1. Tanja Horn University of Maryland
Nonequilibrium Dynamics of Quark Gluon Plasma: Boltzmann-Langevin (Boedeker) Equation
Tanja Horn
University of Maryland

2. Outline Introduction Nonperturbative Physics Langevin Dynamics
Quark Gluon Plasma
Relevant Particles and Scales
Nonperturbative Physics
Kinetic Theory
Hydrodynamics
Effective theories: HTL, LLO, NLLO
Langevin Dynamics
Boedeker equation (LLO)
Origin of noise term in Boedeker equation
Classical Transport Theory
Fluctuation-Dissipation Relation
Master Equation
Linearized Boltzmann equation with forcing term
Fokker-Planck equation
Particle Dynamics
Summary

3. Quark Gluon Plasma (QGP)
State of matter in which quarks and gluons are unconfined Manifest of asymptotic freedom in Quantum Chromodynamics (QCD)
Characteristic phase transition at s in the cosmological standard model.
Different models of heavy ion collisions are studied at RHIC/Brookhaven and LHV/CERN.
Lifetime of a plasma state is short, so a large fraction of QGP lifetime spent in nonequilibrium states.
Need for an understanding of nonequilibrium in QGP

4. Relevant Particles Gluons spin 1 gauge fields, color
Length
Description/
Classification
Particle Wavelength
Hard Modes
Particle Separation
Thermal Wavelength
Plasma Frequency
Semihard Modes
Debye Length
Screening of static fields at large distances
Nonperturbative magnetic scale
Unscreened magnetic fields
Soft modes
Degrees of Freedom in QCD
Gluons spin 1 gauge fields, color
Quarks spin matter particles
Particles in QGP are characterized by the temperature T and the gauge coupling constant g.
Characteristic energy scales
QCD: T ~ 200 MeV
QGP: T ~ 150 MeV
Typical scales for QGP are shown in Table 1.

5. Transport Coefficients
View Boltzmann equation as an effective theory with Quantum Fluctuations integrated out.
On the scale O(T), the dispersion relation of the hard particles is suppressed by O(g2) and collisions can be treated as local collisions.
Hard modes are weakly interacting and can be treated perturbatively.
Method: Linearized Boltzmann equation with Driving Term

6. Transport Coefficients
Driving Fields:
Conductivity:
Diffusion:
Shear Viscosity:

7. Nonperturbative Physics
Perturbation Theory:
At high temperature (T >>mq,
T>> ), g << 1.
Long distance modes of hot non-Abelian gauge fields are strongly coupled resulting in breakdown of perturbation theory.
Nonperturbative Physics:
Lattice calculations for static situations.
Kinetic Theory: for large times and distances compared to particle energy and momentum.
Hydrodynamics: particles propagate classically between collisions, which are independent, uncorrelated events.
Mean free time
Perturbation Theory
Quantum Field Theory
Kinetic Theory
Soft, non-perturbative dynamics
Hydro-dynamics
Transport Coefficients
(Arnold & Yaffe)

8. Effective Theory Effective Theories depend on scale separation.
Expansion parameter:
Eliminates the high energy scale from the equations of motion, which implies that the high energy scale is not dynamical anymore
Obtain an effective Lagrangian in which the information about the high energy scale is only contained implicitly.

9. Classical Field Approximation
Soft field modes have a large occupation number
Classical Field Approximation
Hard field modes (p~T) have occupation number unity, so to use classical field approximation
Integrate out hard field modes
Can think of this as hard field modes propagating in a slowly varying classical background field.
Effective Hard Thermal Loop Theory (HTL)

10. HTL Effective Theory
Infrared divergences due to massless modes in standard thermal theories.
Found that physical observables are gauge dependent.
Resummation of all 1-loop diagrams with hard internal momenta and soft external momenta (Braaten & Pisarski).
HTL corrections to the propagator and vertices lead to gauge invariant results for physical observables.
Resulting effective theory for the soft modes contains non-local interactions.

11. NLLO : Next To Leading Logarithmic Order
Dynamics of low frequency gauge fields.
Effective Leading Logarithmic Order (LLO) obtained by Boedeker after integrating out
Quantum Fluctuations: T
Thermal Fluctuations: gT
Goal: Effective Theory valid beyond LLO
All higher order corrections are suppressed by one or more powers of
(Yaffe)
Form of Effective Theory unchanged to NLLO
Need only to calculate the single parameter to NLLO.

12. Langevin Dynamics: Boedeker
Soft field modes affect gauge field dynamics on the magnetic scale
Interactions between higher and lower
frequency modes
Semihard modes
(P~gT)
Soft modes
(P~ )
Goal: Effective Theory for soft modes in QGP
Treat soft modes as the system, semihard modes as the environment
Coarse-grained effective action (coarse-grain parameter are loop expansions)

13. Classical Effective Theory
Open system introduced through coarse-graining condition.
Procedure:
(i) Integrate out hard field modes
HTL effective theory
(ii)Integrate out degrees of freedom with p~gT.
Resulting Effective Theory
Boltzmann-Langevin Equation

14. Origin of noise term
To leading logarithmic order equations of motion can be approximated by a Langevin equation.
White, Gaussian
Stochastic forcing term
Color conductivity

15. Origin of the noise term
Coarse-graining in closed system reduces one of the subsystems to environment (semihard modes)
Introduce Open System
System
(P~ )
Environment
(P~gT)
Averaged Effect of environment backreacts on the system.
Random Force due to thermal fluctuations of semihard field modes
Dissipative behaviour of the system.
Dissipation of system
Fluctuations in environment

16. Classical Transport Theory
Boltzmann-Langevin equation can be obtained from full QFT as the kinetic limit if the full stochastic Dyson equations (Calzetta/Hu)
But, in the semiclassical limit the soft modes have a large occupation number, so can arrive at the Boltzmann-Langevin equation even in classical transport theory.
Classical colored point particles
Phase Space trajectories determined by Lorentzian type equations
Litim and Manuel introduce a Boltzmann-Langevin equation including QCD effects
non-Abelian: Color charge evolves in time.

17. Classical Transport Theory: Origin of Noise Term
Phenomenological Langevin Approach.
Basically stipulating that equation of motion is incomplete. To include the effect of the environment need to include the noise term.
Fluctuation-Dissipation Theorem for classical linear dissipative systems (Landau&Lifshitz).
Assume: Dissipative process is known:
(i) Central idea: kinetic entropy and thermodynamical forces:
(ii) Near equilibrium, linear response:
(iii) Using fluctuation-dissipation-relation:
The fluctuations near equilibrium can be identified without detailed knowledge of underlying microscopic dynamics responsible for dissipation.

18. Mean Field Fluctuations
Fluctuations at equlibrium, equilibrium distribution function.
soft and semihard modes can be treated classically due to the large occupation number
Quantum Statistics: Bose-Einstein, Fermi-Dirac
Fluctuation-Dissipation-Relation reduced to classical relation.
Nonequilibrium fluctuations
Fluctuations still present in nonequilibrium

19. Fluctuation-Dissipation-Relation
Dissipative Linear System (Landau&Lifshitz)
Dissipation
Fluctuation
Key idea: Kinetic entropy S(x), thermodynamic force
Dynamics :
White,
Gaussian
Random,microscopic fluctuations, induced by environment
dissipative

20. Fluctuation-Dissipation
Near equilibrium: Phenomenological linear response
Equal time statistics of fluctuations Einstein’s Law
Assuming: White, Gaussian noise
Noise-noise Autocorrelation function
Symmetrized dissipative function

21. Master Equation Consider : Homogeneous gas
Set of velocity states available to N particles divided into K cells
State of the system defined by complete set of occupation numbers
Goal: Treat evolution of the system as a stochastic process
Assumptions:
Thermodynamic Limit
stationary
Markovian

22. Master Equation Process entirely determined by Master Equation
Transition Probability
Assume n changes only be a small amount in :
Transition Matrix

23. Master Equation C can be calculated using: Only for allowed
Only for
allowed
transitions
No transition
Transition m->n in
As obtain a Master Equation
N-particle stochastic process, position coordinates suppressed

24. Master Equation Probability of collision between point particles
Occupation numbers before collision
microreversibility

25. Boltzmann Equation
Introduce a characteristic moment generating function (Fourier transform of the probability density)
Summing over all but 1,2…of the occupation numbers equations obtain the BBGKY hierarchy
microreversibilty
Molecular Chaos assumption: Neglect compared to

26. Boltzmann Equation For small deviations from equilibrium:
Langevin idea, Linear response, then a linearized Boltzmann equation with a white, Gaussian noise term is obtained:

27. Fokker-Planck EquationB
Drift Term
Diffusion Term
Comparing terms A and B to the stochastic Boltzmann equation B A
Then the Fokker-Planck equation is equivalent to the linearized Boltzmann equation with forcing term.

28. Particle Dynamics Fermions? Quark Dynamics?
Consider: Worldline Influence Functional Method (Johnson/Hu)
Langevin Equation for QED
Relativistic particles moving in an electromagnetic field.
Quantum/Vacuum fluctuations – Thermal Fluctuations
Decoherence – Dissipation
Key Concept: Quantum Open System
Closed vs. open system
Method = Influence Functional Formalism
System
Collection of Worldlines
Environment
Gauge Field

29. Summary Degrees of Freedom and scales in Quark Gluon Plasma
Semihard field modes backreact on the soft field modes
The Boltzmann equation needs to be “upgraded” with a noise term
Boltzmann-Langevin (Boedeker) equation
Three ways to derive the noise term in kinetic theory:
Boedeker: HTL, coarse-grained effective action, open system
Litim/Manuel: Phenomenological Langevin idea, Fluctuation-Dissipation relation
Kac/Logan: Master equation, stationary Markovian transition process
White, Gaussian noise in all three cases
Nonlocal noise? Fermions?
Gluons (p~g2T, p~gT)
Quarks (p~T)

30. Effective Theories