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

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Boltzmann kinetic equation distribution of particles in the phase space binary collisions! drift term Between collisions “particles” move along characteristic determined by an external force F ext 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 increaseentropy for BKE is

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

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

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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 Dyson equation: Wick theorem Wigner transformation. Separation of slow and fast variables (Fourier trafo.) Gradient expansion Pathway to the kinetic equation 1 st gradient approximation Poisson brackets:

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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 Retarded Green’s function spectral density 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 [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902] +…

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Kinetic equation in the 1 st gradient approximation Drift operator: 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] for non-relat. part.

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Equilibrium relation: Three forms of the kinetic equation. Kadanoff-Baym equation Conservation: Noether currents and energy-momentum tensor (if a conservative Phi-derivable scheme is applied) Entropy: 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. (+ memory terms and derivative terms) [Ivanov, Knoll Voskresensky, NPA672 (00) 313; PAN66(03)1902] (Markovian part) H-theoremnot proven in general case, only for group velocity

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Three forms of the kinetic equation. Botermans-Malfliet Assume small deviation from the local equilibrium replace in the Poisson-bracket term Separation of particle drift and collision terms is possibletest-particle method 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] [Cassing, Juchem NPA672; NPA665, 385; Leupold NPA672,475] There exists a well defined hydrodynamic limit [Voskresensky, NPA849 (11) 120]

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

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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 can be positive (delay) near resonance and negative (advance) far from resonance forward-time in KE

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

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

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

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Problems

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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

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

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

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Conclusion KBE BME NLE 1 st gradient expansion Three forms of kinetic equations have different relaxation times! On large time scale >>1/ all froms are equivalent. KBE=NLE=BME+ phase-space shifts in the collision term Resonance life time does not follow from BME since BME cannot be used for studying resonance decays. One cannot put in =0 in BME Time shift in the collision integral for NLE can 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

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