Geometric Gyrokinetic Theory 几何回旋动力论
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Transcript of Geometric Gyrokinetic Theory 几何回旋动力论
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Geometric Gyrokinetic Theory几何回旋动力论
Hong Qin 秦宏Princeton Plasma Physics Laboratory, Princeton University
Workshop on ITER Simulation
May 15-19, 2006, Peking University, Beijing, China
http://www.pppl.gov/~hongqin/Gyrokinetics.php
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Acknowledgement
Thank Prof. Ronald C. Davidson and Dr. Janardhan Manickam for their continuous support.
Thank Drs. Peter J. Catto, Bruce I. Cohen, Andris Dimits, Alex Friedman, Gregory W. Hammet, W. Wei-li Lee, Lynda L. Lodestro, Thomas D. Rognlien, Philip B. Snyderfor, and William M. Tang for fruitful discussions.
US DOE contract AC02-76CH03073.
LLNL’s LDRD Project 04-SI-03, Kinetic Simulation of Boundary Plasma Turbulent Transport.
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Classical gyrokinetics: average out gyrophase
Magnetized plasmas fast gyromotion.
“Average-out" the fast gyromotion– Theoretically appealing– Algorithmically efficient
2
0
1[ Valsov Maxwell eqs ]
2d
pq
p- .ò
3
3
( ) 0
4 1( )
4 ( )
0
p
p
é ùæ ö¶ ¶ ´ ¶÷çê ú+ × + + × , , = ,÷ç ÷çê úè ø¶ ¶ ¶ë û
¶Ñ´ = , , + ,
¶
Ñ × = , , ,
Ñ × = .
òå
òå
Vlasov-Maxwell Equations
j j
j jj
j jj
e f tt c
e d p f tc c t
e d pf t
v Bv E x p
x p
B v x p E
E x p
B
, ?f f
c
v Bv
x p
¶ ´ ¶× ׶ ¶
Highly oscillatory characteristics
dependentq
Does not work
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Modern gyrokinetics: all about gyro-symmetry
Gyro-symmetry
Decouple gyromotion
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What is symmetry?
Coordinate dependent version:
Geometric version:
L dShg=
Lie derivative
Symmetry vector field
Poincare-Cartan-Einstein 1-form
210 0
2
Problem, what is ?
L d LL v
dtf
q q
q
æ ö¶ ¶ ÷ç= , = , = × + -÷ç ÷çè ø¶ ¶A v&
Symmetry groupS. Lie (1890s)
.Advantage: general, stronger, enables techniques to find q
Disadvantage: it takes longer to explain.
Symmetry is group
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What is Poincare-Cartan-Einstein 1-form?
2
2
( )2
( )
( 2 ),
, four vector potential, 1-form
1-form momentum
vA p d dt
A
p v
g f
f
é ùê ú= + = + × - + ,ê úë û
º - ,
º - / ,
A v x
A
p
( )
2
2
1
21
2
L v
v
f
f
= × + -
æ ö÷ç= + × - + ÷ç ÷çè ø
A v
A v v
dt´
On the 7D phase space
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What is the phase space?
1 2 2{( ) ( ) }xP x p x M p T M g p p mc* -= , | Î , Î , , = -
7D
spacetime inverse of metric
Phase space is a fiber
bundle P Mp : ¾® .
Hamilton's Eq. 0 .i dt g =
World-line
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Symmetry is invaraint
( )L d i i d dSh h hg g g= + =
Noether’s Theorem (1918)
Cartan’s formula
( )
is conversed.
d dS
S
g h t t
g h
× × = ×
× -
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What is gyrosymmetry?
1 1x y
y x
v vB x v B y v
hæ ö æ ö¶ ¶ ¶ ¶÷ç ÷ç÷ ÷ç= + + -ç÷ ÷ç ç÷ ÷ç÷¶ ¶ ¶ ¶ç è øè ø
Noether’s Theorem
Gyrophasecoordinate
Coordinate perturbation to find .
Q: How does the dynamics transform?
A: It transforms as an 1-form!
h
2 2
2x yv v
Bm
+=
hq
¶=
¶
Amatucci, Pop 11, 2097 (2004).
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Dynamics under Lie group of coordinate transformation
[ ]
1 1 2( )
2( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) .
G Z
G Z
Z g z g Z Z L Z O
Z i d Z d G Z O
g g g g e
g g g e
- * -é ùG = = = - +ë û= - - × +
0
10
( )
Continuous Lie group,
Vector field (Lie algbra),
,
g z Z g z
G dg d
G dg d
e
e
e
e
e=
-=
: = ,
= / |
- = / | .
a
Zg in Pullback
Cartan’s formula Insignificant
No need for Poisson bracket
Taylor expansion
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Lie perturbations
0 1 2
0 1
( )
0, 0,
Noncanonical coordinates:
Z u w q
g g g g
g gq q
= , , ,
= + + +...,
¶ ¶= ¹
¶ ¶
X
: ( ) 0 .g Z Z g Z such that g q® = ¶ / ¶ =
1
1
1 2
20 1
0 0
1 1 ( ) 0 1
2 2 ( ) 1
2( ) ( ) 0 2
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ,
( ) ( ) ( )
1( ) .
2
G Z
G Z
G Z G Z
Z Z Z O
Z Z
Z Z i d Z dS
Z Z L Z
L L Z dS
g g g e
g g
g g g
g g g
gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø
= + + ,
= ,
= - -
= -
+ - +
1 2 1 2, , , G G S S
Gyrocenter gauges
0 1 0 1
0 00 1 1 1 1 0
0 0 0 00
0 0 0 0
1 1 1 1
1 1 1 1
1 1
1 11
e e
r r e
r r
= + , = + ,
´ ´, , ,
æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø
æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷W ¶ W ¶è ø è ø
: : :
: :
: :
c c
E E B B
E E t B B t
E E B B
E E t B B t
E E E B B B
v B v BE E B B
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Perturbation techniques — quest of good coordinates
Peanuts by Charles Schulz. Reprint permitted by UFS, Inc.
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20 1 ( )Og eg g= + + ,
( )22 2 2
0 0002 2
Dw u wA u d d dtB
q fgæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø
+ += + + × + - + ,b D X
0 110 0
3
0 1 130 0 0
1 0
0
1( )
2 2
1( ) ( )
2
1( )
2 2
w w wu d
B B
ww d dwB B B
w wu dt
t B t
r r rg
r q r
f r r r r
é ùæ ö÷çê ú= Ñ × + + × ´ - Ñ × + + ×Ñ÷ç ÷çê úè øë ûé ù é ùê ú ê ú+ - × × + + × + + ×Ñê ú ê úë û ë ûé ùæ ö¶ ¶÷çê ú- + + × - ×Ñ × - + × .÷ç ÷çê úè ø¶ ¶ë û
ca b D a A X XB
a b A X c A X aB
D c aX E b
11 0 1( ) ( )
dgZ g Z G Z
d eee == , , | = , 11 ( ) 0 11( ) ( ) ( ) ( )G ZZ Z i d Z dS Zg gg= - + ,
under coordinate perturbationg
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01 1 0 1 1
0
1 11 1 1
0 0
31
01 1 30 0 0
1 0 1
0
1( )
2 2
( )
1( )
2
( )
u
w
w wZ G S u
B
S Sw wd du G
B u B w
Sw wdw G
B B B
wd
B
q
g r r
r
rq
r q
é æ ö÷çê= ´ - +Ñ + Ñ × + + × ´Ñ÷ç ÷çê è øëù éé ù¶ ¶ú êê ú- Ñ × + + × + × + + + +ú êê ú¶ ¶ë ûû ë
ù é ¶ú ê+ + × + - + - × ×Ñú ê ¶û ëùú+ + × + - × +úû
X
X
X
cG B b a b B
D a A X X G b
A X a a bB
A X c E G 11 1 1
0
0
( )
1
2 2
u w
SuG wG
t
w wu dt
t B t
f r
r r r
é ¶ê + + - +ê ¶ë
ùæ ö¶ ¶÷ç ú- × + ×Ñ × + + × .÷ç ÷ç úè ø¶ ¶ û
X
D c aE b
1 0g q¶ / ¶ = .
+
1 1, G S
FreedoomGyro-center gauges
under coordinate perturbationg
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( )
( )
0 1
2 2 2 2
0 0 0
0
2
1 10
( ) ( )
2 2
( )2
Z Z Z
w u w DA u d d dt
B
wZ d H dt
B
g g g
g q f
g
æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø
= + ,
+ += + + × + - + ,
=- × - ,
b D X
R X
( )
2 2
01 0 130 0
2 2
0 0 020 0
0
2 4
4 2
w w uH
B B
w wR
B B
Rt
y= × + ×Ñ´ +Ñ
- Ñ × - :Ñ - ,
¶º Ñ × , º - × ,
¶
E b bB
E bb E
cR c a a
°
011 1 1
0
2
0
( ) ( ) ( )
1
2
wB
d
E bX A X Xc A
p
y f r r r
a a q a a ap
^´º + - × + - + ,×
º , º - .ò
Gyrocenterdynamics
in gyrocenter coordinate, / =0g g q¶ ¶
0
( )
( )
( ) ( )
( )
Gyro-gauge invariant
R R X
X
R R X t
t
d
q q d
¢
¢
¾® +Ñ ,
¾® + .
= , , = , ,
Ñ = - ¶/ ¶ ,Ñ .
R X
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Gyrocenter dynamics
0i dt g= .
† †
† †
† †
†
0 00 0 2
0
1 2 0 00
2
0
( )2
2
1
2
02
du
dt
du
dtBd d u
B Rdt dt B
B
d w
dt B
m
q
y ym
mm
é ùê úë û
´= + ×Ñ´ - ,
× ××
= ,×
×Ñ= + × - + + ×Ñ´
¶+ + - Ñ × - :Ñ
¶
= , º ,
X B b Eb b
b B b B
B E
B bEX
R b b
E bb E
( )†0
2†
0 0 1 2
2†
0 0 1 2
†0
†† † † †
2
2
u
DB u
t t
DB
u
t
m y y
ff m y y
f
é ùê úê úê úê úê úë û
º Ñ´ + + ,
¶ ¶º - Ñ + + + - - .
¶ ¶
º + + + + ,
º + + ,
¶=Ñ´ , =- Ñ - .
¶
B A b D
b DE E
A A b D
AB A E
Banos drift
, driftB´ ÑE B
drift by spacetime
inhomogeneties of 0E
Curvature drift
& curvature driftDÑ D
Effective potentials
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Gyrokinetic equations
( )0 0 6
.
¶= , £ £ .
¶
=
j
j
dZ Fj
dt Z
F F
0dZ
dtq
æ ö¶ ÷ç = ,÷ç ÷çè ø¶
0¶ ¶
+ ×Ñ + = ,¶ ¶
F Fd duF
t dt dt uX
X
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Pullback of distribution function
[ ] ( )
pullback from ( ) :
( ) ( ) ( )
F Z
f z g F Z F g z*= = .
Particle distribution
Don't know ( ).f z
Gyrocenterdistribution
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Physics of pullback transformation — Spitzer paradox
3 30
2 3
2 3
( ) ( )
( ) ( ) ( )
1
y y y
y
yy
j v g F Z dv v F x v dv
v F x F x O dv
F pv dv b
x B B
r
r e
* é ù= = + ,ë û
é ù= + ×Ñ +ë ûæ ö¶ Ñ ÷ç= = ´ .÷ç ÷çè ø¶
ò ò
ò
ò
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Finally, gyrokinetic equations
0¶ ¶
+ ×Ñ + = ,¶ ¶
F Fd duF
t dt dt uX
X
( )
( )
0
0
22 31 1 2
( )
22 31 1 2
( )
14 ( ) ( ) ( ) ( )
2
14 ( ) ( ) ( ) ( )
2
ss Z g z
ss Z g z
q F Z G F Z G F Z G F Z d
q F Z G F Z G F Z G F Z d
f p
p
®
®
é ùê úÑ =- + ×Ñ + ×Ñ + ×Ñ .ê úë û
é ùê úÑ = + ×Ñ + ×Ñ + ×Ñ ,ê úë û
òå
òå
v
A v v
Gyrokinetic Maxwell’sequations
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Gyrokinetic Poisson equation
( ) ( )
( ) ( )
0 1
0
2
0
1 1 2 220
( ) 4
( ) 2
1( ) ( ) ( )
ff
f
f p
p r
é ùê úê úê úê úë û
^
Ñ =- + + ,
º Ñ , , ,
é ùº + Ñ + ×Ñ ,ê úë û
å
ò
P
ss
q N N N
N wdwdu I F w u
N n VB
x
x x
x e e e e x D x b D
( )
( )( ) ( )
1
2
1 2 221 0
2 2
12 220 0 0
0
2( ) ( ) ( )
2
2( ) ( )
2 2
i
ii
i j
i ji ji j
iN M
i
i jM
i j
f f
f
¥^
-=
¥^ ^
+ -, =+ ¹
æ öÑ ÷ç ÷çº - ÷ç ÷ç ÷W! è ø
é ùæ ö æ ö+ Ñ Ñê ú÷ ÷ç ç÷ ÷ç ç+ ,ê ú÷ ÷ç ç÷ ÷ç ç÷ ÷ê úW W!! è ø è øê úë û
å
å
x x x
x x
( ) 4 412
0
1n O
Bf ræ ö÷ç ÷ç ÷÷ç^ ^ ^è øÑ × Ñ + Ñ
Polarization density2 2 1r ^Ñ =
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Physics of pullback transformation — polarization density
1Xx X Gr= + + ,
01 ( ) 22 X Z g z
e ep G dq f
p ® ^
-= = Ñ .
Wò
1 1
1 XG S
B= ´ Ñ .
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Conclusions
Gyrokinetic theory is about gyro-symmetry.
Decouple the gyro-phase, not "averaging out".
Pullback transformation is indispensable.
http://www.pppl.gov/~hongqin/Gyrokinetics.php
Decouple. Not average.
PullbackPullbackPullback
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Gyrokinetic Poisson equation
( ) ( ) ( )0 1N N N= + ,x x x
20 0( ) 4 s
s
q Nf pÑ =- ,åx
0 1
21 1( ) 4 s
s
q N N Nfff pé ùê úê úê úê úë û
Ñ =- + + .åx
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1-form, 2-form, 3-form and all that …
Vector ii
v vx
¶=
¶
1-form iip p dx= 2-form j ii
j
pdp dx dx
x
¶=
¶
d
Exterior
derivative
X
iip v Î ¡
Vector ii
v vx
¶=
¶
X
*j iijj i
ppv dx T M
x x
æ ö¶¶ ÷ç ÷- Îç ÷ç ÷ç¶ ¶è ø
Inner product
Pullback
Push forward
vL g
Vector field
Any tensor
Contra-variant base
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Vector field and flow
®: ( )g z Z t
zZ
( )G z TgVector field Î
Lie symmetry group
Phase Space
Lie algebra is the Infinitesimal generator of Lie group
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Lie perturbations
0 1 2
0 1
( )
0, 0,
Noncanonical coordinates:
Z u w q
g g g g
g gq q
= , , ,
= + + +...,
¶ ¶= ¹
¶ ¶
X
: ( ) 0 .g Z Z g Z such that g q® = ¶ / ¶ =
1
1
1 2
20 1
0 0
1 1 ( ) 0 1
2 2 ( ) 1
2( ) ( ) 0 2
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ,
( ) ( ) ( )
1( ) .
2
G Z
G Z
G Z G Z
Z Z Z O
Z Z
Z Z i d Z dS
Z Z L Z
L L Z dS
g g g e
g g
g g g
g g g
gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø
= + + ,
= ,
= - -
= -
+ - +
1 2 1 2, , , G G S S
Gyrocenter gauges
0 1 0 1
0 00 1 1 1 1 0
0 0 0 00
0 0 0 0
1 1 1 1
1 1 1 1
1 1
1 11
c c
E E B B
E E t B B t
E E B B
E E t B B t
e e
r r e
r r
= + , = + ,
´ ´, , ,
æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø
æ ö æ öÑ ¶ Ñ ¶÷ ÷ç ç÷ ÷, , ,ç ç÷ ÷ç ç÷ ÷ç çW ¶ W ¶è ø è ø
E E E B B B
v B v BE E B B: : :
: :
: :
Time dependentPedestal dynamics
Large Er shearingOrbit squeezing
Micro turbulence ELM
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Leading order gyrocenter coordinates
[ ][ ]
[ ]0
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) 1
( ) ( ) ( )
( ) ( ) ( )( )
( )
x x
x x x
x x
x x
xx
u y y y y y
w y y y y y
y y
y y y
y y yy
B yr
, º - × ,
, , º - ´ ´ ,
, × , = ,
, º ´ , ,
´ -, º .
v b v D b b
v c v v D b b
c v c v
a v b c v
b v Dv
[ ]
0
1
0 0 02
00
( ) ( )
( ) ( ) ( ) sin ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
( )( )
x x x x
g z t Z u w t
u u w w t t
y y u y y w y y
y y yy y
B yB y
q
r q
: = , , = , , , ,
º + , , º , , º , , º - × , º ,
º + , + , , .
´º , º ,
x v X
x X X v X v X v c X e X
v D v b v c v
E B BD b
a
( ) ( ) ( )u w= + + .v D X b X c X
Lab phase space coords
Time-dependentBackground EXB drift
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( )
[ ]
21
01 3 2 20 0 0
1 1
0
2
01 1 120 0 0
20 1
101 20
0 111
( )2
( )
( )2
( )2
1( )
u
w
S w wu w
u B B B
S
B
w wu wG S
B B B
B S wG
w B
B SG
w w wq
r
r
rq
r
¶=- + + Ñ × ´ - Ñ × ´×Ñ
¶
Ñ + ++ ´
= × + ×Ñ × - ×Ñ × - ×Ñ + + ,×Ñ
¶= - × + + ,×Ñ ×
¶
¶=- - + .×
¶
XG b a b b D a baa B
A Xb
c b a b b D a b A Xa B
b Xa B c A
Xa A
±
( ) °
² ±
201 1 1
1 0 0 0 030 0
2 2
02 2 20 0 0 0
3 2
0 1 02 20 0 0 0 0
2
( )2
2 2
S S S wu S E B B
t B u B
wu w w u wu
B B B B
wu w w w uw
B B B t B B t
q
y
éêêê^ êêë
æ ö´¶ ¶ ¶÷ç ÷+ + ×Ñ + + = × ×Ñç ÷ç ÷ç¶ ¶ ¶è ø
ùú+ Ñ × ´ - Ñ × ´ - : - ×Ñ ×Ñúû
¶ ¶+ ×Ñ × + × + × + - Ñ : + × .×Ñ
¶ ¶
E bb E aa
a b b D a b ca b a bB
D bb D a b a E aa aa B
P
1 1 and G S
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( )
2 2
† † † †2 0 1 1 1 1 1 1
†1 1 1 1
0 0
† 11 1 1
1 1
2 2
w
dt
u w
wG G
B B
t
q
g y
y
f y
æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç ç^ è ø è ø
= ,
é ùº × ´ ´ - + × ´ + ×ê úë û
º + + ,
¶º - Ñ - - Ñ .
¶
x
E G B b b c G B E G
a cG G
AE
2 1 2 1: ( ) ( ) g g Z Z g g Z® = o
†22 2 1 1
0 0
†2 2 1 1
†0 22 1 1
†0 22 1 1
1 1
2
1
21
21
2
u
w
SS
u B B
G S
B SG
wB S
Gw wq
q
æ ö÷ç ÷ç ÷÷çè ø
æ ö÷ç ÷ç ÷÷çè ø
æ ö÷ç ÷ç ÷÷çè ø
æ ö÷ç ÷ç ÷÷çè ø
¶=- + Ñ ´ + ´ ´ ,
¶
= ×Ñ + × ´ ,
¶= + × ´ ,
¶¶
=- + × ´ ,¶
XG b b G B b
b b G B
c G B
a G B
²02 2 21 0 0 2
0
S S Su S E B
t B uy
q
æ ö´¶ ¶ ¶÷ç ÷+ + ×Ñ + + = .ç ÷ç ÷ç¶ ¶ ¶è ø
E bb P
2 2 2, ,Second order, and g G S
1
1 2
2 2 ( ) 1
2( ) ( ) 0
( ) ( ) ( )
1( ) .
2
G Z
G Z G Z
Z Z L Z
L L Z
g g g
gæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø
= -
+ -
2 0gq
¶=
¶
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Curvature drfit
( )
† 22 2
†
2 2 2
2 2
1 ( )
( )
( )
B Bu u b b b bu ub u u O
b B B ub b B B
b b bub u b u O
B Bb
ub u OB
e
e
e
æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø
^
æ ö+ Ñ´ Ñ´ ×Ñ´ ÷ç= = + - +÷ç ÷çè ø× + ×Ñ´
Ñ´ ×Ñ´= + - +
Ñ´= + + .
0D case=
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Pullback of distribution function
3( ) ( ) ( ) [ ( ) ( )] [ 1 ]
( ). ( ).
,
Problem: don't know Solution: pullback from
j x f z v d j x n x x v
f z F Z
= , = - , º - ,ò v j v
[ ] ( )( ) ( ) ( )f z g F Z F g z*= = .
( )
( )0
0 1 2 0 2 1
2
1 10
32
13
2
1 2( )
( ) ( ) ( ) ( ( ))
1( ) ( ) ( )
2
( ) ( )
( ) ( )( )1
( ) ( )2 Z g z
f z g F Z g g g F Z g F g g Z
F Z G F Z G F Zg
G F Z O
F Z G F ZO
G F Z G F Z
e
e
* * * * *
*
®
é ù= = = ë û
é ùê ú+ ×Ñ + ×Ñê ú=ê úê ú+ ×Ñ +ë û
é ù+ ×Ñê úê ú= + .ê ú+ ×Ñ + ×Ñê úë û
o o
Needed for Maxwell’q eqs.
Definition of pullback2 1 0
0
2 1
g g g g
g z Z
g g Z Z
= ,
: ,
: ,
o o
a
o a
0Pullback of
nonperturbative
g