H ˆ - Hamiltonian of the system, N ˆ set of conserved number operators and β = i T
All other standard thermodynamic properties can be determined from the partition function
Z = Z(V ,T ,μ1 ,μ2 ,...)
Pressure: The 1st law of thermodynymics Î conservation of energy E:
Entropy:
Particle number: (2)
5 ThermalThermal model:model: nonnon--interactinginteracting gasgas
¾ Consider a non-interacting gas of bosons or fermions of the same type.
The many-body eigenstateβ Ψ is specified by the single-particle levels εi and the set of ε occupation numbers ni where iμruns over all basis states: Ψ (3) N = ∑ni , EΨ = ∑ε ini β ε i i μ Summing over all states Ψ gives
β ε − ∑ ( i − )ni − ( − )n ⎧ − ( −μ )n ⎫ Z(T ,V ,μ ) = ∑e i = ∑∏e i i = ∏∑⎨ e i i ⎬ (4) n1 ,n2 ,... ni1 ,n2 ,... in⎩ ⎭
For the fermions: the occupation numbers n can only be 0 or 1 Î the partition function is
β ε Z = (1+ e− ( i −μ )ni ) F ∏ (5) i
β ε For the bosons: μ any occupation number n is allowed Î the partition function is
∞ − ( − )n 1 Z = ( e i i ) = β ε (6) B ∏∑ ∏ − ( i −μ ) i0n= i 1− e 6 ThermalThermal model:model: nonnon--interactinginteracting gasgas
Replacing the sum over the eigenstates by an integral → V( 2π )−3 d 3 p ∑i h ∫ the partition function for either fermions(„+“) or bosons („-“) becomes
β d 3 p ε ln Z = gV ln(1± e− ( ( p )−μ ) )±1 (7) ∫ 3 ( 2πh ) We have added in (7) g –a degeneracy factor to account for degenerate single-particle levels.
β ∂T ln Z d 3 p ε Pressure: P = = gT ln(1± e− ( ( p )−μ ) )±1 (8) ∫ 3 ∂V ( 2πh ) ∂T lnμ Z d 3 p 1 Particle number: N = = gV β ε ∫ 3 − ( ( p )−μ ) (9) ∂ μ ( 2πh ) e ± 1
3 d πp β ε( p ) Energy: E = −PV +TS + N = gV ε ∫ 3 − ( ( p )−μ ) (10) ( 2 h ) e ± 1
(11) The dispersion relation for single-particle energy: ε( p ) = p2 + m2 7 ThermalThermal model:model: thermalthermal fitfit
ρParticle density for the hadrons of type i 3 Ni d p β ε 1 i = = g (12) ∫ 3 − ( i ( p )−μi ) V ( 2πh ) e ± 1
i – particle species
Thermal fit of the experimental data –
from particle abundances (Ni) or particle ratios (in order to exclude the volume V) one can find the Lagrange parameters T, μ
Good description of the hadron abundances by the thermal hadron gas model Î The hadron abundances are in rough agreement with a thermally equilibrated system !
Î Partial equilibration is approximately reached in central heavy-ion collisions at high energies! 8 ThermalThermal model:model: thermalthermal fitsfits
The phase diagram of QCD
From the thermal fits of the particle ratios
Î chemical freeze-out line T(μ)
Îphase boundary –phase transition from hadronic matter to a quark-gluon-plasma
9 IdealIdeal hydrodynamicshydrodynamics
Basic ideas: assumption of local thermal and chemical equilibrium conservation laws + equation of state
o For an isotropic system in local equilibrium, in the local rest frame (LR) of the fluid, the energy momentum tensor and conserved currents
(13)
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 (14)
μ where is the four-velocity of the fluid ( u uμ = 1)
10 IdealIdeal hydrodynamicshydrodynamics
uμ(x) is fixed by energy-momentum and charge conservation (15)
μ 4 + I equations for 5 + I unknowns (ε , p,u ,{ni })
(in heavy-ion physics, typically I ≤ 1, we only follow baryon charge nB, if at all)
(16)
! Missing ingredient: equation of state
The set of partial differential equations (15) can then be solved for given initial conditions and freeze-out conditions 11 IdealIdeal hydrodynamics:hydrodynamics: initialinitial conditionsconditions
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.
12 IdealIdeal hydrodynamics:hydrodynamics: EoSEoS
Lattice QCD EoS - equation of state Bag-model EoS - equation of state
In the hadronic phase - a resonance gas jump in effective degrees of freedom – equation of state is taken, while from hadronic to partonic - at a critical temperature TC= 170-190 MeV in the plasma phase p = (ε-4B)/3, where B = (0.23 GeV )4 is the bag constant
13 IdealIdeal hydrodynamics:hydrodynamics: freezefreeze--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):
(17)
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)
14 IdealIdeal hydrodynamicshydrodynamics
Success of ideal hydro - spectra
15 IdealIdeal hydrodynamics:hydrodynamics: collectivecollective flowflow
Success of ideal hydro –v2
v2 versus centrality Ideal hydro (open boxes) reproduces the integrated charge particle 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.
v2 versus pT
In the bulk (low pT) : hydrodynamics works ! (full hydro or blastwave parametrization) Î
The system behaves like a strongly interacting liquid (of low viscosity). The system is likely to be partonic but not ‘plasma-like’ (weakly interacting). 16 DynamicalDynamical transporttransport modelsmodels
¾ Dynamics of heavy-ion collisions – a many-body problem! Ψ General Schrödinger equation: ∂ i ()()rr ,rr ,...,rr ,t = Hˆ Ψ rr ,rr ,...,rr ,t (18) h ∂t 1 2 N 1 2 N many-body wave-function many-body Hamiltonian
Approximation - time dependent Hartree-Fock (TDHF) theory:
• many-body wave-function Î antisymmetrized product of single particle wave functions N (19) r r r α α r Ψ ()r1 ,r2 ,...,rN ,t ⇒ A ∏ ψ ()rα ,t =1 • many-body Hamiltonian Î kinetic energy + potential with two-body interactions: α α α N ˆ r r r ˆ r 1 ˆ r r H( r1 ,r2 ,...rN ,t )⇒ T ( r ,t )+ V ( r − r ,t ) (20) ∑ α∑ αβ 2 ≠β α non-local localβ
17 DynamicalDynamical transporttransport modelmodel
TDHF equation for single-particleψα state α : ∂ i ( rr,t ) = h( rr,t )ψ ( rr,t ) (21) h ∂t α β ψ where h(r,t) is the single-particle Hartree-Fockβ Hamiltonian:
h( rr,t ) = T( rr )+ d 3r′ * ( rr′,t ) V( rr − rr′,t )ψ ( rr′,t ) (22) ∑∫ β occ sum over all occupied states *in (22) we neglected the exchange (Fock) term (to reduce complexity in notation) 2 r h r 2 Kinetic energy: T( r ) = − ∇r 2m ρ Introduce the single particle density matrix: (23) β ψ β r r′ * r′ r ( r ,r ,t ) ≡ ∑ ( r ,t )ψ β ( r ,t ) occ Thus, (22) can be written as h( rr,t ) = T( rr )+ ∑∫ d 3r′ V( rr − rr′,t )ρ( rr′,rr′,t ) (24) βocc local potential 18 DynamicalDynamical transporttransport modelmodel α
Multiply (1) by: ψα
* r ψα ψα ( r′,t )⋅( 21) : * r ∂ r * r r r ψ ( rψ′,t )i ( r ,t ) = ( r′,t )h( r ,t )ψα ( r ,t ) (25) α h ∂t ψ + r α ( 21) |for r′ ⋅ψα ( r′,t ) : ψ ⎡ ∂ ⎤ α * r′ r r * r′ r (26) − ih⎢ ( r ,t )⎥ ( r ,t ) = h( r ,t ) ( r ,t )ψα ( r ,t ) ⎣∂t ρ ⎦ ∑()( 25 )− ( 26 ) : α ∂ iρ ( rr,rr′,t ) = []h( rr,t )− h( rr′,t ) ρ( rr,rr′,t ) (27) h ∂t
β ψ β r r′ * r′ r ( r ,r ,t ) ≡ ∑ ( r ,t )ψ β ( r ,t ) occ h( rr,t ) = T( rr )+ ∑∫ d 3r′ V( rr − rr′,t )ρ( rr′,rr′,t ) βocc = T( rr )+U( rr,t ) kinetic term + potential (local) term 19 ρ DynamicalDynamical transporttransport modelmodel
Rewrite (27) using x instead of r
∂ i ⎡ r r ⎤ r r′ h 2 r h 2 r′ r r′ (28) ( x, x ,t )+ ⎢ ∇ x +U( x,t )− ∇ x′ −U( x ,t )⎥ρ( x, x ,t ) = 0 ∂t h ⎣2m 2m ⎦ r r Instead of considering the denisity ρ ( x , x ′ , t ) , let‘s find the equation of motion of its Fourier transform, i.e. Wigner transform density:
⎛ i ⎞ ⎛ sr sr ⎞ (29) f ( rr, pr,t ) = ∫ d 3s exp⎜− prsr⎟ ρ⎜rr + ,rr − ,t ⎟ ⎝ h ⎠ ⎝ 2 2 ⎠ New variables: old : xr xr′ xr + xr′ rr = , sr = xr − xr′ 2 r r f ( r , p,t ) is the single particle phase-space distribution function ρ 1 Density in coordinate space: ( rr,t ) = d 3 p f ( rr, pr,t ) (30) 3 ∫ ( 2πh ) Density in momentum space: g( pr,t ) = ∫ d 3r f ( rr, pr,t ) (31) 20 DynamicalDynamicalρ transporttransport modelmodel
Make Wigner transformation of eq.(28) r r 3 ⎛ i rr⎞ ∂ ⎛ r s r s ⎞ ∫ d s exp⎜− ps ⎟ ⎜r + ,r − ,t ⎟ (32) ⎝ h ⎠ ∂t ⎝ 2 2 ⎠ ρ 2 ⎡ ⎤ r r i h 3 ⎛ i rr⎞ r 2 r 2 ⎛ r s r s ⎞ + d s exp⎜− ps ⎟ ⎢∇ sr − ∇ sr ⎥ ⎜r + ,r − ,t ⎟ ∫ r + r − 2m h ⎝ h ⎠ ⎣ 2 2 ⎦ ⎝ 2 2 ⎠ r r r r i 3 ⎛ i rr⎞ ⎡ r s r s ⎤ ⎛ r s r s ⎞ + ∫ d s exp⎜− ps ⎟ ⎢U( r + ,t )−U( r − ,t )⎥ρ⎜r + ,r − ,t ⎟ = 0 h ⎝ h ⎠ ⎣ 2 2 ⎦ ⎝ 2 2 ⎠
r 2 r 2 r r (33) Use that ∇ sr − ∇ sr = 2∇r ⋅∇sr r + r − 2 2 sr sr Consider U( rr + ,t )−U( rr − ,t ) 2 2 Make Taylor expansion around r; sÆ0 ∞ n ∞ n n 1 ⎛ 1 r r ⎞ 1 ⎛ 1 r r ⎞ ⎛ 1 r r ⎞ (34) = ∑ ⎜ s ⋅∇r ⎟ U |s=0 −∑ ⎜− s ⋅∇r ⎟ U |s=0 = 2∑⎜ s ⋅∇r ⎟ U |s=0 n=0 n!⎝ 2 ⎠ n=0 n!⎝ 2 ⎠ odd ⎝ 2 ⎠ r r r terms even in n cancel ≈ s ⋅∇rU( r ,t ) Classical limit: keep only the first term n=1 21 DynamicalDynamical transporttransport modelmodel
From (32) and (33),(34) obtain 2 r r ∂ r r i h 3 ⎛ i rr⎞ r r ⎛ r s r s ⎞ f ( r , p,t )+ d s exp⎜− ps ⎟ 2∇ r ⋅∇ r ⎜r + ,r − ,t ⎟ ∂t 2m ∫ r s 2 2 (35) h ⎝ h ⎠ ρ ⎝ ⎠ r r i 3 ⎛ i rr⎞ v r r ⎛ r s r s ⎞ + d s exp − ps s ⋅∇ r U( r ,t ) ρ r + ,r − ,t = 0 ∫ ⎜ ⎟ r ⎜ ⎟ h ⎝ h ⎠ ⎝ 2 2 ⎠
Vlasov equation - free propagation of particles in the self-generated HF mean-field potential:
r ∂ r r p r r r r r r r r (36) f ( r , p,t )+ ∇ r f ( r , p,t )− ∇ rU( r ,t ) ∇ r f ( r , p,t ) = 0 ∂t m r r p
Eq.(36) is entirely classical (lowest order in s expansion). Here U is a self-consistent potential associated with f phase-space distribution: 1 U( rr,t ) = d 3r′d 3 pV( rr − rr′,t ) f ( rr′, pr,t ) (37) ( 2π )3 ∑∫ h βocc
22 DynamicalDynamical transporttransport modelmodel
Eq.(36) is equivalent to:
d r r ⎡ ∂ rr r r ⎤ r r (38) & r & r f ( r , p,t ) = 0 = ⎢ + r∇r + p∇ p ⎥ f ( r , p,t ) = 0 dt ⎣∂t ⎦
Î Classical equations of motion:
drr pr rr& = = dt m r (39) r dp v r p& = = −∇ rU( r ,t ) dt r
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
trajectoty: rr( t )
1 2 23