Three Forms of Quantum Kinetic Equations

1. Three Forms of Quantum Kinetic Equations Evgeni Kolomeitsev Matej Bel University, Banska Bystrica Based on: Ivanov, Voskresensky, Phys. Atom. Nucl. 72 (2009) 1168 Kolomeitsev, Voskresensky J Phys. G 40 (2013) 113101 (topical review)

2. binary collisions! Boltzmann kinetic equation distribution of particles in the phase space drift term Between collisions “particles” move along characteristic determined by an external force Fext collision term: Assumptions: “Stosszahlansatz” (chaos ansatz) -- valid for times larger than a collision time; -- sufficiently long mean free path, but not too long. local in time and space! Conservation: Entropy increase entropy for BKE is

3. Modifications of the Boltzmann Kinetic Equation: Vlasov equation: plasma Landau collision integral for Coulomb interaction (divergent) Balescu-Lenard (1960) and Silin-Rukhadze(1961) finite collision integral medium polarization Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy Derivation of kinetic equations: Bogoliubov’s principle of weakening of correlations Quantum kinetic equation: Pauli blocking, derivation of QKE

4. Important assumption behind the above KE: fixed energy-momentum relation In heavy-ion collisions many assumption behind the BKE are not justified At high energies many new particles are produced Resonance dynamics Spectral functions for pions, kaon, nucleons and deltas in medium How to write kinetic equations for resonances?

5. Pathway to the kinetic equation non-equilibrium field theory formulated on closed real-time contour [Schwinger, Keldysh] non-eq. Green’s functions (only 2 of them are independent) weakening of initial and all short-range correlations Wick theorem Dyson equation: Wigner transformation. Separation of slow and fast variables (Fourier trafo.) Gradient expansion 1st gradient approximation Poisson brackets:

6. Physical notations We introduce quantities which are real and, in the quasi-homogeneous limit, positive, have a straightforward physical interpretation, much like for the Boltzmann equation. Wigner functions 4-phase-space distribution functions [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902] spectral density Retarded Green’s function Quasi-particle limit +… weight factor phase space distribution as in Boltzmann KE Weight factor is usually dropped It can be hidden in the collision integral

7. Kinetic equation in the 1st gradient approximation Drift operator: for non-relat. part. Collision term: Gain term (production rate): Loss term (absorption rate): The “mass” equation gives a solution for the retarded Green’s function mass function: width: [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902]

8. Equilibrium relation: H-theorem not proven in general case, only for Three forms of the kinetic equation. Kadanoff-Baym equation group velocity Conservation: Noether currents and energy-momentum tensor (if a conservative Phi-derivable scheme is applied) (+ memory terms and derivative terms) Entropy: (Markovian part) Because the term does not depend on F explicitly, in the KBE one cannot separate a collision less propagation of a test particle and a collision term. [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902]

9. Three forms of the kinetic equation. Botermans-Malfliet replace in the Poisson-bracket term Assume small deviation from the local equilibrium group velocity Conservation: effective current Entropy: (Markovian part) For Markovian part the H-theorem is proven [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902] Separation of particle drift and collision terms is possible test-particle method [Cassing, Juchem NPA672; NPA665, 385; Leupold NPA672,475] There exists a well defined hydrodynamic limit [Voskresensky, NPA849 (11) 120]

10. Three forms of the kinetic equation. Non-local KE [Ivanov, Voskresensky, Phys.Atom.Nucl 72 (2009)1215] suggested to rewrite KBE as drift term as for BME shifted collision term shifts in time-coordinate and energy-momentum spaces: If we replace CNL with the usual collision integral we obtain BME If we expand the non-local collision term up to 1st gradients we arrive at KBE Conservation: the same as for KBE (up to 1st gradients) Entropy: (Markovian part)

11. Characteristic times KBE: relaxation time or an average time between collisions BME: time delay/advance of the scattered wave in the resonance zone NLE: BME with forward-time in KE can be positive (delay) near resonance and negative (advance) far from resonance

12. Spectral density normalizations Ergodicity density of state Noether particle density. Conserved by KBE exactly and by BME approximately Current conserved by BME exactly [Delano, PRA 1(1970) 1175] unstable particle gas Current conserved only approximately Wigner time [Weinhold,Friman, Nörenberg PLB 433 (98) 236]

13. Consider behavior of a dilute admixture of uniformly distributed light resonances in an equilibrium medium consisting of heavy-particles. Thereby, we assume that SR is determined by distribution of heavy particles, Light resonance production by heavy-particles is determined by the equilibrium production rate. Three solutions coincide only for . However this condition may hold only in very specific situations. For Wigner resonances it holds only for Examples of solutions of kinetic equations collision integral: KBE: BME: NLE: different answers!! where a is the solution of equation

14. Resonance life time [Leupold, NPA695 (2001 377] Spatially uniform dilute gas of non-interacting resonances produced at t<0 and placed in the vacuum at t=0. Production of new resonances ceases for t>0 : From the BME Leupold got the solution: On the contrary, from the KBE one finds: NLE However, the BME does not hold for Gin=0, since its derivation is based on the equation Gin=G f !

15. Problems

16. Time shifts in the collision integral collision term last collision happened in the remote past resonances “feel the future collision” Advances and delays in time are quite common in classical and quantum mechanics Scattering of particles on hard spheres

17. Time shifts in classical damped oscillator Damped 1D-oscillator under the action of an external force Green’s function: carrier wave envelop function(signal) driving force: signal fading time approximate solution: signal delay phase shift

18. Quantum scattering in 3D partial wave Wigner time delay: Difference of flight times w. and wo. potential analogue of the signal delay quasi-particle and Noether currents different waves

19. KBE NLE 1st gradient expansion BME Resonance life time does not follow from BME since BME cannot be used for studying resonance decays. One cannot put Gin=0 in BME Conclusion KBE=NLE=BME+ phase-space shifts in the collision term Three forms of kinetic equations have different relaxation times! On large time scale >>1/G all froms are equivalent. Time shift in the collision integral for NLEcan be >0 (delay) or <0 (advance) Time delays/advances are quite common in classical and quantum mechanics, QFT and quantum kinetics. see [J Phys. G 40 (2013) 113101] for examples