Between Green's Functions and Transport Equations

1
Between Green's Functions and Transport Equations
PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS
III Kiel August 22 25, 2005

• B. Velický, Charles University and Acad. Sci. of
  CR, Praha
• A. Kalvová, Acad. Sci. of CR, Praha
• V. Špicka, Acad. Sci. of CR, Praha

2
Between Green's Functions and Transport
Equations Reconstruction Theorems and
the Role of Initial Conditions
PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS
III Kiel August 22 25, 2005

• B. Velický, Charles University and Acad. Sci. of
  CR, Praha
• A. Kalvová, Acad. Sci. of CR, Praha
• V. Špicka, Acad. Sci. of CR, Praha

3
Between Green's Functions and Transport
Equations Correlated Initial Condition for
Restart Process
Topical Problems in Statistical Physics TU
Chemnitz, November 30, 2005

• Kalvová, Acad. Sci. of CR, Praha
• B. Velický, Charles University and Acad. Sci. of
  CR, Praha
• V. Špicka, Acad. Sci. of CR, Praha

4
Between Green's Functions and Transport
Equations Correlated Initial Condition for
Restart Process Time Partitioning for NGF
Topical Problems in Statistical Physics TU
Chemnitz, November 30, 2005

• Kalvová, Acad. Sci. of CR, Praha
• B. Velický, Charles University and Acad. Sci. of
  CR, Praha
• V. Špicka, Acad. Sci. of CR, Praha

5

• Prologue

6
(Non-linear) quantum transport
non-equilibrium problem
many-body Hamiltonian many-body density
matrix additive operator
Many-body system Initial state External
disturbance

7
(Non-linear) quantum transport
non-equilibrium problem
Many-body system Initial state External
disturbance Response
many-body Hamiltonian many-body density
matrix additive operator one-particle density
matrix

8
(Non-linear) quantum transport
non-equilibrium problem
Many-body system Initial state External
disturbance Response
many-body Hamiltonian many-body density
matrix additive operator one-particle density
matrix

Quantum Transport Equation
generalized collision term
9
(Non-linear) quantum transport
non-equilibrium problem
Many-body system Initial state External
disturbance Response
many-body Hamiltonian many-body density
matrix additive operator one-particle density
matrix

Quantum Transport Equation
interaction term
10
(Non-linear) quantum transport
non-equilibrium problem
Many-body system Initial state External
disturbance Response
many-body Hamiltonian many-body density
matrix additive operator one-particle density
matrix

Quantum Transport Equation
interaction term
11
This talk orthodox study of quantum transport
using NGF
TWO PATHS
INDIRECT
use NGF to construct a Quantum Transport
Equation
12
Lecture on NGF
This talk orthodox study of quantum transport
using NGF
DIRECT
TWO PATHS
use a NGF solver
INDIRECT
use NGF to construct a Quantum Transport
Equation
13
Lecture on NGFcontinuation
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Lecture on NGFcontinuation
15
This talk orthodox study of quantum transport
using NGF
DIRECT
TWO PATHS
use a NGF solver
INDIRECT
use NGF to construct a Quantum Transport
Equation
16
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
17
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
? Specific physical approximation --
self-consistent form
18
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
? Specific physical approximation --
self-consistent form
? Elimination of by an Ansatz widely
used KBA (for steady transport), GKBA
(transients, optics)
19
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
? Specific physical approximation --
self-consistent form
? Elimination of by an Ansatz GKBA
Lipavsky, Spicka, Velicky, Vinogradov,
Horing Haug Frankfurt team, Rostock
school, Jauho,
20
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
? Specific physical approximation --
self-consistent form
? Elimination of by an Ansatz GKBA
Resulting Quantum Transport Equation
21
Standard approach based on GKBA
? Real time NGF our choice
? GKBE
? Specific physical approximation --
self-consistent form
? Elimination of by an Ansatz GKBA
Resulting Quantum Transport Equation

• Famous examples
• Levinson eq.
• (hot electrons)
• Optical quantum
• Bloch eq.
• (optical transients)

22

• Act I
• reconstruction

23
Exact formulation -- Reconstruction Problem
GENERAL QUESTION conditions under which a
many-body interacting system can be described in
terms of its single-time single-particle
characteristics
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Exact formulation -- Reconstruction Problem
GENERAL QUESTION conditions under which a
many-body interacting system can be described in
terms of its single-time single-particle
characteristics
Reminiscences BBGKY, Hohenberg-Kohn Theorem
25
Exact formulation -- Reconstruction Problem
GENERAL QUESTION conditions under which a
many-body interacting system can be described in
terms of its single-time single-particle
characteristics
Reminiscences BBGKY, Hohenberg-Kohn Theorem
Here time evolution of the system
26
Exact formulation -- Reconstruction Problem
New look on the NGF procedure
Any Ansatz is but an approximate
solution Does an answer exist, exact at least
in principle?
27
Reconstruction Problem Historical Overview
INVERSION SCHEMES
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Reconstruction Problem Historical Overview
INVERSION SCHEMES
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Parallels
G E N E R A L S C H E M E
Postulate/Conjecture typical systems are
controlled by a hierarchy of times separating
the initial, kinetic, and hydrodynamic stages. A
closed transport equation holds for
LABEL Bogolyubov
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Parallels
G E N E R A L S C H E M E
Postulate/Conjecture typical systems are
controlled by a hierarchy of times separating
the initial, kinetic, and hydrodynamic stages. A
closed transport equation holds for
LABEL Bogolyubov
31
Parallels
G E N E R A L S C H E M E
Runge Gross Theorem Let be local.
Then, for a fixed initial state , the
functional relation is bijective and can
be inverted. NOTES U must be sufficiently
smooth. no enters
the theorem. This is an existence
theorem, systematic implementation
based on the use of the closed
time path generating functional.
LABEL TDDFT
32
Parallels
G E N E R A L S C H E M E
Runge Gross Theorem Let be local.
Then, for a fixed initial state , the
functional relation is bijective and can
be inverted. NOTES U must be sufficiently
smooth. no enters
the theorem. This is an existence
theorem, systematic implementation
based on the use of the closed
time path generating functional.
LABEL TDDFT
33
Parallels
G E N E R A L S C H E M E
Closed Time Contour Generating Functional
(Schwinger) Used to invert the relation
EXAMPLES OF USE Fukuda et al. Inversion
technique based on Legendre
transformation ? Quantum kinetic
eq. Leuwen et al. TDDFT context
LABEL Schwinger
34
Parallels
G E N E R A L S C H E M E
Closed Time Contour Generating Functional
(Schwinger) Used to invert the relation
EXAMPLES OF USE Fukuda et al. Inversion
technique based on Legendre
transformation ? Quantum kinetic
eq. Leuwen et al. TDDFT context
LABEL Schwinger
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Parallels Lessons for the Reconstruction Problem
G E N E R A L S C H E M E

• Bogolyubov importance of the time hierarchy
• REQUIREMENT Characteristic times
  should
• emerge in a constructive manner during the
  
• reconstruction procedure.
• TDDFT analogue of the Runge - Gross Theorem
  
• REQUIREMENT Consider a general non-local
  disturbance
• U in order to obtain the full 1-DM ? as
  its dual.
• Schwinger explicit reconstruction procedure
• REQUIREMENT A general operational method
  for the
• reconstruction (rather than inversion in
  the narrow
• sense). Its success in a particular case
  becomes the
• proof of the Reconstruction theorem at the
  same time.

LABEL NGF Reconstruction Theorem
36
Reconstruction Problem Summary
INVERSION SCHEMES
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Reconstruction Problem Summary
INVERSION SCHEMES
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Reconstruction theorem Reconstruction equations
Keldysh IC simple initial state permits to
concentrate on the other issues
DYSON EQUATIONS
Two well known reconstruction equations easily
follow
RECONSTRUCTION EQUATIONS
LSV, Vinogradov application!
39
Reconstruction theorem Reconstruction equations
Keldysh IC simple initial state permits to
concentrate on the other issues
DYSON EQUATIONS
Two well known reconstruction equations easily
follow
RECONSTRUCTION EQUATIONS

• Source terms the Ansatz
• For tt' tautology

? input
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Reconstruction theorem Coupled equations
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Reconstruction theorem operational description
NGF RECONSTRUCTION THEOREM determination of the
full NGF restructured as a DUAL PROCESS
quantum transport equation
? ?
reconstruction equations Dyson eq.
42
Reconstruction theorem formal statement
NGF RECONSTRUCTION THEOREM determination of the
full NGF restructured as a DUAL PROCESS
quantum transport equation
? ?
reconstruction equations Dyson eq.
"THEOREM" The one-particle density matrix and
the full NGF of a process are in a bijective
relationship,
43

• Act II
• reconstruction
• and initial conditions
• NGF view

44
General initial state
For an arbitrary initial state at
start from the NGF Problem of determination of
G extensively studied Fujita ? Hall ?
Danielewicz ? ? Wagner ? MorozovRöpke
Klimontovich ? Kremp ? ? BonitzSemkat Take
over the relevant result for The
self-energy

depends on the initial state (initial
correlations)
has
singular components
45
General initial state
For an arbitrary initial state at
start from the NGF Problem of determination of
G extensively studied Fujita ? Hall ?
Danielewicz ? ? Wagner ? MorozovRöpke
Klimontovich ? Kremp ? ? BonitzSemkat Take
over the relevant result for The
self-energy

depends on the initial state (initial
correlations)
has
singular components
Morawetz
46
General initial state Structure of
Structure of
47
General initial state Structure of
Structure of
Danielewicz notation
48
General initial state Structure of
Structure of
Danielewicz notation
49
General initial state A try at the reconstruction
50
General initial state A try at the reconstruction
To progress further, narrow down the selection of
the initial states
51
Initial state for restart process
To progress further, narrow down the selection of
the initial states
Special situation
Process, whose initial state coincides
with intermediate state of a host process
(running)
Aim to establish relationship between NGF of the
host and restart process
52
Restart at an intermediate time
Let the initial time be , the initial
state . In the host NGF the Heisenberg
operators are
53
Restart at an intermediate time
We may choose any later time as the
new initial time. For times
the resulting restart GF should be consistent.
Indeed, with we have
54
Restart at an intermediate time
We may choose any later time as the
new initial time. For times
the resulting GF should be consistent. Indeed,
with we have
whole family of initial states for varying t0
55
Restart at an intermediate time
NGF is invariant with respect to the initial
time, the self-energies must be related in a
specific way for Important difference
causal structure of the Dyson equation
develops singular parts at as a condensed
information about the past
56
Restart at an intermediate time
NGF is invariant with respect to the initial
time, the self-energies must be related in a
specific way for Important difference
causal structure of the Dyson equation
develops singular parts at as a condensed
information about the past
57
Restart at an intermediate time
NGF is invariant with respect to the initial
time, the self-energies must be related in a
specific way for Important difference
causal structure of the Dyson equation
develops singular parts at as a condensed
information about the past
58
Restart at an intermediate time
NGF is invariant with respect to the initial
time, the self-energies must be related in a
specific way for Important difference Objective
and subjective components of the initial
correlations The zone of initial correlations of
wanders with our choice of the initial
time if we do not know about the past, it looks
to us like real IC.
causal structure of the Dyson equation
develops singular parts at as a condensed
information about the past
59

• Intermezzo
• Time-partitioning

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Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation
61
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past future notion in reconstruction equation
62
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past future notion in reconstruction equation
RECONSTRUCTION EQ.
63
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

-past future notion in reconstruction equation
RECONSTRUCTION EQ.
64
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

-past future notion in reconstruction equation
RECONSTRUCTION EQ.
future
65
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation for Glt
66
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation for Glt
- past - future notion in corrected semigroup
rule GR
67
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation for Glt
- past - future notion in corrected semigroup
rule GR
CORR. SEMIGR. RULE
68
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation for Glt
- past - future notion in corrected semigroup
rule GR
69
Time-partitioning general method
Special position of the (instant-restart) time t0

• Separates the whole time
• domain into the past and the
  future

- past - future notion in reconstruction
equation for Glt
- past - future notion in corrected semigroup
rule GR
- past - future notion in restart NGF
unified description time-partitioning formalism
70
Partitioning in time formal tools
Past and Future with respect to the initial
(restart) time
71
Partitioning in time formal tools
Past and Future with respect to the initial
(restart) time
Projection operators
72
Partitioning in time formal tools
Past and Future with respect to the initial
(restart) time
Projection operators
Double time quantity X
four quadrants of the two-time plane
73
Partitioning in time for propagators
1. Dyson eq.
2. Retarded quantity
only for
3. Diagonal blocks of
74
Partitioning in time for propagators
continuation
75
Partitioning in time for propagators
continuation
76
Partitioning in time for propagators
continuation
77
Partitioning in time for propagators
continuation
78
Partitioning in time for propagators
continuation
time-local factorization
vertex correction universal form (gauge
invariance) link past-future non-local in time
79
Partitioning in time for propagators
continuation
Corrected semigroup rule
time-local factorization
vertex correction universal form (gauge
invariance) link past-future non-local in time
80
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
(diagonal) past blocks only
81
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
82
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
diagonals of GFs
83
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
off-diagonals of selfenergies
84
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
85
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
86
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
87
Partitioning in time for corr. function
Question to find four blocks of
2. Propagators
by partitioning expressions
88
Partitioning in time for corr. function
Question to find four blocks of
Exception!!! Future-future diagonal
2. Propagators
by partitioning expressions
89
Partitioning in time restart corr. function
HOST PROCESS
RESTART PROCESS
90
Partitioning in time restart corr. function
HOST PROCESS
RESTART PROCESS
91
Partitioning in time restart corr. function
HOST PROCESS
RESTART PROCESS
future
memory of the past folded down into the future by
partitioning
92
Partitioning in time restart corr. function
HOST PROCESS
RESTART PROCESS
future
memory of the past folded down into the future by
partitioning
93
Partitioning in time initial condition
Singular time variable fixed at restart time
94
Partitioning in time initial condition
95
Partitioning in time initial condition
96
Partitioning in time initial condition
97
Partitioning in time initial condition
98
Partitioning in time initial condition
99
Partitioning in time initial condition
omited initial condition,
100
Partitioning in time initial condition
with uncorrelated initial condition,
101
Partitioning in time initial condition
with uncorrelated initial condition,
102
Partitioning in time initial condition
103
Restart correlation function initial conditions
continuous time variable t gt t0
singular time variable fixed at t t0
104
Restart correlation function initial conditions
uncorrelated initial condition ... KELDYSH
singular time variable fixed at t t0
105
Restart correlation function initial conditions
correlated initial condition ... DANIELEWICZ
106
Restart correlation function initial conditions
host continuous self-energy ... KELDYSH initial
correlations correction MOROZOV RÖPKE
107

• Act III
• applications
• restarted switch-on processes
• pump and probe signals
• ....

108
NEXT TIME
109

• Conclusions
• time partitioning as a novel general technique
  for treating problems, which involve past and
  future with respect to a selected time
• semi-group property as a basic property of NGF
  dynamics
• full self-energy for a restart process including
  all singular terms expressed in terms of the host
  process GF and self-energies
• result consistent with the previous work
  (Danielewicz etc.)
• explicit expressions for host switch-on states
  (from KB -- Danielewicz trajectory to Keldysh
  with t0 ? - ?
• ....

110

• Conclusions
• time partitioning as a novel general technique
  for treating problems, which involve past and
  future with respect to a selected time
• semi-group property as a basic property of NGF
  dynamics
• full self-energy for a restart process including
  all singular terms expressed in terms of the host
  process GF and self-energies
• result consistent with the previous work
  (Danielewicz etc.)
• explicit expressions for host switch-on states
  (from KB -- Danielewicz trajectory to Keldysh
  with t0 ? - ?
• ....

111
THE END