Nonequilibrium Dynamics of Quark Gluon Plasma: Boltzmann-Langevin (Boedeker) Equation Tanja Horn University of Maryland Outline Introduction Nonperturbative Physics Quark Gluon Plasma Relevant Particles and Scales Kinetic Theory Hydrodynamics Effective theories: HTL, LLO, NLLO Langevin Dynamics Boedeker equation (LLO) Classical Transport Theory Fluctuation-Dissipation Relation Master Equation Origin of noise term in Boedeker equation Linearized Boltzmann equation with forcing term Fokker-Planck equation Particle Dynamics Summary 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 10 5 10 6 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 Relevant Particles Length Degrees of Freedom in QCD Gluons spin 1 gauge fields, color 1 Quarks spin 2 matter particles Particles in QGP are characterized by the temperature T and the gauge coupling constant g. Characteristic energy scales Description/ Classification QCD: T ~ 200 MeV QGP: T ~ 150 MeV Typical scales for QGP are shown in Table 1. Particle Wavelength Particle Separation T T Hard Modes 1 Thermal Wavelength 1 Plasma Frequency ( gT ) 1 Semihard Modes Debye Length ( gT ) 1 Screening of static fields at large distances Nonperturbative magnetic scale (g T ) Soft modes (g T ) 2 1 2 1 Unscreened magnetic fields Transport Coefficients [ v Fext ] f ( x , p , t ) C [ f ] t x p •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 f ( x , p ) f 0 ( x , p ) f 0 ( x , p )[ 1 f 0 ( x , p )] f 1 ( x , p ) C [ f ] C [ f 0 ] Cf i Transport Coefficients [ t p x Fext p ] f 0 ( x , p , t ) ( Cf 1 )( x , p , t ) Driving Fields: (i) Conductivity: Ei (ii) Diffusion: i (iii) Shear Viscosity: 1 6 [ i u j j u i 2 3 ij u ] Nonperturbative Physics •Perturbation Theory: •At high temperature (T >>mq, T>> QCD ), g << 1. •Long distance modes of hot nonAbelian 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 Kinetic Theory Hydrodynamics T Quantum Field Theory 1 4 ( g T ln( g 4 (g T ) 1 1 )) 1 Soft, nonperturbative dynamics Transport Coefficients (Arnold & Yaffe) Effective Theory Effective Theories depend on scale separation. Expansion parameter: LowEnergyS cale HighEnergy Scale •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. 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) 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. NLLO : Next To Leading Logarithmic Order •Dynamics of low frequency gauge fields. D B . color conductivi ty •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 (ln( g 1 )) 1 (Yaffe) Form of Effective Theory unchanged to NLLO Need only to calculate the single parameter to NLLO. Langevin Dynamics: Boedeker Soft field modes affect gauge field dynamics on the magnetic scale Interactions between higher and lower frequency modes Soft modes (P~ g 2T ) Semihard modes (P~gT) •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) g T gT 2 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 D F a 2 mD W A ( x ) A ( v D )W v E BoltzmannLangevin Equation ( x , v ) gf D igA ( x ) b c (v at ( x ) w ( x, v ) Tr (T T ) a b ( x )T 1 a ab 2 W ( x , v ) W ( x , v )T a Fa abc a a A a A a gf abc A v (1, v ) b A c Origin of noise term To leading logarithmic order equations of motion can be approximated by a Langevin equation. ijk E F 0i i D B . White, Gaussian ( x ) ia m ( x ) 4 i jb m 1 D 2 2 D 2 i ij Ng T log( g 4 2 ( x ) 2T 2 0 (x) ab ( x x ) 4 i 1 ) Stochastic forcing term 1 3 Ng T log( g 1 ) 1 B Color conductivity F jk Origin of the noise term Coarse-graining in closed system reduces one of the subsystems to environment (semihard modes) Introduce Open System System Environment (P~ g 2T ) (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. i Dissipation of system Fluctuations in environment 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 p [ x gf abc A b m m Q c dp Q a gQ a F gQ Fa a d dQ d a p ] f ( x, p, Q ) C [ f ] ( x, p, Q ) p a gf abc p A b Q c non-Abelian: Color charge evolves in time. 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: S (i) Central idea: kinetic entropy and thermodynamical forces: F x F x (ii) Near equilibrium, linear response: (iii) Using fluctuation-dissipation-relation: dx i ij F i i j i ij dt j i The fluctuations near equilibrium can be identified without detailed knowledge of underlying microscopic dynamics responsible for dissipation. Mean Field Fluctuations Fluctuations at equlibrium, equilibrium distribution function. f ( x , p , Q ) f ( x , p , Q ) f eq ( p ) ( x x ) ( p p ) ( Q Q ) 3 3 f 0 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 f gf (1 ) f f g f (1 ) f 0 f ( x , p , Q ) f ( x , p , Q ) f eq ( p ) ( x x ) ( p p ) ( Q Q ) g f 3 3 Fluctuations still present in nonequilibrium 2 (1 ) ( x, p, Q ) f (1 ) ( x , p , Q ) Fluctuation-Dissipation-Relation Dissipative Linear System (Landau&Lifshitz) Dissipation Fluctuation Key idea: Kinetic entropy S(x), thermodynamic force i ( t ) j ( t ) ( t t ) ij Dynamics : dx i dt F j i ij dissipative White, Random,microscopic fluctuations, induced by environment Gaussian Fluctuation-Dissipation Near equilibrium: Phenomenological linear response F i ij x j Equal time statistics of fluctuations Einstein’s Law x ( t ) F j ( t ) i i j Assuming: White, Gaussian noise Noise-noise Autocorrelation function ik [ ik ] ki Symmetrized dissipative function 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 K n ( t ) ( n1 ( t ) n 2 ( t )... n K ( t )) n (t ) N i i 1 Goal: Treat evolution of the system as a stochastic process Assumptions: N n Thermodynamic Limit N K K P ( n ; t | m ; t ) P ( n ; | m ;0 ) stationary P ( n K ; t | n1t1 ... n K 1t K 1 ) P ( n K t K | n K 1t K 1 ) Markovian Master Equation Process entirely determined by P (m | 0 ) P ( m | n ; ) Transition Probability Master Equation P ( n ; ) P ( m | n ; ) P ( m ;0 ) m Assume n changes only be a small amount in t : t t P ( n ; t | m ) P ( n ; t ) P ( n ; | m ; ) t[ |t 0 C ] (t ) Q (n | m ) Transition Matrix Master Equation Q (n | m ) 0 C can be calculated using: P (m | n; t ) 1 n Q ( m | m ) Q (m | n ) i j k l mn P (m | n; t ) (m | n ) Q (m | n )t ij 0 kl Only for allowed transitions ij kl No transition Transition m->n in t kl ji ij ii ij 0 kl kk ij kl kl As t 0 obtain a Master Equation dP ( n ; t ) P ( m ; t )Q ( m | n ) dt m ij N-particle stochastic process, position coordinates suppressed 1 kl ij Q ( m | n ) ( ij m i m j n i m i 1 n j m j 1 n k m k 1 n l m l 1 n r m r ij m i m j n r m r ) 4 ij kl ij r ijk r lk Master Equation Probability of collision between point particles P ( ij kl ) dt d ( ij kl ) m i m j dt d ( ij kl ) ij kl ij kl kl ji Occupation numbers before collision ij lk ii ij 0 kl kk ij kl kl ij microreversibility Boltzmann Equation Introduce a characteristic moment generating function (Fourier transform of the probability density) ( x; t) K n n xi i P ( x ; t ) i 1 n s xs | x1... x K 1 Summing over all but 1,2…of the occupation numbers equations obtain the BBGKY hierarchy d ns dt 1 2 ks ij ni n j ijk d ns dt 1 2 ks ij ni n j ij ks Molecular Chaos assumption: Neglect to n i n j 1 2 ij ks nk ns microreversibilty k ( n i n i )( n j n j ) compared Boltzmann Equation d ns dt 1 2 ij n i n j n k n s ks ij ks For small deviations from equilibrium: d ns dt 1 2 ks ij n i n i (1 hi ) n i n j ( hi h j h k h s ) ys ij ks d ys dt 1 ns ns ns g s j n n j ks g s y s 0 jk 1 2 s ij ni n j jk 1 2 s ks ij ni n j jk Langevin idea, Linear response, then a linearized Boltzmann equation with a white, Gaussian noise term is obtained: d ys dt 1 s g s y s c~s ( t ) s ks ij ns s N N ks ij ns N Fokker-Planck Equation dP ( x ; t ) dt s A 1 [ a s ( x ) P ( x )] xs 2 Drift Term x x s s B 2 [ b s ( x ) P ( x )] Diffusion Term Comparing terms A and B to the stochastic Boltzmann equation d ys dt 1 s A g s y s c~s ( t ) B Then the Fokker-Planck equation is equivalent to the linearized Boltzmann equation with forcing term. Particle Dynamics Fermions? Quark Dynamics? Consider: Worldline Influence Functional Method (Johnson/Hu) Quantum/Vacuum fluctuations – Thermal Fluctuations Decoherence – Dissipation Key Concept: Quantum Open System Langevin Equation for QED Relativistic particles moving in an electromagnetic field. Closed vs. open system System Environment Collection of Worldlines Gauge Field Method = Influence Functional Formalism Summary Degrees of Freedom and scales in Quark Gluon Plasma Gluons (p~g2T, p~gT) 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: Quarks (p~T) 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? Effective Theories ys ns ij 0 kl ns g s n n j ks j jk 1 2 s ij ni n j 1 jk s 2 ks ij d ns ni n j jk dt 1 2 ks ij ni n j ij ks 1 n s 2 ij ks nk ns xs | x1... x K 1 n i n j k P ( n ; t | m ; t ) P ( n ; | m ;0 ) P ( n ; t | m ) P ( n ; t ) P ( n ; | m ; ) t[ |t 0 C ] (t ) n (t ) n (t ) P ( m | n ; t ) 1 n Q ( m | m ) Q ( m | n) n n (1 h ) i i d ys P (m | n; t ) (m | n ) Q (m | n )t dP ( n ; t ) 1 dt g s ns N t y s 0 d ns 1 2 dt n i 1 kl K ks ij kl d ys ni (t ) N i 1 dt s lk kk kl ij K P (m | 0 ) n i n j ( hi h j h k h s ) dt 1 2 n i n j n k n s P ( n K ; t | n1t1 ... n K 1t K 1 ) P ( n K t K | n K 1t K 1 ) ks ij ij ks 1 s g s P ( m | n ; ) y s c~s ( t ) P ( n ; ) P ( m | n ; ) P ( m ;0 ) s m ij ij kl ij ks 1 kl ij Q ( m | n ) ( ij m i m j n i m i 1 n j m j 1 n k m k 1 n l m l 1 n r m r ij m i m j n r m r ) 4 ij kl ij r ijk r dP ( x ; t ) kl ji ii ij 0 d ns ij N P ( ij kl ) dt d ( ij kl ) m i m j dt n xi i P ( x ; t ) kl Q (n | m ) t t dt K d ( ij kl ) ij P ( m ; t )Q ( m | n ) i j k l ( x; t) t2 n ( t ) ( n1 ( t ) n 2 ( t )... n K ( t )) m dt n K m n mn i s t1 1 [ a s ( x ) P ( x )] xs 2 2 x x s s [ b s ( x ) P ( x )] ( n i n i )( n j n j ) d ns dt 1 2 ijk ks ij ni n j ks ij ns N N ks ij ns N f 0 f gf (1 ) f f ( x , p , Q ) f ( x , p , Q ) f eq ( p ) ( x x ) ( p p ) ( Q Q ) 3 f ( x , p , Q ) f ( x , p , Q ) f eq ( p ) ( x x ) ( p p ) ( Q Q ) g f 3 2 m ( x ) 4 i 2 ( x ) ia 1 D 3 0 (x) i 1 2 Ng T log( g ) E F i jb Fi dp gQ Fa a d S x d ij ] mD 2 d 4 v ab ( x , p , Q ) ( x x ) f 0 4 i M D igA ( x ) Tr (T T ) a b 1 a g T gT a A ( t ) A ( t ) D A ( t t ) ( v D )W v E ab 2 Fext Fenv v v Aext A fluct W ( x, v) ( x )T dv dt F i ij x 2 A ( x ) A [ v D , W ( x , v )] v E ( x , v ) m ( x ) 2T (1 ) 2 g T p i dQ ( x, p, Q ) f 0i [D , F m (1 ) 3 ( x , v ) gf a abc j b c (v at ( x ) w ( x, v ) x ( t ) F j ( t ) i W ( x , v ) W ( x , v )T a f g f a (1 ) B i 1 ijk F jk i j 2 a gf abc p A b Q c Fa 4 m 2 3 Ng T log( g b A v (1, v ) c D F 2 mD W i ( t ) j ( t ) ( t t ) ij ik D B . 1 ) p [ dt 1 D 2 dx i A a A a gf abc A F j i ij x gf abc A b Q c Q a gQ a F a p ] f ( x, p, Q ) C [ f ] ( x, p, Q ) [ ik ] ki