1. The R.H.I.C. Transport Challenge
Berndt Mueller (with Steffen A. Bass)
Modeling Methodology Working Group
SAMSI, November 23, 2006

2. Some Like It Hot… Nucleons + mesons Quark-gluon plasma
Genre: Comedy / Crime / Romance / Thriller
Nucleons + mesons
Quark-gluon plasma
Melting nuclear matter (at RHIC / LHC / FAIR)

3. Elements of matter and force
Matter Particles
Quarks
Leptons
Force Particles
Photon (γ), gluon (g), weak bosons (W/Z)
Higgs boson (H), graviton (G)

4. Transitions Normal (atomic) matter: Nuclear matter:
Electrons and atomic nuclei are bound into atoms
With sufficient heat (~ 3000 K) electrons can be set free; atomic matter becomes a electron-ion plasma.
Nuclear matter:
Quarks and gluons are bound into protons and neutrons
With sufficient heat (~ 21012 K) quarks and gluons are liberated; nuclear matter becomes a quark-gluon plasma.

5. When the Universe was hot…
Atoms form and Universe becomes transparent
Quarks acquire QCD mass and become confined

6. Why Heat Stuff Up? What heat does to matter:
Increases disorder (entropy)
Speeds up reactions
Overcomes potential barriers
States / phases of matter:
Solid [long-range correlations, shear elasticity]
Liquid [short-range correlations]
Gas [few correlations]
Plasma [charged constituents] (solid / liquid / gaseous)

7. Interlude about units
Energy (temperature) is usually measured in units
1 MeV  105  binding energy of H-atom
10-3  rest energy of proton
Time is usually measured in units
1 fm/c = 310-24 s  time for light to traverse a proton

8. QCD (Nuclear) Matter
Matter governed by the laws of QCD can also take on different states:
Solid, e.g. crust of neutron stars
Liquid, e.g. all large nuclei
Gas, e.g. nucleonic or hadronic gas (T  7 MeV)
Plasma - the QGP (T > Tc  150 – 200 MeV)
The QGP itself may exist in different phases:
Gaseous plasma (T  Tc)
Liquid plasma (T,m near Tc,mc ?)
Solid, color superconducting plasma (m  mc)

9. QCD phase diagram B T RHIC Quark- Gluon Plasma restored Hadronic
matter
Critical end point
Plasma
Nuclei
Chiral symmetry
broken
restored
Color
superconductor
Neutron stars T
1st order line
Quark-
Gluon
RHIC

10. QCD equation of state RHIC quarks gluons Indication of weak coupling?
color
spin
flavor
MeV
RHIC
Indication of weak coupling?

11. QGP properties
The Quark-Gluon Plasma is characterized by two properties not normally found in our world:
Screening of color fields ( it’s a plasma!):
Quarks and gluons are liberated
Disappearance of 98% of (u,d) quark masses:
Chemical equilibrium among quarks is easily attained

12. Color screening fa Induced color density
Static color charge (heavy quark) generates screened potential

13. Quark masses change Higgs field quark Quark condensate quark
Quark consendate “melts” above Tc and QCD mass disappears: chiral symmetry restoration

14. The practical path to the QGP…
STAR
…is hexagonal and 3.8 km long
Relativistic Heavy Ion Collider

15. RHIC results Some important results from RHIC:
Chemical and thermal equilibration (incl. s-quarks!)
u, d, s-quarks become light and unconfined
Elliptic flow
rapid thermalization, extremely low viscosity
Collective flow pattern related to valence quarks
Jet quenching
parton energy loss, high color opacity
Strong energy loss of c and b quarks (why?)
Charmonium suppression is not increased compared with lower (CERN-SPS) energies

16. Collision Geometry: Elliptic Flow
Reaction
plane x z y
Bulk evolution described by relativistic fluid dynamics,
assumes that the medium is in local thermal equilibrium,
but no details of how equilibrium was reached.
Input: e(x,ti), P(e), (h,etc.).
Elliptic flow (v2):
Gradients of almond-shape surface will lead to preferential expansion in the reaction plane
Anisotropy of emission is quantified by 2nd Fourier coefficient of angular distribution: v2
prediction of fluid dynamics

17. Elliptic flow: early creation
momentum
anisotropy
spatial
eccentricity
Time evolution of the energy density:
initial energy density distribution:
Anisotropic observables are of special interest as they reflect the dynamics of the earliest stages of the reaction, and thus the dynamics of the hottest phases, eventually the plasma phase that we hope to achieve in heavy ion collisions at RHIC. The overlap region of two nuclei at finite impact parameter is strongly deformed, which you can see here characterized by contours of constant wounded nucleon density in the plane orthogonal to the beam axis.
If the system is driven apart by the pressure in the system, then you see that it will expand faster into the direction of the strongest pressure gradients, and that is in the direction of the reaction plane. Upon building up an anisotropic flow field, the system will therefore get rounder with time, as you can see in those subsequent shots of the transverse plane at different times.
This reduction of the spatial eccentricity can be quantified by averaging the difference of the width in x direction and the width in y direction over this transverse distribution function. You see how this leads to a rapid decrease of the initial large geometric anisotropy over timescales relevant for heavy ion dynamics, once displayed here for a system with a very hard equation of state of a ideal gas, where the transition is very rapid, and once for an equation of state that involves a phase transition from hadronic to partonic degrees of freedom.
While the system becomes isotropic in coordinate space, it builds up anisotropies in momentum space, which you can characterize by the difference of the diagonal components of the energy momentum tensor. And here you see nicely how, as the system gets round in coordinate space, no further momentum anisotropy is built up. Note that for a system with a large mixed phase, the generation of further anisotropies is stalled significantly earlier. The momentum anisotropy you observe is really built up in the plasma stage!
The timescale suggested from all model calculations, no matter if macroscopic or microscopic is generally below 5 fm/c. Anisotropies thus probe the earliest stages of the collision.
Flow anisotropy must generated at the earliest stages of the expansion, and matter needs to thermalize very rapidly, before 1 fm/c.

18. v2(pT) vs. hydrodynamics
Failure of ideal hydrodynamics tells us how hadrons form
Mass splitting characteristic property of hydrodynamics

19. Quark number scaling of v2
In the recombination regime, meson and baryon v2 can be obtained from the quark v2 :
Chiho Nonaka
T,m,v
 Emitting medium is composed of unconfined, flowing quarks.

20. Phenomenology provides the connection
Investigative tools
Detectors
Computers
BG-J
Phenomenology provides the connection

21. Purpose of dynamic modeling
initial state
pre-equilibrium
QGP and
hydrodynamic expansion
hadronization
hadronic phase
and freeze-out
Lattice-Gauge Theory:
rigorous calculation of QCD quantities
works in the infinite size / equilibrium limit
Experiments:
only observe the final state
rely on QGP signatures predicted by Theory
Transport-Theory:
full description of collision dynamics
connects intermediate state to observables
provides link between LGT and data

22. Transport theory for RHIC
hadronization
initial state
pre-equilibrium
QGP and
hydrodynamic expansion
hadronic phase
and freeze-out

23. Observables / Probes Two categories of observables probing the QGP:
Fragments of the bulk matter emitted during break-up
Baryon and meson spectra
Directional anisotropies
Two- particle correlations
Rare probes emitted during evolution of bulk
Photons and lepton pairs
Very energetic particles (jets)
Very massive particles (heavy quarks)
Both types of probes require detailed transport modeling

24. RHIC transport: Challenges
Collisions at RHIC cover a sequence of vastly different dynamical regimes
Standard transport approaches (hydro, Boltzmann, etc.) are only applicable to a subset of the reaction phases or are restricted to a particular regime
Hybrid models can extend the range of applicability of conventional approaches
The dynamical modeling of the early reaction stage and thermalization process remain special challenges

25. Microscopic transport
Microscopic transport models describe the temporal evolution of a system of individual particles by solving a transport equation derived from kinetic theory
The state of the system is defined by the N-body distribution function fN
In the low-density limit, neglecting pair correlations and assuming that f1 only changes via two-body scattering, the time-evolution of f1 can be described by a Boltzmann equation:

26. Relativistic fluid dynamics
Transport of macroscopic degrees of freedom based on conservation laws: μTμν=0, μ jμ=0
For ideal fluid: Tμν= (ε+p) uμ uν - p gμν and jiμ = ρi uμ
Equation of state closes system of PDE’s: p=p(e,ρi)
Initial conditions are input for calculation
RFD assumes:
local thermal equilibrium
vanishing mean free path

27. Hybrid transport models
Rel. Hydro
Q-G-Plasma
Micro
time
Hadron Gas
Monte Carlo
Hadronization
Hydrodynamics microscopic transport
Ideally suited for dense systems
model early QGP reaction stage
Well defined Equation of State
Parameters:
initial conditions
equation of state
Ideally suited for dilute systems
model break-up/ freeze-out stage
describe transport properties microscopically
Parameters:
scattering cross sections
matching condition:
same equation of state
generate hadrons in each cell using local conditions

28. Analysis challenge Models Analysis Parameters: Observables:
Initial conditions
Equation of state
Transport coefficients
Reaction rates
Scattering cross sections
Emission source
Etc.
Observables:
Hadron spectra
Angular distributions
Chemical composition
Pair correlations
Photons / di-leptons
Jets
Heavy quarks
Etc.
Models
Analysis

29. Estimate of challenge
Optimization of parameters (with errors) involves:
20 – 30 parameters.
Large set of independent observables (10s – 100s).
Calculation for each parameter set: 1 – 10 h CPU time.
y(x,q) is highly nonlinear.
Output of MC simulations is noisy.
Estimate of required resources:
 104 simulations for each point in parameter space.
MC sampling of O(105) points in parameter space.
O(1011) floating point op’s per simulation.
Total numerical task O(1020) floating point op’s.
Efficient strategy is critical.

30. RHIC Transport Initiative
Modeling Relativistic Heavy Ion Collisions
Proposal to DOE Office of Science
Scientific Discovery through Advanced Computing Program
10 PI’s from 5 institutions led by Duke, including 4 Duke faculty members (S.A. Bass, R. Brady, B.Mueller, R. Wolpert)
Proposed budget ($4.5M over 5 years)

31. RTI structure

32. Optimization strategy
Use Bayesian statistical approach.
Vector of observables {yO(x,q)} with known system parameters x and model parameters q.
Compare with vector of modeled values {yM(x,q)} as
yO(x,q) = yM(x,q) + b(x) + e ,
with bias b(x) and mean-zero random e describing experimental errors and fluctuations.
Create Gaussian random field surrogate zM(x,q) of yM(x,q) for efficient MCMC simulation of posterior probability distribution P(q|yM).

33. Visualization framework

34. IT Infrastructure

35. Outlook The first phase of the RHIC science program has shown that:
equilibrated matter is rapidly formed in heavy ion collisions;
wide variety of probes of matter properties available;
systematic study of matter properties is possible.
The Quark-Gluon Plasma appears to be a novel type of liquid with unanticipated transport properties.
The successful execution of the next phase of the RHIC science program will require:
sophisticated, realistic modeling of transport processes;
state-of-the-art statistical analysis of experimental data in terms of model parameters.
Exciting opportunities for collaborations between physicists and applied mathematicians!

36. THE END