Transcript ppt - Frankfurt Institute for Advanced Studies

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)
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.
Conservation:
Entropy increase
entropy for BKE is
local in time and space!
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
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
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?
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
Poisson brackets:
1st gradient approximation
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
Weight factor is usually dropped
It can be hidden in the collision integral
phase space distribution
as in Boltzmann KE
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]
Three forms of the kinetic equation. Kadanoff-Baym equation
group velocity
Equilibrium relation:
Conservation: Noether currents and energy-momentum tensor
(if a conservative Phi-derivable scheme is applied)
(+ memory terms and derivative terms)
Entropy: (Markovian part)
H-theorem
not proven in general case, only for
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]
Three forms of the kinetic equation. Botermans-Malfliet
Assume small deviation from the local equilibrium
replace in the Poisson-bracket term
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]
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)
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
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]
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 SR is determined
by distribution of heavy particles,
Light resonance production by heavyparticles is determined by the equilibrium production rate.
collision integral:
KBE:
BME:
NLE:
different
answers!!
where a is the solution of equation
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:
NLE
However, the BME does not hold for Gin=0,
since its derivation is based on the equation Gin=G f !
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 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
Quantum scattering in 3D
partial wave
Wigner time delay:
analogue of the signal delay
quasi-particle and Noether currents
different waves
Difference of flight times
w. and wo. potential
Conclusion
KBE
1st
gradient
expansion
NLE
KBE=NLE=BME+ phase-space shifts
in the collision term
BME
Three forms of kinetic equations have different relaxation times!
On large time scale >>1/G all froms are equivalent.
Resonance life time
does not follow from BME since BME cannot be used for
studying resonance decays. One cannot put Gin=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