three forms of quantum kinetic equations evgeni kolomeitsev matej bel university, banska bystrica...

Three Forms of Quantum Kinetic Equations

Evgeni Kolomeitsev

Matej Bel University, Banska Bystrica

Ivanov, Voskresensky, Phys. Atom. Nucl. 72 (2009) 1168Kolomeitsev, Voskresensky J Phys. G 40 (2013) 113101 (topical review)

Based on:

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 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

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

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

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

1st gradient approximation

Poisson brackets:

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]

+…

Kinetic equation in the 1st 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:


for non-relat. part.

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)


(Markovian part)

H-theorem not proven in general case, only for

group velocity

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 possible test-particle method

group velocity

Conservation: effective current

Entropy: (Markovian part)

For Markovian part the H-theorem is proven


[Cassing, Juchem NPA672; NPA665, 385; Leupold NPA672,475]

There exists a well defined hydrodynamic limit [Voskresensky, NPA849 (11) 120]

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 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:

shifts in time-coordinate and energy-momentum spaces:

(Markovian part)

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 resonanceand negative (advance) far from resonance

forward-time in KE

Spectral density normalizations

Ergodicitydensity 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

Examples of solutions of kinetic equationsConsider 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

differentanswers!!

Three solutions coincide only for . However this condition

may hold only in very specific situations. For Wigner resonances it holds only for

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:

However, the BME does not hold for in=0, since its derivation is based on the equation in= f !

NLE

Problems

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

Time shifts in classical damped oscillatorDamped 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

quasi-particle and Noether currents different waves

Quantum scattering in 3D

Wigner time delay:partial wave

analogue of the signal delayDifference of flight times w. and wo. potential

Conclusion

KBE

BME

NLE1st gradientexpansion

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 forstudying resonance decays. One cannot put in=0 in BME

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

three forms of quantum kinetic equations evgeni kolomeitsev matej bel university, banska bystrica...

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