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Beam-plasma interaction in randomly inhomogeneous - Tel

IntégréTéléchargement
Beam-plasma interaction in randomly inhomogeneous
solar wind
Andrii Voshchepynets
To cite this version:
Andrii Voshchepynets. Beam-plasma interaction in randomly inhomogeneous solar wind. Solar and Stellar Astrophysics [astro-ph.SR]. Université d’Orléans, 2015. English. <NNT :
2015ORLE2035>. <tel-01341677>
HAL Id: tel-01341677
https://tel.archives-ouvertes.fr/tel-01341677
Submitted on 4 Jul 2016
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+&(% ',,%)&( (!"$" ".",($ (!" 8+#', /"2('9*)&,(+%& +& ":)'(+%& ;6<= $!%)2/ -" #"12',"/
-0 $% ,'22"/ #"$%&'&," *)&,(+%& R(ω − kv, ∆ω)>
?;6<@A
4!"#" ∆ω +$ (!" 2+*"(+3" %* (!" 4'B" +& (!" #"$%&'&,"6 C%(!D R(vk − ωk , ∆ω) '&/
∆ω D /"1"&/ %& (!" '11#%1#+'(" 3",!'&+$3 (!'( 1#%B+/"$ #"$%&'&," -#%'/"&+&56 E%#
+&$('&,"D #","&(20 (!+$ *#'3"4%#7 4'$ '112+"/ (% ('7" +&(% ',,%)&( ".",($ %* $,'(("#+&5
F'&53)+# 4'B"$ +& G),()'(+&5 12'$3'$ %& (!" 4'B"91'#(+,2" +&("#',(+%& HC+'& "( '26D
;IJ=K6
πδ(vk − ωk ) = R(vk − ωk , ∆ω),
!"!#
$%&'(')*+, +- ).% %&%/)0+, 1%'2
!" $0$("3 %* ":)'(+%&$ ;6<= '&/ ;6<L ,'& -" )$"/ (% /"$,#+-" (!" +&("#',(+%& %* (!"
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P
&+"&( (% 3'7" $2+5!( 3%/+N,'(+%& +& (!" ":)'(+%&$6 E+#$(D (!" $)3D k +& (!" ":)'(+%&
R
P
;6<= ,'& -" #"12',"/ -0 (!" +&("5#'(+%& %B"# k> k = (1/π)
dk 6 O",%&/D %&" ,'&
k>0
2
)$" 4'B" $1",(#'2 "&"#50 /"&$+(0D Ws (k)D +&$("'/ %* ("#3 Ek (t) 6 E%# 1'#(+,)2'# ,'$"
2
%* F'&53)+# 4'B"$ +( ,'& -" 4#+(("& '$ *%22%4$> Ws (k) = Ek (t) /(8π)6 P( +$ 4%#(!
&%(+&5 (!'( Ws (k)dk +$ (!" 4'B" "&"#50 /"&$+(0 ,%&,"&(#'("/ +& (!" 1'#( %* (!" $1",9
(#)3 *#%3 k (% k + dk6 !+#/D +&("5#'(+%& %B"# 4'B" B",(%#D kD ,'& -" #"12',"/ -0 (!"
+&("5#'(+%& %B"# 4'B" 1!'$" B"2%,+(0D V D -0 3'7+&5 )$" %* #"2'(+%& k = ωk /V D '&/ (!)$
dk ∼ (ωk /V 2 )dV 6
Q$ 4" '2#"'/0 $""&D (!" (01+,'2 "2",(#%& -"'3$ +& (!" +&("#12'&"('#0 $1'," '&/ +&
(!" B+,+&+(0 %* ,%22+$+%&2"$$ $!%,7$ !'B" /"&$+(+"$D nb D 3),! $3'22"# (!'& (!" -',75#%)&/
12'$3' /"&$+(0 '&/ (!"0 1#%1'5'(" '2%&5 (!" 3'5&"(+, N"2/ 2+&"$6 !+$ R)$(+N"$ '&
'$$)31(+%& (!'( (!" 1#%-2"3 ,'& -" ,%&$+/"#"/ '$ %&"9/+3"&$+%&'26 Q$$)3+&5 (!'(
(!" ,%&/+(+%& %* (!" %B"#2'11+&5 %* (!" #"$%&'&,"$ +$ $'(+$N"/D (!" ":)'(+%&$ *%# (!"
-"'3 "2",(#%&$ B"2%,+(0 /+$(#+-)(+%& *)&,(+%& '&/ 4'B"$ $1",(#'2 "&"#50 /"&$+(0 ,'& -"
%-('+& *#%3 ":)'(+%&$ ;6<= '&/ ;6<L '$ *%22%4$>
′
′
∂
∂f (v, t)
∂f (v, t)
=
D(v, t)
∂t
∂v
∂v
?;6<SA
∂Ws (V, t)
= 2γ(v, t)Ws (V, t)
∂t
?;6<TA
!"#"D D +$ (!" B"2%,+(0 /+.)$+%& ,%"U,+"&(D /"("#3+&"/ '$ *%22%4$>
8π 2 e2
D(v, t) =
m2
<=
W (V, t)
V
V =v
?;6<VA
!" γ(v, t) #$ %&' ( )'$ *+,(%& + %'-
π nb
γ(v, t) = ωp
2 N0
v
2 ∂f (v, t)
∂v
./0123
V =v
(&'+' 4' 5 '6'7%+,! )'6,7#%8 "#$%+#49%#,! :9!7%#,!; f (v, t); #$ !,+5 6#<'" %, ,!'-
10
R∞
0
f (v, t) =
=$ ,!' 7 ! $'' :+,5 %&' "'>'!"'!7' ,: γ 9>,! '6'7%+,! "#$%+#49%#,! :9!7%#,! %&'
( )'$ $% +% *+,(#!* :+,5 #!#%# 6 %&'+5 6 !,#$' 6')'6 ,!68 #! %&' +'*#,! ,: %&' )'6,7#%8
$> 7' (&'+' ∂f (v, t)/∂v > 00 ? 5'68; %&#$ 5' ! %& % %&' '6'7%+,! 4' 5 >+,> * %#!*
#! %&' >6 $5 ; *'!'+ %'$ ,!68 %&' @ !*59#+ ( )'$ (#%& >& $' )'6,7#%#'$; V ; 6'$$ %& !
%&' 4' 5 )'6,7#%8 vb 0 A&#$ ( )'$ (#66 "'%'+5#!'
4 7B( +" +' 7%#,! ,! %&' 4' 50
C% #$ (,+%& !,%#!* %& % %&' 7,!%+#49%#,! ,: ( )'$ (#%& V > vb 7 ! 4' !'*6'7%'"0 A&#$
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A&9$; #! %&#$ >>+,F#5 %#,! %&' >+,7'$$ ,: %&' +'6 F %#,! (#66 +9! ,!68 %,( +" 6,('+
)'6,7#%#'$0
A&' $ %9+ %#,! ,: %&' 4' 5 #!$% 4#6#%8 ,779+$ (&'! %&' ( )' *+,(%& + %' 4'7 5'
76,$' %, 0 ,+ !'* %#)' :,+ 66 >& $' )'6,7#%#'$0 A&#$ #5>6#'$ %& % %&' $% 46' 'G9#6#4+#95
$% %' :,+ %&' '6'7%+,! )'6,7#%8 "#$%+#49%#,! :9!7%#,! #! %&' + !*' ,: >& $' )'6,7#%#'$ ,:
( )'$ #$
$
>6 %' 9; $#!7' %&#$ "#$%+#49%#,! 6' "$ %, ) !#$&#!* "'+#) %#)' ,: f (v, t) !"
+'$96% %, γ = 00 A&9$ %&' +'6 F %#,! >+,7'$$ 7,!$#$%$ ,: >6 %' 9 :,+5 %#,! ,:
%&' )'6,7#%8 "#$%+#49%#,! :9!7%#,! #! %&' + !*' ,: )'6,7#%#'$ 6'$$ %& ! vb .$,6#" 6#!' %
H#*9+'/0/30 I!' 7 ! J!" %& % #! %&' '!" ,: %&' +'6 F %#,! f (v, t) 7 ! 4' "'$7+#4'" $
:,66,($-
nb fend =
(
nb
vb
0
:,+ v < vb
:,+ v > vb
./01K3
C% #$ (,+%& !,%#!* %& % "9+#!* %&' +'6 F %#,! >+,7'$$ "'$7+#4'" 48 %&' ,: 'G9 %#,!$
/0LM !" /0LN
$8$%'5 ,: ( )'$ !" > +%#76'$ 7,!$'+)'$ #%$ %,% 6 '!'+*80 O,+',)'+ %&'
$8$%'5 & $ ""#%#,! $>'7#J7 G9 $#P6#!' + 7,!$% !% ,: 5,%#,!-
2ωp ∂ W (V, t)
∂
= 0.
nb f (v, t) −
∂t
m ∂v
V3
V =v
./01/3
C: (' 7,!$#"'+ %&' @ !*59#+ ( )'$ $ ! '!$'546' ,: %&' G9 $#P> +%#76'$ (#%& '!'+*8
"#$%+#49%#,! W (V )/V ; %&' 'G9 %#,! /01/ 7 ! 4' #!%'+>+'%'" $
7,!$'+) %#,! ,: %&'
%,% 6 !954'+ ,: > +%#76'$ !" G9 $#P> +%#76'$ #! %&' $8$%'50 Q8 5 B#!* $94$%#%9%#,! ,:
fend #!%, 'G9 %#,! /01/ ,!' 7 ! ,4% #! %&' $>'7%+95 ,: %&' ( )'$ '!'+*8 "'!$#%8 #! %&'
'!" ,: +'6 F %#,!-
Wend (V ) =
(
mnb V 4
2ωp vb
:,+ V < vb
0
:,+ V > vb
LR
./01L3
!"#$% &'&( )%*+ ,-.%/( *0$12 0* +3% %/%4+$0. 5!2+$!6#+!0. *#.4+!0. -+ +3% 6%"!..!."
75-23%5 /!.%8 -.5 !. +3% %.5 0* +3% $%/-9-+!0. ,$04%22 720/!5 /!.%8' :!"3+ ,-.%/( 2.-,;
230+2 0* +3% %/%4+$0. <%/04!+= 5!2+$!6#+!0. *#.4+!0. !. +3% 40.2%>#%.+ 101%.+2 0* +!1%(
t1 < t2 < t3 ' ?3% 40$% 5!2+$!6#+!0.@ +3-+ 40$$%2,0.52 +0 +3% 40/5 %/%4+$0.2 0* 6-4A;
"$0#.5 ,/-21- 3-2 .0+ 6%%. 40.2!5%$%5 B-5-,+%5 *$01 C%5%.0< -.5 :=#+0<@ DEFGH'
?3% -2=1,+0+!4 20/#+!0.2 fend (v) -.5 Wend (V ) $%,$%2%.+ - I%// A.0I. $%2#/+ *0$
10.043$01-+!4 %/%4+$0. 6%-1 $%/-9-+!0. !. 3010"%.%0#2 ,/-21-( +3% 6%-1 ,$%2%$<%2
0./= +3!$5 ,-$+ 0* !+2 !.!+!-/ *$%% %.%$"=@ I3!/% +I0;+3!$52 0* +3% ,-$+!4/%2 %.%$"= !2
+$-.2*%$$%5 +0 +3% )-."1#!$ I-<%2'
!"!
#$%&%'()&*+(*' +'%,)+ -. ($) &),%/%(*-0
J2 0.% 4-. .0+!4% *$01 +3% %>#-+!0. &'KL@ +3% "-!. 0* +3% I-<% %.%$"= 5%.2!+=@ ∆W @
!2 ,$0,0$+!0.-/ +0 +3% I-<% %.%$"= 5%.2!+= W !+2%/*' ?3% 2#62+-.+!-/ "-!. 0* +3% I-<%2
%.%$"= 044#$2 !. 201% $%"!0.2 0* +3% <%/04!+= 2,-4% I3%$% +3% "$0I+3 $-+% !2 3!"3 %.0#"3
+0 %.2#$% 2#M4!%.+ -1,/!N4-+!0.' O-1%/=@ +3% /%*+ 60$5%$ 0* +3!2 $%"!0.@ u@ !2 5%+%$1!.%5
6= +3% !.!+!-/ +3%$1-/ I!5+3 0* +3% 6%-1@ ∆vb ' P!.4% +3% !.!+!-/ 2,%4+$-/ %.%$"= 5%.2!+=
0* +3% I-<% !2 <%$= /0I@ Win /Wend ≪ 1 !" "#$ %$&'(('(& )* "#$ +$,!-!"')( "#$ .!/$0
.'"# 1#!0$ /$,)2'"'$0
V
"#!" 0!"'0*3 2)(4'"')(
.'"# 1#!0$ /$,)2'"'$0 ,$00 "#!(
u < V < vb
.',, &+).
u .',, %$ 5!'("!'($4 )( ! ,$/$, !%)6" Win 7
$-1$2" "#!" "#$ %!29.!+4 +$!2"')( )* "#$ .!/$0 )( "#$ %$!5
4'0"+'%6"')( *6(2"')( '( "#$ +!(&$ )* /$,)2'"'$0 *+)5
%)+4$+
>7>?
u(t)
.#',$ "#$ .!/$0
u
")
8#60 )($ 2!(
.',, 1,!"$!6 "#$ /$,)2'"3
vb 7
:6%0$;6$(",3
"#$ ,$*"
)* "#$ 1,!"$!6 .',, 5)/$ ").!+4 "#$ ,).$+ /$,)2'"'$0 <+'&#" 1!($, )* ='&6+$
!(4 "#$ "$51)+!, $/),6"')( )* "#$
nb f (v, t) =
8#$ *6(2"')(
u(t)
f (v, t)


0
*)+
v < u(t)
u(t) < v < vb
*)+
v > vb
*)+
nb
 vb −u(t)

2!( %$ +)6&#,3 4$02+'%$4 !0 *),,).0@
0
2!( %$ *)6(4 *+)5 $;6!"')( >7BC %3 06%0"'"6"'(& 0),6"')(
BD
<>7AA?
f (v, t)
!"#$% &'() *+,!-!+. +/ -0% 1%/- 2+$3%$ +/ -0% 3!,-$!2#-!+. /#.4-!+. 5, 5 /#.4-!+. +/ -0%
-!6% 75358-%3 /$+6 9%3%.+: 5.3 ;<#-+:= >?@AB'
3%C.%3 2% %D#5-!+. &'EE 5.3 !.-%"$5-!+. +:%$ V ' F0!, 8$+4%3#$% 511+G, +.% -+ G$!-%
%D#5-!+. /+$ u(t) 5, /+11+G,)
du(t)
π nb
u2 (t)
= − ωp
·
dt
Λ N0 vb − u(t)
H&'EAI
0%$% Λ !, G%11 J.+G. K+#1+62 1+"5$!-06' !"#$% &'( ,0+G, 5 ,+1#-!+. +/ -0% %D#5-!+.
&'EA' F0% $%15L5-!+. ,-+8,= G0%. -0% :%1+4!-< 3!,-$!2#-!+. /#.4-!+. +/ -0% 2%56 %1%4-$+.,
$%540%, 5 $%"!+. +/ -0% :%1+4!-< ,854% +44#8!%3 2< -0% 254J"$+#.3 815,65' M, +.% 45.
.+-!4%= -0% 405$54-%$!,-!4 -!6% +/ -0% $%15L5-!+.= tt = 45. 2% %,-!65-%3 5, /+11+G,)
tr ∼
Λ 1 N0
.
π ωp nb
H&'ENI
O. -0% ,!681% 65..%$= 5. %,-!65-% +/ 5 405$54-%$!,-!4 ,85-!51 ,451% +/ -0% $%15L5-!+.=
rs = 45. 2% +2-5!.%3 2< 6#1-!81<!." tr 2< 5 !.!-!51 2%56 :%1+4!-<= vb
rs ∼
Λ v b N0
.
π ωp nb
H&'E@I
!" #$%&'% ()*+ ',-,*,-).%$$/ 01.21%&)-3 01-&)*/ 42,5$1
P%- #, 4+.,!3%$ 5 8$+85"5-!+. +/ -0% P5."6#!$ G5:% 854J%- "%.%$5-%3 2< -0% 2%56 !.
5 815,65 G!-0 6+.+-+.!4511< 3%4$%5,!." 3%.,!-< 51+." -0% G5:%Q, 85-0' R%4$%5,!." +/
-0% 3%.,!-< $%,#1-, !. -0% 3%4$%5,!." +/ -0% 815,65 /$%D#%.4< 5.3 -0#, -0% 8$+85"5-!+.
!. ,#40 5 815,65 3!S%$, /$+6 -0% 45,% +/ 0+6+"%.%+#, 815,65' F0% %S%4- $%15-%3 -+ -0%
3%.,!-< !.0+6+"%.%!-< ,0+#13 2% !.41#3%3 !. -0% 3!,8%$,!+. $%15-!+. /+$ -0% P5."6#!$
G5:%' O/ -0% ,85-!51 405."% +/ -0% 3%.,!-< !, ,1+G= -0% k(x) :%4-+$ +/ G5:% G!11 405."% !.
(@
!"# $ %$& '#$' '#( )* +(, *-. ,(/$'*-. #-!/) 0( $'* 1() %*'# '#( "-.)*'*-. '#$' '#(
%$2( 3,(4!(."& ,(5$*. "-. '$.' %#*/( +/$ 5$ 3,(4!(."& 2$,*( $""-,)*.6 '- 2$,*$'*-.
k
-3 '#( )(. *'&7 8#( "#$.6( -3 '#(
2("'-,9 *. '#(*, '!,.9 ,( !/' *. '#( "#$.6( -3 '#(
%$2( +#$ ( 2(/-"*'&7 :!66( '*.6 '#$' '#( "-.)*'*-. $0-2( *
$'* 1()9 (4!$'*-. ;7<= "$.
0( ,(%,*''(.>
3 vt2
ω = ωp (x) 1 +
2 V 2 (x)
.
?;7@AB
C& 5$D*.6 ! ( -3 (4!$'*-. ;7@A -.( "$. ( '*5$'( '#( 2$,*$'*-. -3 '#( +#$ ( 2(E
/-"*'& -3 '#( %$2(9
∆V 9
3,(4!(."& -3 '#( %$2(9
ω
$3'(, +$
*.6 '#( )* '$."(
∆x7
F *.6 '#( "-.)*'*-. '#$' '#(
,(5$*. "-. '$.' $/-.6 '#( ',$G("'-,&9 '#( (4!$'*-. ;7@A /($)
'- 3-//-%*.6>
(1 +
vt2 ∆V
3 vt2 dωp
)
∆x
−
3ω
= 0.
p 2
2 V 2 dx
V V
?;7@HB
I -.( "$. .-'*"(9 $ )(",($ ( -3 '#( +/$ 5$ 3,(4!(."& ,( !/' *. $ )(",($ ( -3 '#(
+#$ ( 2(/-"*'& -3 '#( %$2(7 I $ ,( !/'9 *. '#( +/$ 5$ %*'# 5-.-'-.*"$//& )(",($ *.6
)(. *'& +,-1/(9 '#( J$.65!*, %$2( $,( #*3'() 3,-5 '#( ,(6*-. -3 '#( 2(/-"*'& +$"(9
%#(,( '#(& %(,( 6(.(,$'() 0& '#( 0($5 *. '$0*/*'&9 '-%$,) /-%(, +#$ ( 2(/-"*'*( 7 8#! 9
!"# *.#-5-6(.(*'& )- .-' "$! ( $. $++($,$."( -3 '#( %$2( %*'# +#$ ( 2(/-"*'*( /$,6(,
'#$. vb $.) '#( $ &5+'-'*" -/!'*-. 3-, '#( (/("',-. )* ',*0!'*-. 3!."'*-. fend (v) * '#(
$5( $
3-, #-5-6(.(-!
+/$ 5$7
"/- ( '- '#( 2(/-"*'& -3 '#( 0($59
I !5*.6 '#$' +#$ ( 2(/-"*'*(
V ∼ vb
$.) '#$'
dωp /dx ∼ ωp /Lun 9
∆V
"#$,$"'(,* '*" "$/( -3 '#( )(. *'& *.#-5-6(.(*'&9
-3 '#( %$2(
%#(,(
Lun
$,(
* $
"$. 0( -0'$*.() 3,-5 (4!$'*-.
;7@H $ 3-//-% >
∆V ∼ vb
I
%( #$2( $/,($)&
∆x vb2
.
Lun vt2
((. *. '#( +,(2*-!
?;7KLB
("'*-.9 '#( (M("'*2( 6$*. -3 '#( %$2(
(.(,6& -""!, *. '#( ,$.6( -3 '#( +#$ ( 2(/-"*'*( 3,-5
vb − ∆vb
'-
vb 7
N!( '- '#( #*3'
-3 '#( +#$ ( 2(/-"*'&9 '#( %$2( /($2( '#* ,(6*-. $.) '#( $5+/*1"$'*-. -3 %$2( $5+/*'!)(
'-+ 7 8#* #$++(.
-. $ "#$,$"'(,* '*" +$'*$/
"$/(
∆xc
'#$' "$. 0( -0'$*.() 3,-5
(4!$'*-. ;7KL $ 3-//-% >
∆xc ∼ Lun
I "#$,$"'(,* '*" '*5(
"$/( -3 '#(
∆vb vt2
.
vb vb2
#*3' -3 '#( %$2(
?;7K<B
3,-5 '#( ,(6*-. -3 (M("'*2(
6,-%'#9 tc 9 "$. 0( ( '*5$'() '$D*.6 *.'- $""-!.' '#$' '#( %$2( +$"D(' +,-+$6$'( %*'#
'#( 6,-!+ 2(/-"*'&
vg = vt2 /vb 7
8#! >
tc ∼
Lun ∆vb
.
vb vb
OA
?;7K;B
! !"#$%& #'( )*!+#' !, +$-(. /0*%&) #'( #%1( tc 02 #! $ 3(-(3 .%)&%45$&#36 3$*)(*
#'$& &!%.( 3(-(3 #'( ,!33!+%&) 5!&/%#%!& .'!03/ "( .$#%.4(/7
89:;<=
! (.#%1$#( #'( )$%& !, #'( +$-( !&( 5$& 0.( #'( .%123%4(/ ,!*1 !, #'( (>0$#%!&
9:?@A $. ,!33!+.7
γtc ≥ Λ
γ ∼ πωp
nb vb2
N0 ∆vb2
89:;?=
B6 .0".#%#0#%&) γ ,*!1 (>0$#%!& 9:;? $&/ tc ,*!1 (>0$#%!& 9:;9A !&( 5$& *(+*%#(
%&(>0$3%#6 9:;< $. ,!33!+.7
nb πωp Lun vb
≥Λ
N0 vb ∆vb
89:;;=
C. !&( 5$& &!#%5( #'( 3(,# .%/( !, #'( %&(>0$3%#6 9:;; /(5*($.(. $. ∆v %&5*($.(.:
D'(& #'( "($1 %. +%/( (&!0)' $&/ #'( 5!&/%#%!& 9:;; %. &!# .$#%.4(/ $&61!*(A #'(
*(3$E$#%!& .#!2.A "(5$0.( #'( )$%& !, #'( +$-( %. -(*6 3!+A &$1(36 $"!0# #'( #'(*1$3
&!%.( 3(-(3: F!**(.2!&/%&) ∆vb 5$& "( (.#%1$#(/ $. ,!33!+.7
π nb ωp
∆vb
∼ Lun
vb
Λ N0 v b
89:;G=
H>0$#%!& 9:;G 5$& "( 0.(/ #! %330.#*$#( $ 2(503%$*%#6 !, #'( "($1 *(3$E$#%!& 2*!5(..
%& #'( 23$.1$ +%#' /(5*($.%&) /(&.%#6 2*!43(: I(# 0. $..01( #'$# #'( 5'$*$5#(*%.#%5
.2$#%$3 .5$3( !, #'( /(&.%#6 %&'!1!)(&(%#6 %. $"!0# #'( /%.#$&5( #'$# #'( I$&)10%*
+$-(. $*( $"3( #! 2$.. /0*%&) #'( #%1( !, #'( *(3$E$#%!& %& '!1!)(&(!0. 23$.1$A tr :
B6 .0".#%#0#%&) L = vg tr %&#! (>0$#%!& 9:;G !&( 5$& !"#$%&7
∆vb
v2
∼ t2 .
vb
vb
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nb
vb
0
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vb A
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4( =5>>5<(B
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∆n(x)
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1 ∆n(x)
1+
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QC
+
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u
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(!+#+(&"#$%&$(% 92 &!" 7"+, +4* &!" %&+&$%&$(+' (!+#+(&"#$%&$(% 92 *"4%$&5 8.(&.+&$94%;
UV
!
"#$%$&'(%)*')& *&$+(* ,- '#( .(/*)'0 12&'2$'),/*
!"!#$% &'()#*$+* ,-$%!, $#! .+)/+ 0#)' ,*12&!, )0 3!$' (%$,'$ &+*!#$-*&)+ &+ 4)')5
6!+!)1, (%$,'$ $+2 (%$,'$ /&*4 ')+)*)+&-$%%7 -4$+6&+6 2!+,&*7 $, -)'($#!2 /&*4
2!+,&*7 &+4)')6!+!&*7 ,-$%!8
94$#$-*!#&,*&- ,-$%!, )0 *4! 3!$' (%$,'$ &+*!#$-*&)+ 2!(!+2 )+ *4! -4$#$-*!#&,*&6#)/*4 #$*! γ 8 :) )3*$&+ *4! 6#)/*4 )0 /$"!, 1( *) $ %!"!% ,&6+&;-$+*%7 %$#6!# *4$+
+)&,! %!"!% )+! ,4)1%2 ,$*&,07 *4! 0)%%)/&+6 -)+2&*&)+<
γ
L
≫ Λ.
vg
=>8?@
A!#! L &, *4! -4$#$-*!#&,*&- ,-$%! )0 *4! 2!+,&*7 &+4)')6!+!&*&!,B vg ∼ vt2 /vb &, *4! 6#)1(
"!%)-&*7 )0 *4! C$+6'1#! /$"!B vt &, *4! *4!#'$% "!%)-&*7 )0 *4! 3$-.6#)1+2 (%$,'$B $+2
vb &, *4! "!%)-&*7 )0 *4! 3!$'B Λ &, *4! ,) -$%%!2 9)1%)'3 %)6$#&*4' $+2 Λ = ln N0 λ3D D
/4!#! λD &, *4! -4$#$-*!#&,*&- E!37! %!+6*48 F0 L &, ,'$%%!# *4$+ (vg /γ)ΛD *4! 6#)/*4
)0 *4! /$"!, &, ,*)((!2 )+ ,1-4 $ %!"!% *4$* +)+%&+!$# (4!+)'!+$ ,1-4 $, /$"! ($#*&-%!
&+*!#$-*&)+, -$+ 3! +!6%!-*!28 :4! %&+!$# 6#)/*4 #$*! )0 *4! 3!$' (%$,'$ &+,*$3&%&*7
#!$2,<
γ ≃ ωp
nb
N0
vb
∆vb
2
,
/4!#! ωp &, *4! -4$#$-*!#&,*&- (%$,'$ 0#!G1!+-7D ∆vb &, *4! -4$#$-*!#&,*&- 3!$' /&2*4
&+ *4! "!%)-&*7 ,($-!D $+2 nb &, *4! 2!+,&*7 )0 *4! 3!$' !%!-*#)+,8 :4!+ *4! -)+2&*&)+
$3)"! -$+ 3! (#!,!+*!2 $, 0)%%)/,<
Lmin =
∆vb
vb
2
N0 v t
ΛλD .
n b vb
H)# ($#$'!*!#, #!%!"$+* *) ,)%$# :7(! FFF !%!-*#)+ 3!$' $+2 3$-.6#)1+2 (%$,'$ $*
1 IJD *4! 0)%%)/&+6 ($#$'!*!#, $#! $((%&-$3%! KL#61+ !* $%8D MNNOP < N0 ≈ 5 × 106 m−3 B
ωp /2π ≈ 20kHz B nb /N0 ≈ 10−5 B λD ≈ 15m B ∆v/vb ≈ 0.05B vb ≈ 15vt $+2 Λ ≈ 15B
Lmin #!,1%*, &+ 250λD )# 3750m8
E!+,&*7 Q1-*1$*&)+, 4$"&+6 "!#7 %$#6! ,($*&$% ,-$%!,D %$#6!# *4$+ *4! 3!$' #!%$R$*&)+
%!+6*4 &+ 4)')6!+!)1, (%$,'$D -$+ +)* ,&6+&;-$+*%7 &+Q1!+-! *4! (#)-!,, )0 *4! 3!$'
(%$,'$ &+*!#$-*&)+8 S!%$R$*&)+ ,*)(, /&*4 $ (%$*!$1 0)#'$*&)+ /4!+ ∆v/vb ∼ 18 :4!+D
Lmax -$+ 3! !,*&'$*!2D $, 0)%%)/,<
Lmax =
N0 v t
ΛλD .
nb v b
H)# *4! ($#$'!*!#, '!+*&)+!2 $3)"!D Lmax #!,1%*, &+ 105 λD )# 1.5 × 106 m8
TT
!" #$% &'(% !) ' *+,%# (!-'" .,/0 .,#$ vSW = 450kms−1 1 Lmin '/0 Lmax &'/ 2,%-0
#.! &$'"'&#%",(#,& )"%*+%/&,%( ,/ ' )"'3% !) (4'&%&"')# fh = 120Hz '/0 fl = 0.3Hz 5
6%"%,/1 .% &!/(,0%" 0%/(,#2 7+&#+'#,!/( .,#$ &$'"'&#%",(#,& (4'#,'- (&'-%(1 L1 #$'# ('#,()2
&!/0,#,!/(8 Lmin < L < Lmax 5
!" #$%&'()$' %* (+,&$'-+./-+%0! 1'2/$/-+%0 %* ,&/3',
9! '44-2 #$% (#'#,(#,&'- '44"!'&$ #! !+" (2(#%31 :%-!. .% 0%(&",:% !+" &!/#,/+!+(
(2(#%31 3';,/< +(% !) #$% 4"!&%0+"% !) 0,(&"%#,='#,!/5 9! #$,( %/01 .% 0,>,0% #$%
&!/#,/+!+( (4'#,'- ,/#%">'- !/ ' (%# !) %*+'--2 (,=%0 (+:,/#%">'-( !) ' -%/<#$ a5 9$% (,=%
a ,( (+44!(%0 #! :% 3'#&$ (3'--%" #$'/ #$% &$'"'&#%",(#,& (&'-% !) #$% &$'/<% !) %-%&#"!/
0,(#",:+#,!/ )+/&#,!/5 ?!"%!>%"1 ,# ,( (+44!(%0 #! :% (3'--%" #$'/ #$% &$'"'&#%",(#,&
(&'-% !) #$% 0%/(,#2 <"'0,%/# :+# (+@&,%/#-2 -'"<%" #$'/ #$% .'>%-%/<#$ !) A'/<3+,"
.'>%( <%/%"'#%0 :2 #$% %-%&#,!/ :%'38
λ ≪ a ≪ L.
B/!#$%" ,34!"#'/# -,3,#'#,!/ )!" a &'/ :% !:#',/%0 :2 &!/(,0%",/< >%-!&,#2 &$'/<%
!/ '/ ,/#%">'- .,#$ '/ ,/$!3!<%/%,#2 $'>,/< -,/%'" 4"!C-%8 N (x) = N0 + ∆nx/L5
9$% 0,(4%"(,!/ "%-'#,!/($,4 )!" #$% A'/<3+," .'>% &'/ :% .",##%/ '( )!--!.(8 ω =
ωp (x)(1 + (3/2)(vt2 /V 2 ))5 6%"% ω '/0 V '"% #$% )"%*+%/&2 '/0 #$% 4$'(% >%-!&,#2 !) #$%
.'>%5 B((+3,/< #$'# #$% )"%*+%/&2 !) #$% .'>% '/0 #$% #$%"3'- >%-!&,#2 !) 4'"#,&-%(
'"% &!/(#'/#1 #$% )!--!.,/< %*+'#,!/ &'/ !:#',/%08
∆ω =
3 vt2
1+
2V2
dωp (x)
v 2 ∆V
∆x − 3ωp t2
= 0,
dx
V V
.$%"% ∆V ,( #$% 0%>,'#,!/ !) #$% 4$'(% >%-!&,#2 !/ #$% (&'-% ∆x &'+(%0 :2 ' 0%/(,#2
<"'0,%/#5 D,/&%
1 dωp (x)
1 d
=
ωp dx
2 dx
'/0 #';,/< ,/#! '&&!+/# #$% )'&# #$'# V
|∆V | ∼ vb
x ∆n
L N0
=
1 ∆n
,
2L N0
∼ vb ≫ vt 1 ∆V
&'/ :% %(#,3'#%0 '( )!--!.(8
∆x |∆n| vb
.
L N0 vt
A'/<3+," .'>%( &'/ <"!. !" 0%&'2 0+% #! #$% ,/#%"'&#,!/ .,#$ #$% %-%&#"!/ :%'31
.$,-% #$% 4$'(% >%-!&,#21 V 1 "%3',/( ,/ #$% ,/#%">'- vb − ∆vb < V < vb + ∆vb 5 9$%
,/#%"'&#,!/ .,-- (#!4 .$%/ ∆V ∼ ∆vb '/0 .$%/ #$% "%(!/'/# &!/0,#,!/( '"% /!# ('#,(C%0
'/23!"% )!" :%'3 4'"#,&-%(5 D+&$ 0%>,'#,!/ !) 4$'(% >%-!&,#2 !&&+"( !/ #$% )!--!.,/<
(4'#,'- (&'-%8
EF
∆xc = L
N0 ∆vb vt2
.
∆n vb vb2
!" #$% &!'()#)!' *%'#)!'%( +%,!"% -∆vb /vb = 0.05. vb = 15vt / 0'( ∆n/N0 = 0.01.
102%3 %3&04% ,"!* "%3!'0'&% !' #$% 3&05% ∆xc ∼ 0.02L6 7' !8" *!(%5. #$% &$0'9% !,
2%5!&)#: !' !'% 3)'95% )'#%"205 3$!85( +% *8&$ 3*055%" #$0' 0 2%5!&)#: 20")0#)!' #$0#
1!85( 5%0( #! 0 2)!50#)!' !, #$% "%3!'0'# &!'()#)!'6 ;$)3 )*45)%3 #$0# #$% &$0"0&#%")3#)&
3&05% ,!" 38+)'#%"205 a )3 38<&)%'#5: 3*055%" #$0' ∆xc 6 7' #$)3 &03% 0 102% &0' +%
&!'3)(%"%( #! "%*0)' )' "%3!'0'&% 1)#$ #$% %5%&#"!' +%0* !' 0 3)'95% )'#%"205 1)#$!8#
0 5!33 !, &!$%"%'&%6
!" #$%&'%()*+ *, (-. /*+(0)1'()*+ ,0*2 (-. %+3'&%0 4),5
,'6)*+
;1! *0=!" 4$:3)&05 %>%&#3 0"% &083%( +: (%'3)#: ?8&#80#)!'3. 5%0( #! &$0'9%3 )' #$%
102% 4$03% 2%5!&)#:6 @!#$ "%385# )' &$0'9)'9 !, #$% &!'()#)!'3 !, "%3!'0'&% 0'( &!'A
3%B8%'#5: 3$),# 102%A40"#)&5% )'#%"0&#)!' )' 0'!#$%" 40"# !, #$% 4$03% 340&%6 ;$83. #$%
102% 1)#$ ,"%B8%'&: ω 0'( 8'4%"#8"+%( 102% 2%&#!" k0 1)55 )'#%"0&# 1)#$ 40"#)&5%3.
#$0# $02% 2%5!&)#)%3 ()>%"%'# ,"!* ω/k0 6
;$% C"3# %>%&# &!'3)3#3 )' 0'9850" ()>83)!' !, #$% ()"%&#)!' !, 102% 4"!4090#)!' 0'(
#$% 0'9850" (%2)0#)!' !, #$% 102% 2%&#!" ,"!* )#3 3844!3%( #"0=%&#!": )' 0 $!*!9%'%!83
4503*06 ;$% 4"!&%33 +%&!*%3 %D#"%*%5: )*4!"#0'#. ,!" (%'3)#: ?8&#80#)!'3 1)#$ 0
340#)05 3&05%3. l. #$0# 0"% 3*055 &!*40"%( #! vg /γ . 0'( 1)#$ 0*45)#8(%3. #$0# 0"% '!#
#! 50"9% |∆n/N0 | ≪ vt2 /vb2 6 ;$% C"3# !, )'%B805)#)%3 )*45)%3 #$0# #$% 0'9850" (%?%&#)!'
!' %0&$ 3&0##%")'9 %2%'# )3 *8&$ 3*055%" #$0' 8')#:. 3! #$0# 4"!&%33 &0' +% (%3&")+%(
+: 0'9850" ()>83)!'6 ;$% 3%&!'( !, )'%B805)#)%3 3#0#%3 #$0# #$% k 2%&#!" *09')#8(% !,
#$% E0'9*8)" 102%3 )3 &!'3%"2%( -3%% F)3$)G010 0'( H:8#!2 IJKLMN 0'( O"03'!3%53G)G$
%# 056 IPQQLN ,!" 0(()#)!'05 (%#0)53/6
;$% 3%&!'( %>%&# &!'3)3#3 !, 0 &$0'9% )' *09')#8(% !, k )' #$% ()"%&#)!' !, 102%
4"!4090#)!'6 R%'3)#: ?8&#80#)!'3 3$!85( $02% 0*45)#8(%3 #$0# 30#)3,: &!'()#)!' |∆n/N0 | ≥
vt2 /vb2 #! +% )'2!52%( )' #$)3 4"!&%336 S3 3$!1' %0"5)%". #$% *!3# )*4!"#0'# )*40&# !'
+%0* "%50D0#)!' )3 4"!2)(%( +: ?8&#80#)!'3 1)#$ 340#)05 3&05%3. L. #$0# 5)% 1)#$)' #$%
"0'9% Lmin < L < Lmax 6 7# )3 1!"#$ '!#)'9 #$0# )'%B805)#: -T6P/ 5%0(3 #! &!'&583)!'
#$0# L )3 *8&$ 50"9%" #$0'. #$% 340#)05 3&05%. l. !, #$% ?8&#80#)!'3 #$0# 0"% #$% *!3#
)*4!"#0'# ,!" #$% 4"!&%33 !, 0'9850" ()>83)!'6
;$%"%+:. #$% 4"!&%33 !, 0'9850" ()>83)!' )3 (!*)'0'#. 1$%' (%'3)#: ?8&#80#)!'3
$02% "%50#)2%5: U3*055U 3&05%3 0'( 0*45)#8(%3. 1$)5% #$% %>%&# !, 0 &$0'9% !, k *09')A
#8(% 4"%20)53 ,!" #$% ?8&#80#)!'3 1)#$ 50"9%" 3&05%3 0'( 0*45)#8(%36 V8" 9!05 $%"%0,#%"
)3 #! 3#8(: #$% "!5% !, #$% (%'3)#: ?8&#80#)!'3 !, #$% 3%&!'( #:4% )' #$% +%0* 4503*0
)'#%"0&#)!'36
WL
!" #$ !%&'#&"! ()*"+,-#",)* .+)/ "0! &*1#'&+ 2,3#$,)* ). "0! $(&'! ∆xc 4 50!$! !3!("
,$ 2!$(+,-!2 -6 "0! 2,3#$,)* !7#&",)* 89,$0,:&;& &*2 <6#")%= >?@A= B+&$*)$!'$:,:0
!" &'4= CDD@E = &*2 "0! ()++!$F)*2,*1 F+)(!$$ ,$ (0&+&("!+,G!2 -6 & 2,3#$,)* ()!H(,!*"
"0&" (&* -! !$",/&"!2 ") -! ). "0! )+2!+ ).I
∆nad 2
1
1
.
D ∼ ωp
lk (k 2 λ2D ) N0 J!+! k ,$ "0! (0&+&("!+,$",( ;&%! %!(")+ ). "0! F+,/&+6 &*1/#,+ ;&%! 1!*!+&"!2 2#! ")
-!&/KF'&$/& ,*"!+&(",)*= ∆nad &*2 l= &/F',"#2! &*2 $F&",&' $(&'! ). &* ,*0)/)1!*!,"6
,*%)'%!2 ,* "0! F+)(!$$ ). "0! &*1#'&+ 2,3#$,)*4 5) $,1*,L(&*"'6 +!2#(! "0! ,*$"&-,',"6
1+);"0 "0! ;&%! %!(")+ $0)#'2 2!%,&"! .+)/ ,"$ ,*,",&' 2,+!(",)* ") "0! &*1'! θ= &$
2!"!+/,*!2 -6 "0! .)'');,*1 ()*2,",)*I
∆vb
.
vb
1 − cos θ =
M. ∆vb /vb ≪ 1 "0!* θ ∼ (∆vb /vb )1/2 4 50#$= "0! (0&+&("!+,$",( ",/! ). &*1#'&+ 2,3#$,)*
,* "0! %!')(,"6 $F&(! (&* -! !$",/&"!2 &$ .)'');$I
Dτad ∼
&*2
τad
1
≃
D
∆vb
vb
=
∆vb
,
vb
2 2
∆vb
−1 (k λD )
ωp .
kl
∆nad 2
vb
N0 9); )*! (&* ()/F&+! "0! ",/! ). "0! &*1#'&+ 2,3#$,)*= τad = &*2 "0! ",/! ). &
F+)F&1&",)*= τpr (∆xc )= ). & &*1/#,+ ;&%! "0+)#10 "0! ,*"!+%&' ∆xc I
l
τad
=
τpr (∆xc )
L
∆n
∆nad
2
k 2 λ2D
N0 v b k
∆n ωp
5&:,*1 ,*") &(()#*" "0&" vb k/ωp ∼ 1= l/L ≤ 1/Λ= ∆n/∆nad ≫ 1= &*2 ∆n/N0 ≥
)*! (&* L*2I τad /τpr (∆xc ) ≫ 14 50!+!-6= "0! !3!(" ). "0! &*1#'&+ 2,3#$,)* ,$ "))
N$');N ") /&:! $,1*,L(&*" ,*O#!*(! )* "0! ;&%!KF&+",('! ,*"!+&(",)*$ )* "0! $#-,*"!+%&'=
a ≪ ∆xc 4 P-$!+%&",)*$ ). "0! &*1/#,+ ;&%!$ ,* "0! $)'&+ ;,*2 $0); "0&" /)$" &*1K
/#,+ ;&%!$ 0&%! &*1'!$ ;,"0 -&(:1+)#*2 /&1*!",( L!'2 $/&''!+ "0&* 20◦ 8Q+1#* !" &'4=
CDDR= S&'&$F,*& &*2 Q+1#*= CDDR= B+&$*)$!'$:,:0 !" &'4= CD>>E= "0&" 1,%!$ &22,",)*&'
&+1#/!*" %&',2&",*1 "0! ;!&:*!$$ ). &*1#'&+ 2,3#$,)* ). ;&%!$4
k 2 λ2D
!" #$%&'&()(*+(, -%./) %0 +1/ &/'- 2)'*-' (3+/$',+(%3 (3
2)'*-' 4(+1 ./3*(+5 67,+7'+(%3*
!" #$ &$$#/! "0&" & (0&*1! ,* "0! 2!*$,"6= ∆n = δn(xin + a) − δn(xin )= .)+ !&(0
$#-,*"!+%&' $&",$L!$ "0! ()*2,",)* ∆n/N0 ≪ 14 9); ;! (&* ()*$,2!+ "0! F+)-'!/ ).
TR
!"# $!%&'()# '*&#%!(&'+* +* &,# '*&#%"!) '&, &,# -#*.'&/ $%+0)# .+ -#&#%1'*#-2 3# .,!))
(+*.'-#% &,# '*&#%!(&'+* +4 ! .1!)) !1$)'&5-# (+,#%#*& !"#6 '&, ! 7*+ * 4%#85#*(/
ω !*- !* !1$)'&5-# Eω 6 '&, ! $!%&'()# '&, 7*+ * "#)+('&/ ve +* .5(, !* '*&#%"!)2
9,#.# !..51$&'+*. !))+ +*# &+ (!)(5)!&# (,!*:#. '* &,# $!%&'()# "#)+('&/6 ∆v !*- '*
&,# !"# !1$)'&5-#6 ∆Eω 6 '&, *#(#..!%/ -#:%## +4 !((5%!(/2
3# *+ (+*.'-#% -#"'!&'+*. +4 &,# $)!.1! -#*.'&/6 δn6 4%+1 N0 !& &,# #*-. +4 &,#
.5;'*&#%"!). !. %!*-+1 !*- '*-#$#*-#*& '&, ! $%#-#&#%1'*#- .&!&'.&'(!) -'.&%';5&'+*6
Pδn (δn)2 9,'. )!.& -'.&%';5&'+* 5*'85#)/ -#&#%1'*#. &,# -'.&%';5&'+* +4 &,# !"# $,!.#
"#)+('&/ Pω (V )2 <=('#*& "!%'!&'+*. 4+% ∆v !*- ∆Eω +*)/ +((5% '4 &,# (+*-'&'+* +4 #>!(&
%#.+*!*(#6 V = ve 6 '. .!&'.0#- '*.'-# &,# .#)#(&#- '*&#%"!)2 ?,!*:# '* &,# "#)+('&/ +4 !*
#)#(&%+* -#$#*-. +* Eω 6 ! $,!.# -'@#%#*(# ;#& ##* !* #)#(&%+* !*- ! !"#6 φ6 !*- &,#
!"# $,!.# "#)+('&/ +* &,# .#)#(&#- .5;'*&#%"!)2 3'&,+5& )+.. +4 :#*#%!)'&/6 &,# '*'&'!)
$,!.# φ (!* ;# .5::#.&#- &+ ;# ! %!*-+1 "!%'!;)# '&, ! 5*'4+%1 -'.&%';5&'+*2 9,5.6
"!%'!&'+*. '* ;+&, ∆v !*- ∆Eω +* #*.#1;)# +4 .5;'*&#%"!). !%# -#&#%1'*#- ;/ &,#
%!*-+1 "!%'!;)#. '&, &,# 7*+ * $%+;!;')'&/ -'.&%';5&'+*.2 A* '1$+%&!*& !--'&'+*!)
!..51$&'+* &,!& !))+ #- 5. &+ .+)"# &,# .&!&'.&'(!) $%+;)#1 '. !* !..51$&'+* &,!& !"#
$!%&'()# '*&#%!(&'+*. +* #!(, .5;'*&#%"!) !%# '*-#$#*-#*& +4 !"# $!%&'()# '*&#%!(&'+*.
+* &,# $%#"'+5. '*&#%"!)2 9,# %#.5)& '*-'(!&#. &,!& &,# $%+(#.. (!* ;# (+*.'-#%#- &+
;# ! .#%'#. +4 %!*-+1 !*- '*-#$#*-#*& '*&#%!(&'+*.6 ,'(, !))+ . +*# &+ -#.(%';# &,#
$%+(#.. +4 ;#!1 %#)!>!&'+* '* &#%1. +4 &,# 5*(+%%#)!&#- B!%(+"'!* $%+(#..2 A*+&,#%
.'1$)'0(!&'+* &,!& # 5.#- '. ;!.#- +* &,# .1!))*#.. +4 &,# (,!*:#. '* #)#(&%+* "#)+('&/C
∆v/ve ≪ 12
D*-#% .5(, (+*-'&'+*.6 +*# (!* -#0*# &,# $%+;!;')'&/ -#*.'&/ 45*(&'+*6 U (v, t|v0 , t0 )6
&,!& -#&#%1'*#. &,# $%+;!;')'&/ U (v, t|v0 , t0 )dv &,!& ! $!%&'()# ,!"'*: ! "#)+('&/ v0 !& !
1+1#*& +4 &'1# t0 ')) ,!"# ! "#)+('&/ v !4&#% Q '*&#%!(&'+*. &,!& +((5% -5%'*: &,# &'1#
'*&#%"!) t − t0 2 E/ .5::#.&'*: &,!& &,# *51;#% +4 .&#$.6 Q6 ;# )!%:# #*+5:, &+ F5.&'4/
.&!&'.&'(!) !"#%!:'*:6 &,# G+77#%HI)!*(7 #85!&'+* (!* ;# 5.#- &+ -#.(%';# &,# #"+)5&'+*
+4 U (v, t|v0 , t0 )6 !. 4+))+ .C
∂
∂U (v, t|v0 , t0 )
= − A(v)U (v, t|v0 , t0 ) +
∂t
∂v
∂2
B(v)U (v, t|v0 , t0 ),
∂v 2
JK2KL
,#%# A(v) !*- B(v) !%# -%'4& !*- -'@5.'+* (+#=('#*&. &,!& '*-'(!&# &,# !"#%!:#(,!%!(&#%'.&'(. 4+% "!%'!&'+*. '* &,# "#)+('&/ !*- '&. -'.$#%.'+*C
A(v) =
< ∆v >
1 < ∆v 2 >
, B(v) =
.
∆t
2
∆t
9+ +;&!'* &,# -%'4& !*- -'@5.'+* (+#=('#*&.6 &,# (,!*:# '* "#)+('&/ &,!& !* #)#(&%+*
5*-#%:+#. '&,'* .5;'*&#%"!) a .,+5)- ;# #.&'1!&#-6 !. -#.(%';#- !;+"#2 9'1# !"#%!:'*:
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R ωp (V )
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PQ
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! "#$%%!& '( )*++*"(, Pr (ω) = ω∞ fωp (ωp )dωp - ./! 0!&!#'%!1 "'2! 3'$&%'$&( $%(
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Pω (V ) %!&1( %*"'#1 %/! V$#'4 1!+%' );&4%$*&6 4*##!(7*&1$&0 %* ' /*3*0!&!*;( 7+'(3'
"/!#! %/! "'2! 4'& /'2! *&+9 ' NO!1 7/'(! 2!+*4$%96 V = ω/k0 -
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XJ
3
probability
2.5
2
1.5
1
0.5
0
5
6
7
8
9
10
velocity [vt ]
!"#$% &'() *+,-./%0 12 34% .$15,5!/!36 7!03$!5#3!18 2#893!180: Pω (V ): 21$ 34% ;,$<
!1#0 =#93#,3!18 /%;%/0' Pω (V )dV !0 34% .$15,5!/!36 34,3 , >,;% >!34 , $,3!1 12
ω/k0 = 7v
t >!// 4,;% , .4,0% ;%/19!36 V ,3 , "!;%8 !83%$;,/' Pω >,0 81$-,/!?%7 31
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1 #0!8") 0 Pω (V )dV = 1' @1/1$0 91$$%0.187 31 ;,$!1#0 =#93#,3!18 /%;%/0: ,0 21//1>0)
(hδn2 i)1/2 /N0 = 0.005 : 5/#%A (hδn2 i)1/2 /N0 = 0.01: "$%%8A (hδn2 i)1/2 /N0 = 0.015: $%7A
(hδn2 i)1/2 /N0 = 0.02: 96,8'
γ=π
nb
N0
v2
∂f (v)
∂v
.
v=ω/k
B&'CD
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"$1>34 $,3% 12 , >,;% 9,8 5% 9,/9#/,3%7 ,0 21//1>0)
nb
hγ(ωi )i = π
N0
Z∞
V2
∂f
Pω (V )dV.
∂V i
B&'ID
0
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#8!L#%/6 7%3%$-!8%7 .4,0% ;%/19!36: Vi = ωi /ki ,87 Pωi = δ(V − Vi ): >4%$% δ !0 34%
M!$,9 7%/3, 2#893!18' M%80!36 =#93#,3!180 9,#0% ;,$!,3!180 12 k ,/18" 34% .,34 12 >,;%
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C&
300
80
200
γ[πωp · 10 −5 ]
γ[πωp · 10 −5 ]
60
40
20
0
100
0
−100
−20
−40
5
10
−200
5
velocity[vt ]
10
15
20
velocity [vt ]
!"#$% &'() *%+, -./%0) 12.3-0%4 5+ ,6% "$57,6 $.,% 5+ 7.8%4 +5$ ,6% 8.$!5#4
9#:,#.,!5/ 0%8%04' ;!<%$%/, :505$4 :5$$%4-5/= ,5 ,6% =!<%$%/, 0%8%04> .4 +500574)
(hδn2 i)1/2 /N0 = 0.005> ?0#%@ (hδn2 i)1/2 /N0 = 0.01> "$%%/@ (hδn2 i)1/2 /N0 = 0.02> :A./@
(hδn2 i)1/2 /N0 = 0.04> 3."%/,.' B0.:C 0!/% :5$$%4-5/=4 ,5 ,6% "$57,6 $.,% 5?,.!/%= !/
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nb /N0 = 10−5 '
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I/ 5#$ 4!3#0.,!5/4> 7% #4% . 4%, 5+ 2000 7.8%4 7!,6 #/!+5$30A =!4,$!?#,%= -6.4%
8%05:!,!%4> Vi > !/ ,6% $./"% +$53 3vt ,5 40vt > !/ 5$=%$ ,5 :5/4,$#:, hγ(ωi )i .4 . +#/:,!5/
5+ ,6% 7.8% +$%J#%/:A' E6% 0%+, -./%0 5+ !"#$% &'( -$58!=%4 %2.3-0%4 5+ hγ(ωi )i +5$
8.$!5#4 0%8%04 5+ ,6% =%/4!,A 9#:,#.,!5/4> 5?,.!/%= ?A 3.C!/" #4% %J#.,!5/ K&'LM' E6%
?0.:C :#$8% :5$$%4-5/=4 ,5 ,6% 7.8% "$57,6 $.,% 5?,.!/%= +$53 ,6% D* .--$52!3.,!5/
#4!/" %J#.,!5/ K&'NM' E5 ,6!4 %/=> 7% #4% . F.#44!./ =!4,$!?#,!5/ 7!,6 ?%.3 8%05:!,A
vb = 6vt ./= ?%.3 ,6%$3.0 8%05:!,A ∆vb = 0.5vt .4 ,6% !/!,!.0 8%05:!,A =!4,$!?#,!5/
+#/:,!5/ +5$ ?%.3 %0%:,$5/4'
H/ !/:$%.4% !/ ,6% 0%8%0 5+ =%/4!,A 9#:,#.,!5/4> (hδn2 i)1/2 /N0 > 0%.=4 ,5 . =%:$%.4% !/
"
,6% 3.2!3#3 7.8% "$57,6 $.,%' 18%/ 9#:,#.,!5/4 7!,6 . 43.00 .3-0!,#=%> hδn2 i 1/2 =
0.005N0 > K?0#% :#$8% !/ !"#$% &'(M $%4#0, !/ . 4#?4,./,!.0 $%=#:,!5/ !/ hγ(ωi )i' E6#4>
5/% 465#0= %2-%:, ,6., ,6% :6.$.:,%$!4,!: ,!3% 5+ ,6% "$57,6 5+ . 7.8% 4!"/!O:./,0A
!/:$%.4%4 !/ . -0.43. 7!,6 $./=53 =%/4!,A 9#:,#.,!5/4' H/5,6%$ /5,.?0% %<%:, !4 .
46!+, !/ ,6% 3.2!3#3 "$57,6 $.,% !/ ,6% 8%05:!,A 4-.:% ,57.$= 057%$ -6.4% 8%05:!,!%4'
NP
!" #!$%& $# '(()*+',$"- ./ ' -"(0"'#" )% &!" 0"1$), 2!"0" &!" 10)2&! 0'&" $# +)#$&$3"4
5# ' 0"#67&8 ', $,(0"'#" $, &!" 7"3"7 )% -",#$&/ 96(&6'&$),# 0"-6("# &!" 3)76*" )% &!"
3"7)($&/ #+'(" 2!"0" 2'3"# ()67- 10)2 ":($",&7/4
5,)&!"0 $*+)0&',& +'0'*"&"0 &!'& ('6#"# ' (!',1" $, &!" 10)2&! 0'&" $# &!" ."'*
&!"0*'7 -$#+"0#$),8 ∆vb 8 )0 &!" 2$-&! )% &!" "7"(&0), 3"7)($&/ -$#&0$.6&$), %6,(&$),4
!" 0$1!& +',"7 )% ;$160" <4= +0)3$-"# &!" ">'*+7"# )% γ ).&'$,"- %0)* &!" ?@ '++0)>A
$*'&$), %)0 ."'*# 2$&! &!" #'*" vb %)0 -$B"0",& 3'76"# )% &!" &!"0*'7 -$#+"0#$),8 ∆vb 4
5# ">+"(&"-8 ', $,(0"'#" $, &!" &!"0*'7 -$#+"0#$), )% &!" ."'* 0"#67&# $, ' -"(0"'#" $,
&!" *'>$*6* 10)2&! 0'&"4 ;0)* "C6'&$), D<4EF8 %)0 &!" ('#" )% ' G'6##$', -$#&0$.6&$),
)% "7"(&0),#8 &!" #$*+7" 0"7'&$),#!$+ ."&2"", γ ',- ∆vb ()67- ." -"&"0*$,"-4 !"
2"77AH,)2, 0"7'&$),#!$+ %)0 H$,"&$( ."'*#8 γ ∼ (vb /∆vb )2 8 0"*'$,# 3'7$- %)0 )60 ('#"4
I"0"8 $& $# 2)0&! ,)&$,18 &!'& ' (!',1" $, &!" &!"0*'7 -$#+"0#$), )% ' ."'* -)"# ,)&
7"'- &) ' (!',1" $, &!" 0"1$), )% &!" 3"7)($&/ #+'(" 2!"0" 2'3"# (', 10)24
) "3'76'&"8 2!$(! C6',&$&/8 ∆vb /vb )0 (hδn2 i)1/2 /N0 8 $# *)0" $*+)0&',& %)0 &!"
+0)("## )% &!" 1","0'&$),8 2" ('7(67'&"- 10)2&! 0'&" )% &!" 2'3"# %)0 -$B"0",& 3'76"#
)% ∆vb /vb ',- (hδn2 i)1/2 /N0 4 J"#67&# #!)2, '& 7"%& +',"7 )% ;$160" <4< 2'# ).&'$,"%)0 ' ."'* 2$&! ."'* 3"7)($&/ vb = 6vt ',- -$B"0",& ∆vb D#!)2, 2$&! 3'0$)6# ()7)0#F4
;)0 *)#& )% ('#"#8 &!" 0'&$) ∆vb /vb $# 7'01"0 &!', (hδn2 i)1/2 /N0 8 ">("+& &!" ('#" )%
∆vb = 0.5vt D#!)2, 2$&! .7'(H ()7)0F4 5# )," (', #""8 %)0 &!$# ('#"#8 ', $,(0"'#" $,
&!" 7"3"7 )% &!" -",#$&/ 96(&6'&$), -)"# ,)& (!',1" &!" 10)2&! 0'&" #$1,$K(',&7/4 ;0)*
&!" )&!"0 !',-8 ', $,(0"'#" $, ∆vb 0"#67& $, #6.#&',&$'7 -"(0"'#" $, γ 4 !6#8 )," *'/
(),(76-"8 &!'& %)0 &!" #7)2 ',- 2$-" ."'*#8 +0"#",(" )% &!" -",#$&/ 96(&6'&$),# !'# ,)
#6.#&',&$3'7 $,96",(" ), &!" 1","0'&$), )% &!" @',1*6$0 2'3"4
L$--7" +',"7 )% ;$160" <4< #!)2# 0"#67&# %)0 ' ."'* 2$&! ."'* 3"7)($&/ vb = 12vt 4
5# +0"3$)6#7/8 ', $,(0"'#" $, (hδn2 i)1/2 /N0 0"#67&# $, -"(0"'#" )% 2'3" 10)2&! 0'&"4 5#
)," (', ,)&$("8 %)0 &!" ('#"# )% ∆vb /vb ≤ (hδn2 i)1/2 /N0 ', $,(0"'#" $, ∆vb 7"'-# &)
$,(0"'#" $, γ 4 ;)0 $,#&',("8 #&'0&$,1 2$&! (hδn2 i)1/2 /N0 > 0.28 ."'* 2$&! ∆vb = 2vt
D.76" 7$,"F !'# *'>$*6* $, γ 7'01"0 &!', ."'*# 2$&! ∆vb = 0.5vt ',- ∆vb = vt
D#!)2, 2$&! .7'(H ',- 0"- ()7)0# 0"#+"(&$3"7/F4 !6#8 &!" &!"0*'7 -$#+"0#$), )% &!"
."'*8 0"-6("# ":($",(/ )% &!" "B"(&# 0"7'&"- &) &!" -",#$&/ 96(&6'&$),#4
J$1!& +',"7 )% ;$160" <4< +0)3$-"# 0"#67&# %)0 ' ."'* 2$&! vb = 16vt 4 M& $# (7"'07/
#"", &!'& &!"0" $# ,) #$1,$K(',& -$B"0",(" $, ' 2'3"# 10)2&! 0'&"#8 &!'& ()00"#+),- &)
-$B"0",& ∆vb 4 5& &!" #'*" &$*"8 (!',1"# $, (hδn2 i)1/2 /N0 8 0"#67& $, #6.#&',&$'7 0"-6("
)% γ 4 ;)0 $,#&',("8 ', $,(0"'#" )% &!" 7"3"7 )% -",#$&/ 96(&6'&$),# %0)* 1% &) 2%8
0"-6("# &!" *'>$*6* )% &!" 10)2&! 0'&" *)0" &!', $, 10 &$*"#4 ;)0 &!" %'#& ."'*#8
2$&! ."'* 3"7)($&$"# !$1!"0 &!'& 15vt 8 &!" +0)("## )% 10)2&! ',- -"('/ )% &!" @',1*6$0
2'3"# $# ()*+7"&"7/ -"&"0*$,"- ./ &!" -",#$&/ 96(&6'&$),#4
EN
−1
1 2 3 4
2 1/2
(hδn i) /N0 · 10−2
−1.5
−1.5
1 2 3 4
2 1/2
(hδn i) /N0 · 10−2
1 2 3 4
(hδn i)1/2 /N0 · 10−2
2
!"#$% &'&( )*"+$!,-. */ ,-% .+0!.#. !1 ,-% +2%$+"%3 "$*4,- $+,% +5 + /#16,!*1 */
,-% 7%2%7 */ 3%15!,8 9#6,#+,!*15' :-% 3$!2%1 ;%+. -+5 + <+#55!+1 2%7*6!,8 3!5,$!;#,!*1
4!,- 3!=%$%1, ;%+. 2%7*6!,!%5( vb = 6vb > 7%/, ?+1%7@ vb = 12vb > .!337% ?+1%7@ vb = 16vb >
$!"-, ?+1%7' A*7*$5 6*$$%5?*13 ,* ,-% 2+$!*#5 ,-%$.+7 2%7*6!,!%5 */ ,-% ;%+.> +5 /*77*45(
∆vb = 0.5vt > ;7+6B@ ∆vb = vt > $%3@ ∆vb = 2vt > ;7#%@ ∆vb = 4vt > "$%%1'
!" #$%&'&()*+ *, (-$ $%$.(/*+ 0$&1 )+ (-$ 2%&31& 4)(5$+3)(6 78.(8&()*+3
:* 6*15!3%$ ,-% !1,%$+6,!*1 */ ,-% ;%+. 4!,- 5%2%$+7 .*1*6-$*.+,!6 4+2%5 4!,- %1%$"8
3%15!,!%5> Wi > +13 /$%C#%16!%5 ωi ,-% 6*1,$!;#,!*15 /$*. %+6- 4+2% !1 DC#+,!*1 E&'FG
5-*#73 ;% 5#..%3' *$ ,-!5 6+5%> ,-% 585,%. */ %C#+,!*15 6+1 ;% 4$!,,%1> +5 /*77*45(
K
∂f
∂ X
∂f
Wi Pωi (v) ,
=
v
∂t
∂v
∂v
E&'HG
i=1
+13
dWi
=
dt
Z∞
Wi V 2
∂f
Pω (V )dV.
∂V i
E&'IG
0
:* ;%,,%$ +3+?, %C#+,!*15 E&'HG +13 E&'IG /*$ ,-% 1#.%$!6+7 5!.#7+,!*15 !, !5 4*$,!1,$*3#6!1" ,-% 3!.%15!*17%55 2+$!+;7%5> +5 /*77*45( t̃ = (1/(πωp0 ))(N0 /nb )t> ṽ = vt v>
f˜ = f /vt > P˜ω (V ) = Pω (V )/vt > +13 W̃ = 2nb mvt2 W ' J% *.!,,%3 ,-% ,!73%5 ,* 5!.?7!/8
/*$. */ ,-% %C#+,!*15'
:-% 585,%. 6*15%$2%5 ,-% ,*,+7 %1%$"8> +5 /*77*45(

d 1
dt 2
Z∞
v 2 f (v)dv +
K
X
i=1
0

Wi  = 0.
E&'KLG
:-% M$5, ,%$. !1 %C#+,!*1 E&'KLG 6*$$%5?*135 ,* ,-% %7%6,$*1 %1%$"8' :-% 5%6*13
,%$. 6*$$%5?*135 ,* ,-% ,*,+7 %1%$"8 */ ,-% 4+2%5 E/*$ 3%,+!75 5%% N??%13!0 N'&G'
OO
!" #$%& &#'() *+ ,+"-!".+( /'.+"%012 &%.'21#%!/& !- #$+ &)&#+. '&%/3 +4'1#%!/&
56789 1/( 567:9 1/( ,"+&+/# "+&'2#& %/ #$+ /+;# &+0#%!/7 <+ '&+( 1 &+# !- 1 : 2000 *1=+&>
*%#$ ω/k0 '/%-!".2) (%&#"%?'#+( %/ #$+ "1/3+ -"!. 2vt #! 38vt 7 @! &!2=+ #$+ &)&#+. *+
1,,2%+( 1 A+1,-"!3 .+#$!(7 <+ '&+( B%.,&!/C& "'2+ %/ !"(+" #! !?#1%/ #$+ %/#+3"1#%!/
%/ +4'1#%!/ 567:97
D+&'2#& !- #$+ /'.+"%012 &%.'21#%!/ -!" (hδn2 i)1/2 /N0 = 0.02 1"+ ,"+&+/#+( %/ %3E
'"+ 67F7 <+ '&+ G1'&&%1/ (%&#"%?'#%!/ ?) +.,2!)%/3 #$+ ?+1. =+2!0%#) vb = 10vt 1/(
#$+ (%&,+"&%!/ ∆vb = 0.5vt 1& #$+ %/%#%12 0!/(%#%!/ -!" #$+ +2+0#"!/ (%&#"%?'#%!/ -'/0E
#%!/7 @$+ 01&+ ,"+&+/#+( 0!""+&,!/(& #! #$+ 0!/(%#%!/ (1/2)((hδn2 i)1/2 /N0 ) = vt2 /vb2 >
*$+/ (+/&%#) H'0#'1#%!/& ,21) 1 &%3/%I01/# "!2+ %/ #$+ "+21;1#%!/ ,"!0+&&7 J& 01/ ?+
&++/> ?+1. "+21;1#%!/ "+&'2#& %/ ,21#+1' -!".1#%!/ %/ #$+ =+2!0%#) "1/3+ !- v < vb > 1&
-!" 1 $!.!3+/+!'& ,21&.17 K/+ 01/ +;,+0# &'0$ 1 "+&'2#> &%/0+ -!" A1/3.'%" *1=+&
*%#$ 21"3+ k =+0#!" 51/( 0!""+&,!/(%/32) &.122 ,$1&+ =+2!0%#)9 #$1# &1#%&-) 0!/(%#%!/
vt2 /V 2 ≫ (1/2)((hδn2 i)1/2 /N0 ) #$+ %/H'+/0+ !- #$+ (+/&%#) H'0#'1#%!/& !/ ?+1. "+E
21;1#%!/ %& /+32%3%?2+7 L!*+=+"> #$+ /'.?+" !- ,1"#%02+& *%#$ =+2!0%#%+& v > vb 3"!*&
('"%/3 #$+ "+21;1#%!/ ,"!0+&&7 @$+ +/+"3) #"1/&-+" #! +/+"3+#%0 ,1"#%02+& %& ,!&&%?2+ ?+E
01'&+ H'0#'1#%!/& !- ,21&.1 (+/&%#) 0$1/3+ #$+ *1=+ ,$1&+ =+2!0%#) 1/( #$+ "+&!/1/#
0!/(%#%!/ -!" *1=+E,1"#%02+ %/#+"10#%!/&7 @$'&> 1 *1=+ 3+/+"1#+( *%#$ 1 ,$1&+ =+2!0%#)
!- V0 01/ ?+ "+1?&!"?+( ?) +2+0#"!/& *%#$ =+2!0%#%+& ve > V0 1/( +=+/ ve ≫ V0 7 @$+
"+1?&!",#%!/ !- 3+/+"1#+( *1=+& 12&! 2+1(& #! 1/ %/0"+1&+ %/ #$+ "+21;1#%!/ #%.+7
%3'"+ 67M ,"!=%(+& #$+ %/%#%12 +2+0#"!/ (%&#"%?'#%!/ -'/0#%!/> fin (v)> 1/( #$+ (%&E
#"%?'#%!/ -'/0#%!/> fend (v)> 1# #$+ +/( !- #$+ "+21;1#%!/ ,"!0+&&7 @! 0!/&%(+" #$+
%/H'+/0+ !- (+/&%#) H'0#'1#%!/& !/ #$+ 100+2+"1#%!/ !- ,1"#%02+&> *+ &+2+0#+( ,1"#%E
02+& *%#$ =+2!0%#%+& 21"3+" #$1/ vb + 3∆vb 7 @$'&> #$+ /'.?+" !- 100+2+"1#+( +2+0E
R
#"!/& *1& (+#+".%/+( 1& 1 (%N+"+/0+ ?+#*++/ #$+ %/#+3"12& v∞b +3∆vb fend (v)dv 1/(
R∞
vb +3∆vb fin (v)dv 7 O/ 1 &%.%21" .1//+"> #$+ +/+"3) !- #$+ 100+2+"1#+( ,1"#%02+& 01/
R
R
?+ -!'/( -"!. 21 [ v∞b +3∆vb v2 fend (v)dv − v∞b +3∆vb v2 fin (v)dv]7
@$+ 2+-# ,1/+2 !- %3'"+ 67P ,"!=%(+& #$+ "1#%! -!" #$+ +/+"3) !- 100+2+"1#+( ,1"#%02+&>
Ea > #! #$+ %/%#%12 +/+"3) !- #$+ ?+1.> Eb0 > -!" ?+1.& *%#$ (%N+"+/# ?+1. =+2!0%#%+& 1/(
(%N+"+/# 2+=+2& !- (+/&%#) H'0#'1#%!/&7 @$+ "%3$# ,1/+2 !- #$+ %3'"+ 67P ,"!=%(+& #$+
"+21#%=+ /'.?+"&> na /nb > !- 100+2+"1#+( +2+0#"!/&7 !" %/&#1/0+> -!" #$+ ?+1. =+2!0%#)
vb = 6vt 1/( (+/&%#) H'0#'1#%!/ 2+=+2 0.005> #$+ +/+"3) !- 100+2+"1#+( ,1"#%02+& %& 2+&&
#$1/ 0.1% -"!. #$+ %/%#%12 +/+"3) !- #$+ ?+1.7 !" #$+ &1.+ 2+=+2 !- H'0#'1#%!/& ?'# -!"
1 ?+1. *%#$ vb = 12vt #$+ "1#%! !- #$+ +/+"3) !- 100+2+"1#+( ,1"#%02+& %/0"+1&+& #! 29%7
J/ %/0"+1&+ %/ #$+ ?+1. =+2!0%#) 2+1(& #! 1/ %/+4'12%#) (1/2)((hδn2 i)1/2 /N0 ) ≫ vt2 /vb2
1/(> #$'&> "+&'2#& %/ 1 .!"+ +N+0#%=+ 100+2+"1#%!/ !- #$+ ,1"#%02+&7 B%.%21" "+&'2#& 01/
?+ !?#1%/+( ?) %/0"+1&%/3 #$+ 2+=+2 !- #$+ (+/&%#) H'0#'1#%!/&7 !" %/&#1/0+> -!" 1 ?+1.
*%#$ vb = 8vt #$+ /'.?+" !- 100+2+"1#+( ,1"#%02+& -!" (hδn2 i)1/2 /N0 = 0.01 %& 7%nb >
1/( 30%nb -!" 1 01&+ *%#$ (hδn2 i)1/2 /N0 = 0.0357 @$+"+-!"+> 122 01&+& 01/ ?+ &+,1"1#+(
PQ
distribution function
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
20
4
4
x 10
15
6
8
10
10
time[ ω1p ]
12
5
velocity [vt ]
!"#$% &'() *+,-#.!,/ ,0 .1% %-%2.$,/ 3!4.$!5#.!,/ 0#/2.!,/' 61% !/!.!7- 2,/3!.!,/ 0,$
5%78 %-%2.$,/4 !4 .1% 97#44!7/ 3!4.$!5#.!,/ :!.1 7 5%78 +%-,2!.; ,0 vb = 10vt 7/3 7
+%-,2!.; 3!4<%$4!,/ ,0 ∆vb = 0.5vt ' 61% 2,$% 3!4.$!5#.!,/= .17. 2,$$%4<,/34 ., .1% 2,-3
%-%2.$,/4= 174 /,. 5%%/ 2,/4!3%$%3' 61% 3!8%/4!,/-%44 %-%2.$,/ 3!4.$!5#.!,/ 0#/2.!,/ :74
/,$87-!>%3 ., 1' ?@%4) +%-,2!.!%4 /,$87-!>%3 ., .1% .1%$87- +%-,2!.; ,0 .1% 572A"$,#/3
<-7487 vt B .1% .!8% /,$87-!>%3 ., .1% 572A:7$3 <-7487 0$%C#%/2; (1/ωp )' 61% 5%78
3%/4!.; nb /N0 = 2.5 × 10−5 ' 61% -%+%- ,0 D#2.#7.!,/4 !4 (hδn2 i)1/2 /N0 = 0.02 E0$,8
F,4121%<;/%.4 %. 7-'= GHIJK'
LM
distribution function
1
vb
0.8
0.6
0.4
vb + 3∆vb
0.2
0
6
7
8
9
10
11
velocity [vt ]
0.8
0.4
0.6
0.3
na /nb
Ea /Eb0
!"#$% &'() *+% %,%-.$/0 1!2.$!3#.!/0 4#0-.!/0 5. .+% 3%"!00!0" /4 $%,565.!/0 73,#% ,!0%8
501 5. .+% %01 7"$%%0 ,!0%8' *+% ,%9%, /4 :#-.#5.!/0 !2 1%' *+% 3%5; 9%,/-!.< vb = 8vt =
501 .+% .+%$;5, 9%,/-!.< /4 .+% 3%5; !2 0.5vt > nb /N0 = 2.5 × 10−5 ' *+% ?,,%1 5$%5
-/$$%2@/012 ./ 5--%,%$5.%1 @5$.!-,%2 A4$/; B/2+-+%@<0%.2 %. 5,'= CDE(F'
0.4
0.2
0
0.2
0.1
0.01
0.02
0.03
0
0.04
(hδn2 i)1/2/N0
0.01
0.02
0.03
0.04
(hδn2 i)1/2/N0
!"#$% &'G) *+% %0%$"<= Ea = 7,%4. @50%,8 501 .+% 0#;3%$= na = 7$!"+. @50%,8 /4 5--%,%$5.%1
@5$.!-,%2 52 5 4#0-.!/0 /4 .+% 1%02!.< :#-.#5.!/0 ,%9%,' Eb0 501 nb 5$% .+% !0!.!5, %0%$"<
/4 .+% 3%5; 501 .+% 1%02!.< /4 .+% 3%5; %,%-.$/02' H/,/$2 -/$$%2@/01 ./ .+% 95$!/#2
9%,/-!.!%2 /4 .+% 3%5;= 52 4/,,/I2) vb = 6vt = 3,#%> vb = 8vt = "$%%0> vb = 12vt = $%1>
vb = 16vt = -<50> 501 vb = 20vt = ;5"%0.5' *+% .+%$;5, 9%,/-!.< /4 5,, 3%5;2 ∆vb = 0.5vt
A4$/; B/2+-+%@<0%.2 %. 5,'= CDE(F'
GJ
!"# "$% &#''#( !) "$*%% +',--%-.
/01
(1/2)((hδn2 i)1/2 /N0 ) ≪ vt2 /vb2 2
$#5#)%!%#6- 3',-5,7 86+"6," #!- "## -5,'' "#
(1/2)((hδn2 i)1/2 /N0 )
∼
vt2 /vb2 2
,!
!86%!+% "$% *%',9," #! 3*#+%--: /;1
!"%*5%4 ,"% *%) 5%7 +$,*,+"%* <%4 => , 3*%-%!+% =6"
+#!", ! !) , -5,'' ?6,!" "> #& ,++%'%*,"%4 3,*" +'%-: ,!4 /@1
vt2 /vb2 2
86+"6," #!- *%-6'" !)
((1/2)((hδn2 i)1/2 /N0 ) ≫
! ,! %A%+" B% %!%*)> "*,!-&%* "# =#"$ +#'4 ,!4 %!%*)%" +
3,*" +'%- ($%*% "$% %!%*)> #& "$% ,++%'%*,"%4 %'%+"*#!- +,! *%,+$ 63 "#
=%,5 %!%*)>C
D#"$ ?6,!" " %-7
!+*%,- !) 86+"6," #! '%B%'-C
4%"%*5 !%47 ,- &#''#(-.
+#**%-3#!4 !) "#
na /nb
,!4
Ea /Eb0 7
70% #& "$%
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E#* +,-%- #& "$% "$ *4 +',-- , - 53'% *%'," #!-$ 3 +,! =%
na /nb , Ea /Eb0 ∼
p
(hδn2 i)1/2 /N0 C
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(1/2)(hδn2 i)1/2 /N0
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vt2 /vb2 C
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ωp t
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0.4
12
11
log tr
Ew /Eb0
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10
9
8
0.01
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2 1/2
(hδn i)
7
0.04
−5
−4.5
2
/N0
log (hδn
−4 −3.5
i)1/2/N0
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tr ∼ ((hδn2 i)1/2 /N0 )2 '
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VI
0.4
0.3
0.5
Ew /Eb0
Ea /Eb0
0.6
0.4
0.3
0.2
0.2
0.1
0.1
0
0.01
0.02 0.03
2 1/2
(hδn i)
0
0.04
/N0
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0.02 0.03
2 1/2
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0.04
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1
0.9
0.8
0.7
tr (∆vb1 )
tr (∆vb2 )
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0.5
0.4
0.3
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tr
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Zt
0
v1 dt sin((kv0 − ω)t + φ).
>?4DA
E-11-$'(& '()!&"+)'-( '( >?4@A -3!" + .2+11 )'2! '()!"3+1 t → 0# )*! =".) )!"2 ,+(
%! "!$"'))!(# +. 0-11-$.8
F@
v1 = −
eE
1
[sin((kv0 − ω)t + φ) − sin(φ)].
m (kv0 − ω)
!"#$
%&'( )* )+ ,-+. *& /012
Zt
v1 dt =
0
eE
1
t
eE
[cos((kv0 − ω)t + φ) − cos(φ)] +
sin(φ).
2
m (kv0 − ω)
m (kv0 − ω)
3&44&')05 +67+*)*6*)&0 &8
Rt
0
v1 dt
)0 !"9$ &0, :-0 &7*-)0
k
dv2
e2 E 2
[cos((kv0 − ω)t + φ) − cos(φ)] sin((kv0 − ω)t + φ) +
=
2
dt
m (kv0 − ω)2
e2 E 2
kt
sin(φ) sin((kv0 − ω)t + φ).
m2 (kv0 − ω)
!";$
<. =,,>)05 *?, *,@A+ *& - +,:&01 &@1,@ &8 -::6@-:.( *?, ,B6-*)&0 8&@ :?-05,+ )0 *?,
,4,:*@&0 ,0,@5. :-0 7, '@)**,0 -+ 8&44&'+2
dv1
dv2
dǫ
= mv1
+ mv0
.
dt
dt
dt
D67+*)*6*)05
v1 ( dv1 /dt
-01
dv2 /dt
!"C$
)0 *& !"C$ &0, :-0 /012
e2 E 2
1
dǫ
=
[sin((kv0 − ω)t + φ) − sin(φ)] cos((kv0 − ω)t + φ) +
dt
m (kv0 − ω)
e2 E 2
vk
[cos((kv0 − ω)t + φ) − cos(φ)] sin((kv0 − ω)t + φ) +
m (kv0 − ω)2
vkt
e2 E 2
sin(φ) sin((kv0 − ω)t + φ).
m (kv0 − ω)
<. *-=)05 )0*& -::&60* *?-*
φ
)+ 0,:,++-@. *& -F,@-5, !"E$ &F,@
dǫ
dt
φ
=
!"E$
)+ - @-01&A F-@)-74, ')*? - 60)8&@A 1)+*@)76*)&0( )*
φ(
-+ 8&44&'+2
1
v0 k
e2 E 2
e2 E 2
sin((kv
−
ω)t)
−
sin((kv0 − ω)t) +
0
2
2
2m (kv0 − ω)
2m (kv0 − ω)2
vkt
e2 E 2
cos((kv0 − ω)t).
2m2 (kv0 − ω)
!"G$
H6, *& I6:*6-*)&0+ &8 *?, >4-+A- 1,0+)*.( *?, >?-+, F,4&:)*. &8 *?, '-F, :-0 ?-F,
F-@)&6+ F-46,+ -* 1)J,@,0* )0*,@F-4+" K)*? - >@&7-7)4)*. 1)+*@)76*)&0 860:*)&0
&0, :-0 -F,@-5, !"G$ &F,@
V(
-+ 8&44&'+2
L9
Pω (V )(
dǫ
dt
φV
e2 E 2
=
2m
Z∞
[
0
v
1
sin(k(v − V )t) −
sin(k(v − V )t) +
k(v − V )
k(v − V )2
vt
cos(k(v − V )t)]Pω (V )dV.
(v − V )
%&'& (& )*+,-.&' /0& 12'/-)3& &+,&453& (-/0 2 6-7&+ .-,/'-58/-*+ 98+)/-*+:
)02+6& -+ /0& &+&'6< *9 2 (27& *7&' /-4&
t
!"#$
f (v)"
;0&
-, &=823 /* /0& /*/23 )02+6& *9 /0& 12'/-)3&
&+&'6< 2/ 2 7*384& /2>&+ (-/0 24 *11*,-/& ,-6+: 2, 9*33*(,?
dW
= −nb
dt
Z∞ dǫ
dt
−∞
f (v)dv.
!"@$
φV
A02+6-+6 /0& *'.&' *9 /0& -+/&6'2/-*+ -+ !"@$ 2+. 42>-+6 2 ;2<3*' ,&'-&, &B12+,-*+
9*'
f (v):
C=82/-*+ !"@$ )2+ 5& '&('-//&+: 2, 9*33*(,?
dW
e2 E 2
=
nb
dt
2m
Z∞
Z∞ 2
V
sin((v − V )kt)
[ f (V )
dv +
ω
(v − V )2
0
−∞
V 2 ∂f (v)
|v=V
ω ∂v
Z∞
−∞
sin(k(v − V )t)
dv]Pω (V )dV.
v−V
;0& '&42-+-+6 /&'4, -+ /0& &B12+,-*+ 2'& 1'*1*'/-*+23 /*
t
!"DE$
2+. .* +*/ 42>& 2+<
)*+/'-58/-*+ /* /0& -+/&6'23," ;0& F',/ /&'4 -+ C=82/-*+ !"DE$ -, &=823 /* G&'* 5&)28,&
/0& -+/&6'2+. -, 2+ *.. 98+)/-*+" !9/&' ,-413& )23)832/-*+,: /0& ,&)*+. /&'4 )2+ 5&
('-//&+ 2,
2+.
ωp0 =
R∞
−∞ (sin(x)/x)dx = π " H< /2>-+6 -+/* 2))*8+/ /0& 92)/ /02/
4πe2 N0 /m: 9*33*(-+6 &=82/-*+ )2+ 5& *5/2-+&.?
dW
nb
= πωp0
W
dt
N0
Z∞
V2
W = E 2 /(8π)
∂f
Pω (V )dV.
∂V
!"DD$
0
!" #$% %&'()*+, -+. )$% %/%0).+, 1*2).*3')*+, -',0)*+,
%&'&: (& -+/'*.8)& 2 1'*525-3-/< .&+,-/< 98+)/-*+
U (v, t|v0 , t0 ) 9*' /0& &3&)/'*+ 7&3*)-/<"
;0& 98+)/-*+ 1'*7-.&, /0& 1'*525-3-/< /02/ 2 12'/-)3& (-/0 2 7&3*)-/<
*9 /-4&
t0
(-33 027& 2 7&3*)-/<
v
2/ /-4&
t
v0
-+ /0& 4*4&+/
2+. ,2/-,F&, /0& 9*33*(-+6 I*>>&'JK32+)>
&=82/-*+?
∂U (v, t|v0 , t0 )
∂
∂2
= − A(v)U (v, t|v0 , t0 ) + 2 B(v)U (v, t|v0 , t0 ),
∂t
∂v
∂v
#M
!"DL$
!"!# $%&'()*&+ A(V ) ,&- B(v) "!."!+!&( (/! ,0!",1!- '/,",'(!")+()'+ $*" 0,"),()*&+ )&
(/! 0!2*')(3 ,&- )(+ -)+.!"+)*&# ,+ $*22*4+5
A(v) =
B(v) =
hv2 iφV
,
t
1 hv12 iφV
.
2 t
4/!"! (/! 6",'7!(+ )&-)',(! ,0!",1)&1 *0!" φ ,&- V 8
9* -!:&! ,& !;%,()*& $*" (/! !2!'("*& -)+(")6%()*& $%&'()*& )( )+ &!'!++,"3 (* <%2().23
=>8?@A 63 f (v0 , t0 ) ,&- )&(!1",(! )( *0!" v0 # ,+ $*22*4+5
∂f (v, t)
∂
∂2
= − A(v)f (v, t) + 2 B(v)f (v, t).
∂t
∂v
∂v
=>8?BA
C%6+!;%!&( (* )&(!1",()*& )& =>8@A# *&! ',& 4")(! v2 # ,+ $*22*4+5
e2 E 2
k
(sin2 ((kv0 − ω)t + φ) − sin2 (φ) +
2m2 (kv0 − ω)3
2 cos(φ) cos((kv0 − ω)t + φ) + 2 sin(φ) sin((kv0 − ω)t + φ) − 2) −
kt
e2 E 2
sin(φ) cos((kv0 − ω)t + φ).
2
m (kv0 − ω)2
v2 (v, φ) =
D*4# )( )+ !,+3 (* -!(!"<)&! hv12 iφ ,&- hv2 iφ # ,+ $*22*4+5
hv2 iφ =
e2 E 2
k
kt
e2 E 2
(cos((kv
−
ω)t)
−
1)
+
sin((kv0 − ω)t),
0
2
3
2
m (kv0 − ω)
2m (kv0 − ω)2
hv12 iφ =
e2 E 2
1
(1 − cos((kv − ω)t)).
2
m (kv − ω)2
E3 (,7)&1 )&(* ,''*%&( (/,( t = a/v# A(v) ,&- B(v) ',& 6! 4")((!& ,+ $*22*4+5
e2 E 2
A(v) =
m2
Z∞
[
0
v
1
(cos((v − V )kt) − 1) +
sin((v − V )kt)]Pω (V )dV,
3
−V)
2k(v − V )2
ak 2 (v
e2 E 2
B(v) =
2m2
Z∞
0
v
(1 − cos((v − V )kt))Pω (V )dV.
ak 2 (v − V )2
!"!# )( )+ ,2+* 4*"(/ 4")()&1 (/! $*22*4)&1 !;%,()*& $*" ∂B(v)/∂v5
FG
e2 E 2
∂B(v)
=
∂v
2m2
Z∞
(
0
V
V +v
(cos((v − V )kt) − 1) +
sin((v − V )kt))Pω (V )dV.
ak 2 (v − V )3
kv(v − V )2
! "#$%&' "() *+," -)+%.#"%.) /0 "() ,)1/&- ")+2 /0 345#"%/& 6789:; #&- ,5<,"%"5"%&'
A(v)= B(v)= #&- ∂B(v)/∂v = /&) 1#& /<"#%& #& )45#"%/& 0/+ "() )>)1"+/& -%,"+%<5"%/&
05&1"%/& %& "() 0/>>/?%&' 0/+2@
e2 E 2 ∂
∂f
=−
[
∂t
2m2 ∂v
Z∞
0
1
(cos((v − V )kt) − 1)Pω (V )dV f (v) +
ak 2 (v − V )2
Z∞
0
Z∞
1
sin((v − V )kt)Pω (V )dV f (v) −
kv(v − V )
tk 2 (v
0
∂f
1
(1 − cos((v − V )kt))Pω (V )dV
].
2
−V)
∂v
B(+/5'( # B#!>/+ ,)+%), )CD#&,%/& /0 Pω (V ) #" V
%& 6789A;= #, 0/>>/?,@
Z∞
−Pω (v)
2
ω
0
1
(cos((V − v)kt) − 1)Pω (V )dV f (v) =
− V )2
kt
π
1
sin2 ((V − v) )dV f (v) = −Pω (v) f (v),
kt(V − v)2
2
2ω
Z∞
0
Pω (v)
1
ω
Z∞
0
0
2v
ω
Z∞
0
1
sin((v − V )kt)Pω (V )dV f (v) =
kv(v − V )
1
π
sin((V − v)kt)dV f (v) = Pω (v) f (v),
(V − v)
2ω
Z∞
Pω (v)
/&) 1#& %&")'+#") "() ")+2,
ak 2 (v
0
Z∞
= v=
6789A;
∂f
1
(1 − cos((v − V )kt))Pω (V )dV
=
tk 2 (v − V )2
∂v
kt
∂f
πv
∂f
1
sin2 ((V − v) )dV
=
Pω (v) .
2
tk(V − v)
2
∂v
2ω
∂v
70")+ ,5<,"%"5"%/& /0 "() +),5>", "/ 6789A;= /&) 1#& *&- "() *&#> )45#"%/& 0/+ "()
f (v, t) 05&1"%/&= #, 0/>>/?,@
?()+) W = E 2 /(8π)8
∂f
2π 2 e2 W ∂
∂f
=
vPω (v) ,
2
∂t
m ω ∂v
∂v
FE
6789E;
!" #$%&'()*+,$% -*.
!"#$%&'( )*+,,- #'. )*+,/- 0&'($%$"$1 $21 3#(%0 (4($15 &6 1!"#$%&'( 6&7 &"7 5&.18+
9& 5#:1 $2%( (4($15 5&71 0&';1'%1'$ 6&7 '"517%0#8 (%5"8#$%&'< 2171< =1 %'$7&."01
$21 6&88&=%'> .%51'(%&'81(( ;#7%#381(? t̃ = (1/(πωp0 ))(N0 /nb )t< ṽ = vt v< f˜ = f /vt <
P̃ω (V ) = Pω (V )/vt < #'. W̃ = 2nb mvt2 W + @1 &5%$ $%8.1( $& (%5A84 $21 6&75 &6 $21
1!"#$%&'(+ 921 (4($15 $#:1( $21 6&88&=%'> 6&75?
∂f
∂
∂f
=
vW Pω (v) ,
∂t
∂v
∂v
dW
=
dt
Z∞
∂f
Pω (V )dV.
∂V
WV 2
)*+,B-
)*+,C-
0
D'1 0#' 1#(%84 .1$175%'1 $2#$ $21 (4($15 2#( # 0&'(17;#$%&' 8#= 6&7 $&$#8 1'17>4<
#( 6&88&=(?
∂ 1
[
∂t 2
Z∞
1
v 2 f (v)dv + W ] =
2
Z∞
∂f
∂
vW Pω (v) dv +
∂v
∂v
Z∞
0
1
2
Z∞
v2
∂f (v)
∂W
dv +
=
∂t
∂t
0
WV 2
∂f
Pω (V )dV =
∂V
0
0
−
v2
Z∞
0
W v 2 Pω (v)
∂f
dv +
∂v
Z∞
WV 2
0
EB
∂f
Pω (V )dV = 0.
∂V
)*+,E-
!"#$%& '
!"#$#%&%'(%) *"+,& "#,$*./&$'*$ %0(,!$)(%"0 %0
!$0+"*&1 %02"*"3,0,"4' '"&$!
5%0+6
!" #$%&'()*%+'$
! "#$ %&$'()*+ ,#-%"$& .$ %&)%)+$/ - +$012,)!+(+"$!" %&)3-3(0(+"(, 4)/$0 "#-" /$+,&(3$+
3$-42%0-+4- (!+"-3(0("5 (! - %0-+4- .("# &-!/)4 /$!+("5 6*,"*-"()!+7
! ,)!"&-+" ")
"#$ 4)/$0 %&)%)+$/ 35 8(+#(9-.- -!/ :5*")' ;<=>?@A /$!+("5 6*,"*-"()!+ .$&$ +*%2
%)+$/ ") 3$ #(B# $!)*B# ") ,-*+$ +(B!(C,-!" ,#-!B$+ (! "#$
k 4-B!("*/$ (! "#$ /(&$,"()!
)1 .-'$ %&)%-B-"()!7 D+ - &$+*0"A "#$ .-'$ %#-+$ '$0),("5 *!/$&B) ,#-!B$+A -00).(!B
"#$ .-'$ ") &$+)!-!"05 (!"$&-," .("# 3$-4 $0$,"&)!+ "#-" #-'$ /(E$&$!" '$0),("($+ .("#(!
- F*("$ 0-&B$ &-!B$7 D! -++*4%"()! "#-" "#$ %#-+$ '$0),("5 (+ - &-!/)4 F*-!"("5 "#-"
)3$5+ - %&$/$"$&4(!$/ /(+"&(3*"()! -00).+ )!$ ") /$+,&(3$ "#$ $!$&B5 $G,#-!B$ 3$".$$!
"#$ .-'$+ -!/ "#$ 3$-4 (! "$&4+ )1 -! -'$&-B$/ (! "#$ '$0),("5 +%-,$ B&)."# &-"$ )1 "#$
.-'$+A
hγiV
hγiV A
-!/
-!/ +(4(0-&05 -'$&-B$/ $0$,"&)! '$0),("5 /(E*+()! ,)$H,($!"A
hDiV
hDiV 7
I#$
/$%$!/ )! "#$ /(+"&(3*"()! 1*!,"()! 1)& "#$ .-'$ %#-+$ '$0),("5 "#-"
(+ /$"$&4(!$/ 35 "#$ /(+"&(3*"()! 1*!,"()! )1 "#$ /$!+("5 6*,"*-"()!+7 I#$ 4)/$0 .-+
(!("(-005 -%%0($/ 1)& - ,-+$ .("# - !)&4-0 /(+"&(3*"()! 1)& "#$ /$!+("5 6*,"*-"()!+7 I#(+
-00).$/ *+ ") (!'$+"(B-"$ #). 9$5 %-&-4$"$&+ )1 "#$ &$0-G-"()! %&),$++ /$%$!/ )! "#$
0$'$0 )1 /$!+("5 6*,"*-"()!+ -!/ )! "#$ (!("(-0 '$0),("5 )1 "#$ 3$-47
" (+ .)&"# !)"(!B
"#-" "#$ &$+*0"+ )3"-(!$/ (! "#$ 4)/$0 -&$ (! - B))/ -B&$$4$!" .("# .$-9 "*&3*0$!,$
"#$)&57 I#$ B)-0 )1 "#$ %&$+$!" ,#-%"$& (+ ") /$"$&4(!$ .#-" "5%$ )1 /(+"&(3*"()! &$%&$2
+$!"+ "#$ 3$+" C" 1)& )3+$&'$/ /$!+("5 '-&(-"()!+ (! "#$ +)0-& .(!/ %0-+4- -!/ #). ("J+
1)&4 -E$,"+ "#$ 3$-4 &$0-G-"()! %&),$++7 K$ +#-00 -0+) ,)4%-&$ "#$ &$+*0"+ )3"-(!$/
.("# "#)+$ 1)& "#$ !)&4-005 /(+"&(3*"$/ /$!+("5 6*,"*-"()!+7
! "#$ %&$+$!" ,#-%"$& .$ %&)%)+$ - "$,#!(F*$ 1)& $'-0*-"(!B "#$ /(+"&(3*"()! 1*!,2
L>
!"# "$ %&#'! ( )*+ *, !"#' *'!#- %&#'! ( )*+ *, !"#' ". ,!#&% $/"0 0&,'*/&0&# '
"#.",/% ', &11! &'2 3& *'& 4& 5&,/'"# &+4#!6*& $"/ +1,''!$(!#- %!7&/&# %!' /!.* !"#'
85&,/'"#9 :;<=> !# "/%&/
" %&?#&
4&
(@& "$ %!' /!.* !"# +"//&'@"#%!#-
"
4& ".A
'&/B, !"#'2 C4& 5&,/'"# +1,''!?+, !"# ,11"D' "#& " ". ,!# ,# ,#,1( !+,1 $"/0 $"/ 4&
%!' /!.* !"# $*#+ !"# "$ %&#'! ( )*+ *, !"#' $"/ ".'&/B&% %!' /!.* !"#' %&@&#%!#- "#
$"*/ ' , !' !+,1 @,/,0& &/'9 #,0&1(9
4& 0&,# B,1*&9
4& B,/!,#+&9
4& 'E&D#&''9 ,#%
4& E*/ "'!'2 F'!#- 4& 5&,/'"# %!' /!.* !"# (@&9 D& %& &/0!#& 4& %!' /!.* !"# $*#+A
!"# "$ D,B& @4,'& B&1"+! !&' ,#% ,@@1( 4!' %!' /!.* !"# !# "*/ 0"%&1 $"/ .&,0 @1,'0,
!# &/,+ !"#2 G!#,11( D& ". ,!# 4& /&'*1 ' "$ 4& #*0&/!+,1 '!0*1, !"#' "$ 4& /&1,H, !"#
"$ 4& &1&+ /"# .&,0' 4,B!#- %!7&/&# .&,0 B&1"+! !&'9 4&/0,1 '@/&,% ,#% , %!7&/&#
1&B&1' "$ 4& %&#'! ( )*+ *, !"#' !# 4& @1,'0, *#%&/ +"#%! !"#' '!0!1,/ " 4"'& !# 4&
'"1,/ D!#%2
!" #$%&'() *+,(+-('.%& '% (/$ &.0-1 2'%3
I@&+ /*0 "$ 4& %&#'! ( )*+ *, !"#' !# 4& '"1,/ D!#% 4,B& .&&# ". ,!#&% $/"0 4& !#
'! * '@,+&+/,$ 0&,'*/&0&# ' 8J&*-&.,*&/9 :<K=> ,#% @4,'& '+!# !11, !"#' "$ 4& '!-#,1'
/,#'0!
&% $/"0 ', &11! &
" L,/ 4 83"" ,#% M/0' /"#-9 :<K<> ,1/&,%( !#
70A
!&'2
N&1#!E!&/ & ,12 8:<;O> ,#,1(P&% %, , $/"0 QILL @/"@,-, !"# &H@&/!0&# ' ,#% /&@"/ &%
4,
4& '@&+ /,1 %&#'! ( "$ )*+ *, !"#'9
@"D&/ 1,D2 Q#
Wk 9
+,# .& @/&'&# &% !# 4& $"/0 "$ , %"*.1&A
4& 1"D $/&6*&#+( R1"#- D,B&1&#- 4'S @,/
"$
4& '@&+ /*09 !
+,# .&
,@@/"H!0, &% 6*! & D&11 *'!#- 4& T"10"-"/"B @"D&/ 1,D9 ,' $"11"D'U
Wk ∼
Q#
∆n
N0
2
∼ k −5/3 .
4& 4!-4&/ $/&6*&#+( /,#-& R'4"/ &/ D,B&1&#- 4'S
4& @"D&/ 1,D D,' &B,1*, &%
"
4,B& %!7&/&# '@&+ /,1 !#%&H9 ,' $"11"D'U
Wk ∼ k −ν , ν ≃ 0.64 ± 0.01.
V&/&
N0
!' 4& .,+E-/"*#% @1,'0, %&#'! ( ,#%
∆n !'
C4& /,#'! !"# "++*/' , , $/&6*&#+( ,/"*#% 6 × 10−2
4& ,0@1! *%& "$ 4& )*+ *, !"#'2
Hz 2
V&/&9 ! !' D"/ 4 0&# !"#!#-
4, !# 4!' $/&6*&#+( /,#-& 4& D,B& %!'@&/'!"# !' ' !11 #&-1!-!.1& ,#% 4,
4& /&1, !"#A
'4!@ .& D&&# $/&6*&#+!&' ,#% D,B&1&#- 4' +,# .& &' ,.1!'4&% .( ,''*0!#- 4& C,(1"/
4(@" 4&'!'2 C4& /&'*1 D,' ". ,!#&% .( N&1#!E!&/ & ,12 8:<;O>9 0,E!#- *'& 4& &+4#!6*&
"$ ,+ !B& '"*#%!#- .& D&&# D" ', &11! &'2 J" &D"/ 4(9 '@&+ /,1 !#%!+&' %&@&#% "# ,#
,#-1& .& D&&# '"1,/ D!#% ,#% , .,+E-/"*#% 0,-#& !+ ?&1% 8G"/0,# &
,129 WX::>2 Q#
'"0& +,'&' 4& 1"D $/&6*&#+( @,/ "$ 4& %&#'! ( )*+ *, !"#' '@&+ /*0 !' 0&,'*/&% "
.& ' &&@&/ 4,#
−5/3
8N&1#!E!&/ & ,129 :<;K>2 Y&+&# 1(9 , '!0!1,/ '@&+ /*0 D,' %& &/A
0!#&% "#.",/% N1*' &/ 8T&11"-- ,#% V"/.*/(9 WXX=>9 M/ &0!' 8N4&# & ,129 WX:W> ,#%
I@&E /AY 8Z,$/[#E"B[ & ,129 WX:=>2
;;
spectrum[cm−6 s−1 ]
0.5
0
Wk ∼ k −5/3
−0.5
−1
Wk ∼ k −0.64
−1.5
−2
−2.5
−3
−2
−1.5
−1
−0.5
0
0.5
1
frequency [s ]
−1
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GF
1.15
1.15
1.1
1.1
1.05
1.05
1
1
0.95
0.95
0.9
0.9
0.85
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
V /V 0
N/N 0
density variation
phase velocity variation
0.85
time [s]
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x−a
dp(x)
p(x),
=
dx
b0 + b1 x + b2 x2
H:*$*
a
&!,
bi
&$* (#!.'&!' I&$&%*'*$. #" ':* , .'$ )+' #!>
12
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#" ':* , .'$ )+' #!
p(x)
µ1 Lµ4 5
&. "#33#H.D
a = b1
µ2 (4µ2 µ4 − 3µ23 )
b0 =
10µ2 µ2 − 18µ32 − 12µ23
−µ3 (µ4 + 3µ22 )
b1 =
10µ2 µ2 − 18µ32 − 12µ23
−2µ2 µ4 + 6µ32 + 3µ23
b2 =
10µ2 µ2 − 18µ32 − 12µ23
EF><G
M!&3A' (&3 .#3+' #! #" *C+&' #! F>6 "#$ ':* (&.* #" (*!'*$*, , .'$ )+' #! (&! )*
#)'& !*, &. "#33#H.D
p(x) = C exp (φ(x))
EF>OG
H:*$*
φ(x) =
Zx
0
M. (&! )* .**!5 ':* "*&'+$*. #" ':*
*C+&' #!D
y − b1
dy
b0 + b1 y + b2
φ(x)
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b0 + b1 y + b2 y 2 = 0>
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)*'&.5
β1
&!,
β2
E.:#H!
! & "#$% #" , &K$&%
β1 =
! ':* '*$%. #" ?*&$.#!P.
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µ23
, β2 = 2
2
µ2
µ2
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FC
β1
=
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Ks
Kb
A
50
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σ-
3%
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2&5/0% B1B C=,/% '/0<%D1 :$% .=#)&"%( (& #0&=/#&." 3) '$)0)'#%0&9%( / &"5 )" )<%0)5%(
β1 = 2 · 10−6
)"(
β2 = 2.861
;''.0(&"5 #. #$% E%)0 ." ',)
&F')#&." GE%)0 ." HIJKL-
#$% (& #0&=/#&." '.00% +."(%( #. ) :!+% MM E%)0 ." (& #0&=/#&."1 :$& (& #0&=/#&." &
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0% /,# &" #$% %4/)#&." B1P ) 6.,,.3 N
m
p(x) = Ac 1 − x2 ab b
3$%0%
ab = 2
)"( 6)'#.0
Ac ')" =% %
µ4
µ2
5µ4
3 − µ4
µ4
/ 2
,
/ 3 − 2 , mb =
µ2
µ22 − 9
µ22
#&*)#%( 60.* #$% ".0*),&9)#&." .6
p(x) #. ."%1
M# & 3.0#$ ".#&"5
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β1 = 0
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β2 = 3
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Kb
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(N f (xi ) − hi )2
χ2ν
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χ2
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ν = K b − K f − 1-
(& #0&=/#&." )"(
&" #$% F##%( 6/"'#&."1 M" %4/)#&." B1K
JP
xi
&
Kf
3$%0%
ν
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#$% '%"#%0 .6 %)'$ =&"-
hi
&
800
700
160
1
540
140
520
2
number
600
500
400
120
500
2
100
−1.8
−0.2
−1.7
P(δ n)
Pearson type II
Gaussian
300
200
0.2
3
10
5
1
2.7
100
0
−5
0
15
−1.6
2.8
2.9
3
−4
−3
−2
−1
0
1
2
3
4
5
density fluctuations
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P
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%#) 1(# ,)1%#!',/1' ( +,%&(&"-"'. *"!',"&/'"%# 0/#1'"%# 0%, '2) +2(!) 5)-%1"'")! %0 '2)
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'2) 0%--%8"#$ !"#$-) 5(-/) 0/#1'"%#(- *)+)#*)#1) 0%, '2) +-(!7( 0,);/)#1. %# +-(!7(
*)#!"'.< ωp2 (δn) = 4πe2 (N0 + δn)/m9 =2/!4 ( *"!',"&/'"%# 0/#1'"%# 0%, '2) +-(!7(
0,);/)#1")! Pω (ωp ) 1%/-* &) 0%/#* /!"#$ ( /#";/) ,)-('"%#!2"+< '2) +,%&(&"-"'. '% >#*
+-(!7( 0,);/)#1. -)!! '2(# ωp (N0 + δn) "! );/"5(-)#' '% +,%&(&"-"'. '% >#* '2) *)#!"'.
3/1'/('"%# -)!! '2(# δn<
p
Pωp (ωp )dωp = Pδn (δn(ωp ))dδn = Pδn (δn(ωp ))
∂δn
dωp .
∂ωp
=2)#4 /!"#$ '2) *"!+),!"%# ,)-('"%#!2"+ 0%, 6(#$7/", 8(5)!4 %#) 1(# %&'("# '2)
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ω 2 = ωp2 (δn)(1 + 3vt2 /V 2 )9 ?. '(@"#$ "#'% (11%/#' '2(' '2) 0,);/)#1. %0 '2) 8(5) "!
1%#!'(#'4 '2) +,%&(&"-"'. *"!',"&/'"%# 0/#1'"%# %0 '2) +2(!) 5)-%1"'")! PV (V ) 7(. &)
*)'),7"#)* "# ( !"7"-(, 7(##),4 (! 8(! *%#) 0%, Pω (ωp )<
p
PV (V )dV = Pωp (ωp (V ))
∂ωp
dV.
∂V
A' "! 8%,'2 ,)7"#*"#$ '2(' PV (V )dV "! '2) +,%&(&"-"'. %0 >#*"#$ ( 8(5) 8"'2 (
+2(!) 5)-%1"'. );/(- '% V (' %#) %0 '2) )*$)! %0 '2) !/&"#'),5(-9 =2) +,%&(&"-"'. %0
>#*"#$ ( 8(5) 8"'2 ( +2(!) 5)-%1"'. );/(- '% !%7) 5(-/)4 Vi 4 "# !%7) +%"#' "#!"*)
!/&"#'),5(-4 Pω (Vi )4 1(# &) %&'("#)* &. 7(@"#$ /!) %0 '2) +,%1)*/,) *)!1,"&)* "# '2)
+,)5"%/! 12(+'),9
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+-(!7(F4 82),) vt "! '2) '2),7(- 5)-%1"'. %0 '2) &(1@$,%/#* +-(!7(4 0%, 1(!)! 8"'2
*"G),)#' -)5)-! %0 *)#!"'. 3/1'/('"%#!4 (hδn2 i)1/2 /N0 E!2%8# 8"'2 1%-%,!F9 H%-"* -"#)!
1%,,)!+%#* '% '2) Pω (V ) %&'("#)* 0,%7 '2) I)(,!%# '.+) AA *"!',"&/'"%# %0 '2) *)#!"'.
3/1'/('"%#!4 82"-) *(!2)* -"#)! 1%,,)!+%#* '% Pω (V ) 1(-1/-(')* 0,%7 '2) #%,7(- *"!',"J
&/'"%#9 K! "#*"1(')*4 &%'2 '.+)! %0 Pδn (δn) ,)!/-' "# ( !"7"-(, 0%,7 0%, '2) +,%&(&"-"'.
*"!',"&/'"%# 0/#1'"%# %0 '2) 8(5) +2(!) 5)-%1"'")!9 A# &%'2 1(!)!4 Pω (V ) !2%8! ( !"7"-(,
&)2(5"%, 8"'2 (# "#1,)(!"#$ -)5)- %0 *)#!"'. 3/1'/('"%#!9 K# "#1,)(!) "# '2) 7($#"'/*)
%0 '2) *)#!"'. 3/1'/('"%#! 1(/!)! 8"*), &,%(*)#"#$ "# '2) +,%&(&"-"'. *"!',"&/'"%# 0/#1J
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!"$#">1(#' 12(#$)! "# '2) 8(5) +2(!) 5)-%1"'.9 K! ( ,)!/-'4 Pω (V )4 0%, ( 1(!) %0 *)#!"'.
3/1'/('"%#! 8"'2 '2) #%#JL(/!!"(# *"!',"&/'"%#4 "! 12(,(1'),"M)* &. 8"*), &,%(*)#"#$
NC
probability
1.5
1
0.5
0
5
6
7
8
9
10
velocity [vt ]
!"#$% &'() *+,-./%0 12 34% .$15,5!/!36 7!03$!5#3!18 2#893!180: Pω (V ): 21$ ;,$!1#0 <#9=
3#,3!18 /%;%/0' Pω (V )dV !0 34% .$15,5!/!36 34,3 , >,;% >!34 , $,3!1 12 ω/k0 = 7vt
>!//
R ∞ 4,;% , .4,0% ;%/19!36 V ,3 , "!;%8 !83%$;,/' Pω >,0 81$-,/!?%7 31 18% #0!8")
0 Pω (V )dV = 1' @1/1$0 91$$%0.187 31 34% ;,$!1#0 <#93#,3!18 /%;%/0: ,0 21//1>0)
(hδn2 i)1/2 /N0 = 0.005 : $%7A (hδn2 i)1/2 /N0 = 0.01: 5/#%A (hδn2 i)1/2 /N0 = 0.015: "$%%8A
(hδn2 i)1/2 /N0 = 0.02: -,"%83,' B1/!7 /!8%0 91$$%0.187 31 7%80!36 <#93#,3!180 153,!8%7
#0!8" 34% C%,$018 36.% DD 7!03$!5#3!18 ,87 34% 7,04%7 /!8% 31 7%80!36 <#93#,3!180 15=
3,!8%7 #0!8" 34% 81$-,/ 7!03$!5#3!18'
12 34% $%"!18 12 $%018,89%' E4#0: 21$ 34!0 9,0%: 34% %F%930 12 7%80!36 !841-1"%8%!3!%0 !8
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34% 7!F%$%89% 5%3>%%8 Pω (V ): 91$$%0.187!8" 31 34% H,#00!,8 ,87 818=H,#00!,8 7!03$!=
5#3!180 12 34% 7%80!36 <#93#,3!180: 7%9$%,0% >!34 ,8 !89$%,0% !8 34% /%;%/ 12 34% 7%80!36
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34,8 34!0 /%;%/ J34% "$%%8 ,87 -,"%83, /!8%0 !8 !"#$% &'(K: 81 0!"8!L9,83 7!F%$%89%
199#$$%7 5%3>%%8 34% .$15,5!/!36 7!03$!5#3!18 2#893!180 12 .4,0% ;%/19!36 21$ 81$-,/
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!" #$% %&'()*+', '- ./,01)+2 3/&%4
E1 %;,/#,3% 34% !-.1$3,89% 12 34% 818=H,#00!,8 21$- 21$ , 7!03$!5#3!18 12 7%80!36
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P
12 34% 313,/ %8%$"6 7%80!36 12 >,;%0: Wt = Wi : "%8%$,3%7 56 ,8 %/%93$18 5%,- >!34
,8 !8!3!,/ H,#00!,8 ;%/19!36 7!03$!5#3!18 2#893!18 21$ vb = 6vt ,87 ∆vb = vt ' E4% /%;%/
12 34% 7%80!36 <#93#,3!180 (hδn2 i)1/2 /N0 >,0 9410%8 31 5% 0.005' E4% %8%$"6 7%80!36
ON
0.1
0.05
0.1
Wt /Eb0
0.08
Wt /Eb0
Wt /Eb0
0.15
0.06
0.04
0.05
time
10
[ ωπp
20
4
· 10 ]
0.03
0.02
0.02
0
0.04
0
10 20 30
time
[ ωπp
4
· 10 ]
0.01
0
10 20 30
time [ ωπp · 10 4 ]
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.!,/' B%0. ;2/%-) (1/2)(hδn2 i)1/2 /N0 = 0.0057 vb = 6vt 7 ∆vb = vt ' C!44-% ;2/%-)
(1/2)(hδn2 i)1/2 /N0 = 0.027 vb = 10vt 7 ∆vb = vt ' D!"1. ;2/%-) (1/2)(hδn2 i)1/2 /N0 =
0.037 vb = 16vt 7 ∆vb = vt ' E-- ,0 .1% ;2/%-5 ;$,+!4% .1% 62+% %/%$"3 4%/5!.3 6!.1
$%5;%:. ., .1% :,$$%5;,/4!/" !/!.!2- %/%$"3 4%/5!.3 ,0 .1% =%2<7 Eb0 '
,=.2!/%4 0,$ .1% F2#55!2/ 4%/5!.3 9#:.#2.!,/5 4!5.$!=#.!,/ !5 51,6/ !/ =-#%7 61!-% .1%
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-,6' ,$ =,.1 :25%57 Wt 51,65 43/2<!:5 5!<!-2$ ., .1,5% 0,$ 2 1,<,"%/%,#5 ;-25<27
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2;;$,P!<2.%-3 $%2:1%5 15% ,0 .1% !/!.!2- %/%$"3 ,0 .1% =%2<7 61!-% :,/+%/.!,/2- RB
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9#:.#2.!,/5 -%+%- (hδn2 i)1/2 /N0 = 0.027 2$% ;$%5%/.%4 !/ .1% <!44-% ;2/%- ,0 !"#$%
&'(' ,$ .1!5 :25%7 .1% .%$< ;$,;,$.!,/2- ., (3/2)(vt /vb )2 !5 5-!"1.-3 2=,+% .1% .%$<
(1/2)(hδn2 i)1/2 /N0 ' O1#57 .1% ;$%5%/:% ,0 4%/5!.3 9#:.#2.!,/5 51,#-4 12+% 2 <,$%
5!"/!N:2/. %8%:. ,/ .1% %+,-#.!,/ ,0 62+%5' E5 0,$ 2 ;$%+!,#5 5.#437 .1% %/%$"3 4%/5!.3
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%/%$"3 4%/5!.37 Wt 7 =%"!/5 ., 4%:$%25%' E 4%:23 !/ 62+% %/%$"3 :2/ =% %P;-2!/%4 !/
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vb = 16vt
<#$ 0 8#" %&!$) *- = 850$ A6B %0*( 2$' "#$ 0$'5)"' -*0 & 4$&1 . "#
∆vb = vt 6
<#$ )$($) *- "#$ 2$!' ", ?5+"5&" *!'
(1/2)(hδn2 i)1/2 /N0
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2$!' ", *4"& !$2 &" "#$ $!2 *- 0$)&H&" *!6
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"#$ 2$%$!2$!+$ *- 1&H 151
0$)&H&" *! %0*+$''/
Wmax /
<#$ 0 8#" %&!$) *- = 850$ A6K %0$'$!"'
! "*"&) .&($ $!$08, 2$!' ", "#&" .&' 0$&+#$2 250 !8 "#$
&' & -5!+" *! *-
(hδn2 i)1/2 /N0 6
L$0$/ .$ '#*. "#$ 0$'5)"' *-
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& 2$+0$&'$ !
Wmax 6
L*.$($0/ "#$ ! " &) "#$01&) 2 '%$0' *! *- "#$ 4$&1 0$25+$' $3$+"'
0$)&"$2 "* "#$ 2$!' ", ?5+"5&" *!'6 =*0 & +&'$ . "# ')*. &!2 . 2$ 4$&1' M
∆vb > vt N/
"#$ %0$'$!+$ *- 2$!' ", ?5+"5&" *!'
vb < 10vt
&!2
! & %)&'1& 2*$' !*" +&5'$ &!, !*"&4)$
$3$+"'6 =*0 -&'" 4$&1' . "# ($)*+ " $' )&08$0 "#&!
10vt / "#$ +#&!8$
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!+0$&'$
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(hδn2 i)1/2 /N0
0$'5)"'
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OP
!"0*25+$2 "#$ -*))*. !8 :5&!" ",
0.01
5
0.008
4
max
log( W
2Win )/tgr [πωp ]
2Wmax /mN0 vt2
−5
0.006
0.004
0.002
0
x 10
3
2
1
1
!
2
3
0
4
" 1/2
/N0 · 10 −2
hδn i
2
1
!
2
3
4
" 1/2
/N0 · 10 −2
hδn i
2
!"#$% &'() *+% ,-.!,#, /0$ 1-2% 303-4 %5%$"6 7%58!369 Wmax 9 $%-:+%7 7#$!5" 3+%
$%4-.-3!05 ;$0:%88 <4%/3 ;-5%4= -57 3+% %>%:3!2% "$013+ $-3% 0/ 1-2%8 <$!"+3 ;-5%4= -8
- /#5:3!05 0/ 3+% 4%2%4 0/ 7%58!36 ?#:3#-3!058' Wmax !8 ;$02!7%7 1!3+ $%8;%:3 30 3+%
%5%$"6 7%58!36 0/ 3+% @-:A"$0#57 ;4-8,-' Win -57 tgr !5 3+% $!"+3 ;-5%4 $%;$%8%53 3+%
!5!3!-4 303-4 %5%$"6 7%58!36 0/ 1-2%8 -57 3+% ;%$!07 0/ 3!,%9 02%$ 1+!:+ 3+% 1-2% %5%$"6
7%58!36 +-8 "$015 /$0, Win 30 Wmax 9 $%8;%:3!2%46' B%8#438 -$% ;$02!7%7 /0$ 2-$!0#8
;-$-,%3%$8 0/ 3+% @%-,' C040$8 :0$$%8;057 30 3+% 2-$!0#8 2%40:!3!%8 0/ 3+% @%-,9 -8
/044018) vb = 6vt 9 @4#%D vb = 10vt 9 $%7D vb = 16vt 9 ,-"%53-' E!>%$%53 ,-$A8 :0$$%8;057
30 2-$!0#8 3+%$,-4 7!8;%$8!058 0/ 3+% @%-,9 -8 /044018) ∆vb = 0.5vt F -83%$!8AD ∆vb = vt F
:!$:4%D ∆vb = 3vt F 7!-,057'
GG
!" #!"$"# %$&'%( !% ")%$"*% &+% ,-$ !% *$-. ! -, .")%(/
.!%$%
Win
"01
Wmax
&01&#" %
γef = log(Wmax /Win )/tgr
!% &0& &"2 "01 +"3&+4+ )"24%( -,
!%
- "2 .")% %05
%$*6 1%0(& 67 $%(8%# &)%267 "01 tgr &( !% &+% 14$&0* .!&#! !% %0%$*6 1%0(& 6 -, .")%(
*$-.( ,$-+
Win
(hδn2 i)1/2 /N
-
Wmax 9
:!% $&*!
8"0%2 -, ;&*4$% <9= (!-.(
vb
0 -> "&0%1 &0 (&+42" &-0( ,-$ >%"+( .& ! 1&?%$%0
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%A#&%0 26 !"0 0"$$-. >%"+( B.& !
∆vb ≤ vt C9
γef
"( " ,40# &-0 -,
"01
∆vb 9
∆vb ≥ 3vt
@ 1%#$%"(% &0
γef 7
"0 &0#$%"(% &0 !% 2%)%2 -, !% 1%0(& 6 D4# 4" &-0(9 E-.%)%$7
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!"017 .!%0 !% 2%)%2 -, !% 1%0(& 6 D4# 4" &-0( &( G4& % !&*!7
!%$% &( 0- 0- ">2% 1&?%$%0#% &0
γef
(hδn2 i)1/2 /N0 ≫ ∆vb /vb 7
∆vb 9
#-$$%(8-01%1 - >%"+( .& ! " 1&?%$%0
;-$
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10vt
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"01
!% &0& &"2
;-$ "22 -, !%(% >%"+(
(vt /vb )2 ≤
!% 8$%(%0#% -, 1%0(& 6 D4# 4" &-0( 82"6(
" #$4#&"2 $-2% &0 !% 8$-#%(( -, .")% *%0%$" &-09 ;-$ "22 #-0(&1%$%1 #"(%(7 "0 &0#$%"(%
&0
!%
(hδn2 i)1/2 /N0
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!"
2%"1(
tgr 9
- "0 &0#$%"(% &0
"0 &0#$%"(% &0
:!% $%(42 ( .%$% %38%# %1 (&0#% .%
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- " 1%#$%"(% &0
!% %?%# &)% *$-. ! $" %9 K01%$ " #-01& &-0 -, (hδn2 i)1/2 /N0 ≫ ∆vb /vb 7 tgr 2&0%"$26
1%8%01( -0 #!"0*%( &0 (hδn2 i)1/2 /(N0 ) .& ! " #-%A#&%0 -, "88$-3&+" %26 -0%9
L% &01&#" %1
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Wt
tdec
"(
!%
tdec 7
"$% (!-.0 &0
&+% &0 %$)"2 >% .%%0
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Wmax /29
!% $&*!
!% +-+%0
Wt
&0
1%#$%"(%( -
@( #"0 >% (%%07 "22 !% $%(42 ( #"0 >% (%8"$" %1 &0 - !% ,-22-.&0*
.- #2"((%(/ BOC L!&2% !% &0& &"2 !%$+"2 1&(8%$(&-0 -, "0 %2%# $-0 >%"+ &( $%2" &)%26
2"$*%
∆vb /vb ≥ (hδn2 i)1/2 /N0 7
"0 &0#$%"(% &0 !% 2%)%2 -, 1%0(& 6 D4# 4" &-0( 2%"1( -
" 1%#$%"(% &0 !% &+% -, 1%#"69 :!&( $%(42 &01&#" %( !" "0 &0#$%"(% &0
(hδn2 i)1/2 /N0
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>4 &-0 +-$% %A#&%0 9 BPC :!% &0%G4"2& 6
$%*&-0 &0
∆vb /vb ≪ (hδn2 i)1/2 /N0
!% 8!"(% (8"#%7 .!%$% .")%( #"0 %?%# &)%26 &0 %$"#
&01&#" %(
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!%
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&0#$%"(% &0 !%
(hδn2 i)1/2 /N0
$%(42 ( &0 !% 2&0%"$ *$-. ! -, !% tdec 9
OQQ
50
20
π
tdec [ ωp
· 10 4 ]
π
tgr [ ωp
· 10 4 ]
40
30
20
15
10
10
2
3
!
4
2
" 1/2
hδn2 i
/N0 · 10 −2
3
!
4
" 1/2
hδn2 i
/N0 · 10 −2
!"#$% &'() * +,-$-+.%$!/.!+ .!0% 12$ "$23.,4 tgr 4 -56 - +,-$-+.%$!/.!+ .!0% 12$ 6%+-74 tdec 4
12$ .,% .2.-8 %5%$"7 6%5/!.7 21 3-9%/ -/ - 1#5+.!25 21 .,% 8%9%8 21 .,% 6%5/!.7 :#+.#-.!25'
;!<%$%5. +282$/ +2$$%/=256 .2 6!<%$%5. >%-0 9%82+!.!%/4 -/ 128823/) vb = 10vt 4 $%6?
vb = 12vt 4 +7-5? vb = 16vt 4 0-"%5.-? -56 vb = 20vt 4 >8-+@' ;!<%$%5. 0-$@/ +2$$%/=256
.2 9-$!2#/ !5!.!-8 .,%$0-8 6!/=%$/!25/ 21 .,% >%-0 A.,% /-0% -/ 12$ !"#$% &'BC
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=$2+%// 21 $%8-E-.!25 12$ +-/%/ 3!., 6!<%$%5. 6!/.$!>#.!25/ 21 .,% 6%5/!.7 :#+.#-.!25/'
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:#+.#-.!25/ !/ 0.005' K,% %928#.!25 12$ .,% .2.-8 %5%$"7 6%5/!.7 21 3-9%/ +2$$%/=256!5"
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$%/#8./ -$% 9%$7 /!0!8-$'
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MNM
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.( 3!,&- *( ! .!)& +"*2 ! ) !$$&, (hδn2i)1/2/N0/ 6(,&('&,1 *2& #7 8&, (4 3!,*".$&)
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-&#)"*% ?7.*7!*"(#) ") (hδn2i)1/2/N0 = 0.03/ F#-&, )7.2 "#"*"!$ .(#-"*"(#)1 -&#)"*%
?7.*7!*"(#) 3$!% ! '&,% " 3(,*!#* ,($& "# *2& 3,(.&)) (4 ,&$!5!*"(#/ 0&,&1 *+( !)3&.*)
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8&! ) 2!'"#9 -"@&,&#* "#"*"!$ 8&! '&$(."*"&) Ovb = 6vt1 vb = 10vt1 !#- vb = 16vtJ !#-"@&,&#* *2&, !$ -")3&,)"(#) O∆vb = 0.5vt1 ∆vb = vt1 !#- ∆vb = 3vtJ/ M# "#.,&!)& "#
*2& $&'&$ (4 *2& -&#)"*% ?7.*7!*"(#) $&!-) *( !# "#.,&!)& "# *2& &#&,9% (4 !..&$&,!*&3!,*".$&)/ F#-&, ! .(#-"*"(# (4 (vt/vb)2 ≤ (hδn2i)1/2/(3N0)1 &'&# -&#)"*% ?7.*7!*"(#)
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4(, )$(+ !#- +"-& 8&! ) O+"*2 vb < 10vt !#- ∆vb > vtJ *2& &@&.* (4 -&#)"*% ?7.*7!*"(#)
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INK
0.4
0.4
tωp /π =1*104
0.4
tωp /π =1.5*104
0.4
tωp /π =2*104
0.4
tωp /π =3*104
tωp /π =4*104
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0
5
0
10
velocity [vt ]
0.4
5
10
0
velocity [vt ]
0.3
tωp /π =2*104
5
10
0
5
velocity [vt ]
tωp /π =3*104
0.2
10
velocity [vt ]
tωp /π =4*104
0.15
5
10
velocity [vt ]
tωp /π =5*104
0.25
0.3
0
0.1
tωp /π =20*104
0.08
0.15
0.2
0.1
0.06
0.2
0.15
0.1
0.04
0.1
0.05
0.1
0.05
0.02
0.05
0
5
10
15
0
20
velocity [vt ]
0.4
5
10
15
20
0
velocity [vt ]
tωp /π =15*104
0.3
tωp /π =20*104
5
10
15
20
0
5
velocity [vt ]
0.15
tωp /π =30*104
10
15
20
velocity [vt ]
0.08
tωp /π =40*104
0.25
5
10
15
20
velocity [vt ]
0.06
tωp /π =80*104
0.05
0.3
0.06
0.2
0.2
0.1
0.04
0.15
0.04
0.1
0.03
0.05
0.02
0.1
0.02
0.05
0
0
10
20
30
velocity [v!"#$%
t]
0
0.01
10
20
30
0
10
20
30
0
10
20
30
0
10
20
30
&'() velocity
*+ %,-.#/!-+
0-$ velocity
%.%1/$-+[vt,%.-1!/2
3!4/$!5#/!-+
0#+1/!-+4'
[vt ]
]
velocity
[vt ]
velocity67%
[vt ] 1-$% 3!4/$!5#8
/!-+ /79/ 1-$$%4:-+34 /- 1-.3 %.%1/$-+4 -0 /7% 591;"$-#+3 :.94<9 =94 +-/ 1-+4!3%$%3'
67% 3!<%+4!-+.%44 %.%1/$-+ 3!4/$!5#/!-+ 0#+1/!-+ =94 +-$<9.!>%3 /- -+%' ?917 :.-/
:$-,!3%4 9 4+9:47-/ -0 /7% %.%1/$-+ 3!4/$!5#/!-+ 0#+1/!-+ 9/ 3!@%$%+/ <-<%+/4 -0 /!<%'
A-.-$4 1-$$%4:-+3 /- 4!<#.9/!-+4 =!/7 ,9$!-#4 3!4/$!5#/!-+4 0-$ /7% 3%+4!/2 B#1/#9/!-+4C
94 0-..-=4) 5.#%C /7% D-$<9. 3!4/$!5#/!-+ 9+3 $%3C /7% E%9$4-+ /2:% FF 3!4/$!5#/!-+'
G%4#./4 =%$% -5/9!+%3 0-$ ,9$!-#4 :9$9<%/%$4 -0 /7% 5%9< 9+3 .%,%.4 -0 /7% 3%+4!/2
B#1/#9/!-+' H: :9+%.) (hδn2 i)1/2 /N0 = 0.005I vb = 6vt I 9+3 ∆vb = vt ' J!33.% :9+%.)
(hδn2 i)1/2 /N0 = 0.02I vb = 10vt I 9+3 ∆vb = vt ' K-//-< :9+%.) (hδn2 i)1/2 /N0 = 0.03I
vb = 16vt I 9+3 ∆vb = vt
LMN
7
0.7
6
h(vb − v)2 i1/2[vt ]
0.6
Ea /Eb0
0.5
0.4
0.3
0.2
4
3
2
1
0.1
0
5
1
!
2
"
2 1/2
hδn i
3
/N0 · 10
4
0
1
!
−2
2
3
4
" 1/2
/N0 · 10 −2
hδn i
2
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3,% <%5; %1%83$7-:' @717$: 5-9 ;5$A: 87$$%:47-9 37 3,% ?5$!7#: !-!3!51 <%5; ?%178!3!%:
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5:3%$!:A:6 ,5: 5 3,%$;51 9!:4%$:!7- 5<7?% 5vt 7-8% $%15=53!7- :3744%9' C: :,7D-/ 27$
5 87-9!3!7- 72 ∆vb /vb ≪ (hδn2 i)1/2 /N0 / -7 :#<:35-3!51 9!E%$%-8% D5: 9%3%$;!-%9 !:!;#153!7-: D!3, <%5;: ,5?!-" 9!E%$%-3 !-!3!51 ∆vb '
!" #$% &'()* +*,(-('.%/
C: 4$%?!7#:1. 9!:8#::%9/ D,%- 3,% 1%?%1 72 9%-:!3. >#83#53!7-: !: ,!", %-7#", 37 :3$7-"1.
5E%83 3,% -7-F1!-%5$ 9!:4%$:!7- $%153!7-:,!4 72 G5-";#!$ D5?%:/ 3,% %1%83$7- ?%178!3.
9!:3$!<#3!7- 2#-83!7- 85- 4$%:%$?% 5 $%"!7- !- ?%178!3. :458% D!3, 5 47:!3!?% :174% 9#$!-"
5 4%$!79 72 3!;% 5<7?% 80 · 104 π/ωp ' +,% $!",3 45-%1 72 !"#$% &'(( 4$7?!9%: 3,% 9!:3$!F
<#3!7- 2#-83!7- 53 3D7 9!E%$%-3 ;7;%-3: 72 3!;%/ t1 = 106 π/ωp 0;5$A%9 !- <1#%6 5-9
t2 = 50 · 108 π/ωp 0;5$A%9 !- $%96' +,% <%5; !-!3!511. ,59 5 ?%178!3. vb = 20vt 5-9 5
3,%$;51 9!:4%$:!7- ∆vb = vt ' +,% 1%?%1 72 9%-:!3. >#83#53!7-: !: 0.04' C: :,7D-/ 27117DF
!-" 3,% ;5H7$ 4,5:% 72 3,% $%15=53!7- 4$78%::/ !-:35<!1!3. :3!11 788#$: 53 :7;% ;5$"!-51
1%?%1 5-9 $%15=53!7- <%87;!-" ?%$. :17D' C: 3,% 9!:3$!<#3!7- 2#-83!7- %?71?%: 37 27$;
5 4153%5#/ 3,% -#;<%$ 72 %-%$"%3!8 45$3!81%: !-8$%5:%:' I7D%?%$/ $%15=53!7- 788#$: ?%$.
:17D1./ ,#-9$%9: 72 3!;%: :17D%$ 3,5- 9#$!-" 3,% ;5H7$ 4,5:% 72 $%15=53!7-' J5:%9 75 $7#", %:3!;53% 72 3,% "$7D3, $53% 27$ 3,!: K;5$"!-51K !-:35<!1!3./ D% 87-81#9%9 3,53
3,% 9!:3$!<#3!7- 2#-83!7- !: 5<1% 37 ;5!-35!- 5 $%"!7- D!3, 5 47:!3!?% "$59!%-3 %?%- 27$
5 3!;% 4%$!79 5<7?% 1010 π/ωp '
()&
!"#$ %&%'()*&+ ",# -%"% ./"%*&#- *& ",# 0*12'%"*.&0 20*&+ -*3#$#&" 4%'2#0 .! vb 5
∆vb 5 %&- h(hδn2 i)1/2 /N0 5 6# -#"#$1*&#- % 0*17'# 8$*"#$*% ",%" *&-*8%"#0 ",# #&- .! ",#
%8"*4# 0"%+#9 ",# 1%:*121 +$.6", $%"# .! 6%4#0 hγi /#8.1#0 01%''#$ ",%& 25·10−9 πωp ;
<,#$#!.$#5 % 0*+&*=8%&" *&8$#%0# *& ",# %17'*"2-# .! 6%4#0 *& ",*0 0"%+# $#>2*$#0 % 4#$(
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0.05
electron distribution function
electron distribution function
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
10
20
30
40
0.045
0.04
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10
velocity [vt ]
20
30
velocity [vt ]
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-*+4/352& +* '(# $9!.#0 1(# !-'(/3$ $(/ #) '(!' '(# /%$#3"#) '+5#6!"#3!8#) 83/ '(
3!'#$ .!* /"#3./5# 10s−1 +* 2/.!2+<#) .2-59$ +'( $9!'+!2 $.!2# /4 !993/;+5!'#2& 100=50
1(# !-'(/3$ !38-#) '(!' !' 2#!$' +* '(# 3#8+/*$ (#3# $.!''#3+*8 +$ */' '// 2!38#: '(#
%#!56)3+"#* 83/ '( 3!'# .!* /"#3./5# '(# 3#43!.'+/* /4 !"#$ /-' 43/5 '(# 83/ '(
3#8+/* +* '(# "#2/.+'& $9!.#0
1(# $'/.(!$'+.683/ '( 5/)#2 ./*$+)#3$ '(# %#!5 93/9!8!'+*8 +* '(# $'!'# .2/$# '/
'(# 5!38+*!2 $'!%+2+'&: 93#)+.'#) %& >7 '(#/3&0 1(# %#!5 9!$$+*8 '(3/-8( '(# '&9#
??? $/-3.# 3#2#!$#$ 9!3' /4 +'@$ 43## #*#38& '/ '(# #*#38& /4 '(# 7!*85-+3 !"#$ )-# '/
'(# %#!5 92!$5! +*$'!%+2+'&: '(!' 3#$-2'$ +* 92!'#!- 4/35!'+/* /* '(# #2#.'3/* )+$'3+%-6
'+/* 4-*.'+/* +'(+* $/5# 3!*8# /4 '(# "#2/.+'+#$0 A/ #"#3: %#' ##* '(# +*'#3!.'+/*$
+'( +*'#*$+"# .2-59$: '(# 9/$+'+"# $2/9# .!* %# 3#./"#3#) %#.!-$# /4 '(# !)"#.'+/*:
(+.( '3!*$4/35$ $9!'+!2 83!)+#*'$ +*'/ "#2/.+'& 83!)+#*'$ !$ '(# #2#.'3/*$ 93/9!8!'#
BC3/8*!3): DEFG: H#23/$# !*) C/2)5!*: DEFI: J/%+*$/* #' !20: DEEKL0
1(# 93#$#*.# /4 '(# )#*$+'& ,-.'-!'+/*$ !M#.'$ '(# !"#$: !22/ +*8 '(#5 '/ 83/
#M#.'+"#2& /*2& +* '(# .#3'!+* .2-59$: (+2# /'(#3 .2-59$ /4 !"#$ ./33#$9/*)+*8 '/
-*4!"/3!%2# )#*$+'& ,-.'-!'+/*$ !3# )!59#)0 1(+$: +* '-3*: 3#$-2'$ +* 9#3'-3%!'+/* /4
'(# %#!5 N+* '(# $#*$# /4 $9!'+!2 +*(/5/8#*#+'+#$ +* '(# #2#.'3/*$ )+$'3+%-'+/* 4-*.'+/*O
!%/-' +'$ !"#3!8#) $'!'# /4 '(# 5!38+*!2 $'!%+2+'& /* '(# '+5# $.!2#$ )#'#35+*#) %& '(#
.(!3!.'#3+$'+. '+5# $.!2#$ /4 '(# .(!*8# /4 )#*$+'& ,-.'-!'+/*$0 1(#$# +*(/5/8#*#+'+#$
93/"+)# '(# ,-.'-!'+/*$ /4 '(# 83/ '( 3!'# /4 '(# 7!*85-+3 !"#$: (+2# '(# !"#3!8#)
83/ '( 3!'# +$ .2/$# '/ <#3/0 P*)#3 $-.( ./*$+)#3!'+/* '(# 83/ '( 3!'# /4 '(# !"#$
+* ! 8+"#* .2-59 N +'( .(!3!.'#3+$'+. $9!'+!2 $.!2# /4 '(# /3)#3 /4 .(!3!.'#3+$'+. $.!2#$
/4 '(# )#*$+'& ,-.'-!'+/*$O -*)#38/ ! 3!*)/5 !2= -*'+2 #+'(#3 '(# !"#$ 2#!"# '(#
.2-59 /3 '(# !"# 83/ '( 3!'# $!'-3!'#$ "+! */*62+*#!3 93/.#$$#$0 1(# .(!3!.'#3+$'+.
'+5# /4 '(# +*'#3!.'+/*: tg : .!* %# /%'!+*#) !$ l/vg : (#3# l +$ '(# .2-59 $+<# !*) vg +$
'(# 83/-9 "#2/.+'& /4 '(# 7!*85-+3 !"#$ N+* '(# '#35$ /4 '(# '(#35!2 "#2/.+'& /4 '(#
%!.=83/-*) #2#.'3/*$: vt : !*) %#!5 "#2/.+'&: vb : vg = 3vt2 /vb O0 Q-3+*8 ! '+5# tg : '(#
%#!5 93/9!8!'+*8 !' "#2/.+'& vb .!* 93/9!8!'# /"#3 ! )+$'!*.# -9 '/ vb tg !*) '3!"#3$#
vb tg /hli .2-59$ +* '3!*$+': (#3# hli +$ ! 5#!* .2-59 $+<#0 1(# !-'(/3$ !$$-5# '(!'
'(# +/*6$/-*) !"#$ 3#$9/*$+%2# 4/3 '(# )#*$+'& +*(/5/8#*#+'+#$ +* '(# $/2!3 +*): !*)
'(-$ '(# .(!3!.'#3+$'+. 2+4#'+5# /4 '(# .2-59$ .!* %# #$'+5!'#) !$ ts = hli/cs : (#3# cs
+$ '(# '&9+.!2 +/*6$/-*) $9##)0 1(+$ #$'+5!'+/* +*)+.!'#$ '(!' ! '&9+.!2 .2-59 4/35$
!*) )+$9#3$#$ htg i/ts '+5#$ *#!3 ! 9!3'+.-2!3 2/.!'+/* )-3+*8 '+5# tg 0 R!.( 4/35!'+/*
!*) )+$9#3$!2 /4 .2-59 3#$-2'$ +* !99#!3!*.# /4 ! ./33#$9/*)+*8 +*(/5/8#*#+'+#$ +*
'(# 9!3' /4 '(# %#!5 '(!' +*'#3!.'$ +'( 8+"#* .2-59 )-3+*8 tg 0 1(# '/'!2 *-5%#3 /4
DDS
!"#$#%&!& ' &( #! '"& &)&*'+#! , ('+ -.' #! /.!*' #! 01( #-'1 !&, 1(2
ng (l) =
8'
' $&
vg tg htg i
v b cs l
=
hli ts
hlivg2
( 0#+'" !#' !% '"1' /#+ '"& '9: *1) (#)1+ 0 !, *#!, ' #!(;
t < tg
'"& '#'1) !.$-&+
/
(
<.+ !% '"&
ng t
t
=
tg
ti
ni (t) =
ti
ng ≫ 15
!"#$#%&!& ' &( '"1' :1(( '+#.%" '"& *).$: *1! -&
&(' $1'&, 1(2
0"&+&
34567
345=7
(hlivg )/(cs vb )5
>"& (#).' #! #/ '"& ? #.@ ))& &A.1' #! /#+ '"& (:&*'+1) &!&+%9 ,&!( '9 #/ '"& ?1!%B
Ws ;
$. + 01@&( :1*C&';
*1! -& #-'1 !&, 1( /#))#0(2
Ws (t) = Ws (t0 ) exp (G)
0"&+&
t0 ;
Ws (t)
1!,
Ws (t0 )
+&(:&*' @&)9; 1!,
G
1+& '"& (:&*'+1) &!&+%9 ,&!( '9 1' '"& $#$&!'( #/ ' $&
( '"& '#'1) %+#0'" #/ '"& 01@& ,.+ !% '"& ' $&
'#'1) %+#0'" #/ 1 % @&! %+#.: #/ ?1!%$. + 01@&(
∆Gi = γi ti
345D7
( 1 (.$ #@&+
1( '"& 01@&( 1+& &!*#.!'&+&, -9 &1*"
G=
X
t − t0 5
!, @ ,.1)
t
1!,
>"&
!*+&$&!'(
!"#$#%&!& '9 #/ '"& -&1$
∆Gi
345E7
i
0"&+&
γi
( '"& %+#0'" +1'& :+#@ ,&, -9 ( !%)&
$#%&!& ' &(5 >"& $&1! 1!, @1+ 1!*& #/
#/
∆G
G
!"#$#%&!& '9 1!, @1+ &( -&'0&&!
1+& '".( +&)1'&, '# *#++&(:#!, !% A.1!' ' &(
1(2
/#+ ' $&
t < tg
hGi = h∆Gini (t) = hγit,
34547
σ 2 (G) = σ 2 (∆G)ni (t) = σ 2 (γ)ti t,
345F7
1!,2
hGi = h∆Ging =
hγil
,
vg
σ 2 (G) = σ 2 (∆G)ng =
/#+
!"#B
σ 2 (γ)l2
,
vb vg
345G7
345H7
t = tg 5
8! '"& '&+$( #/
' #! #/
G
hGi
1!,
σ(G);
'"& (:1' 1))9 1@&+1%&, :+#-1- ) '9 , ('+ -.' #! /.!*B
*1! 1::+#I $1'&)9 ,&J!&, 1( /#))#0(2
P (G) = p
@ 1 *&!'+1) ) $ ' '"&#+&$ 3( !*&
1
2πσ 2 (G)
ng ≫ 175
exp (−
(G − hGi)2
),
2σ 2 (G)
345K7
>".(; -9 $1C !% .(& #/ (#).' #! 45D 1!,
&A.1' #!( 454 1!, 45F #!& *1! #-'1 ! /#))#0 !% &A.1' #!2
664
!"#$% &'() *$+,-,!.!/0 1!2/$!,#/!+3 +4 5-6% -78.!/#1% 4+$ /9% %.%:/$!: ;%.1 +4 /9%
2%.%:/%1 5-6%4+$7 +,2%$6%1 ,0 <-22!3! =>? @=!1%,-31 ?%:%!6%$A' B9% "$-0 1-29%1
.!3% $%8$%2%3/ - ;/ +4 .+"C3+$7-. 1!2/$!,#/!+32' B9% $%1 .!3% 29+52 /9% 9!"9 -78.!/#1% ;/
@4$+7 0.25 /+ 1mV /m A 5!/9 - 8+5%$ .+5 %D8+3%3/' -A E6%$-.. -78.!/#1% 1!2/$!,#/!+3
#2!3" .+"C.+" 2:-.%' ,A F%/-!. +4 1!2/$!,#/!+3 4#3:/!+3 4+$ -78.!/#1%2 ,%/5%%3 0.001 -31
0.1mV /v #2!3" - .!3%-$ 2:-.% G4$+7 *HI- %/ -.'J KL(&M'
1
hW (t)i = exp([hγi + σ 2 (γ)ti ]t)
2
@&'(LA
4+$ t < tg G?+,!32+3J (NNKM' B9% %O%:/!6% "$+5/9 $-/% +4 /9% 5-6%2J Γef f J 5-2 1%;3%1
,0 /9% -#/9+$2 -2)
1
Γef f = hγi + σ 2 (γ)ti
@&'((A
2
B9!2 $%2#./ -..+52 +3% /+ $%2+.6% /9% 8-$-1+D +4 5-6% "$+5/9 1%28!/% /9-/ /9% 7%-3
"$+5/9 $-/% hγi !2 3%"-/!6% ,%:-#2% /9% Γef f :-3 ,% 8+2!/!6% %6%3 4+$ hγi < 0' P3+/9%$
!78+$/-3/ $%2#./ 5-2 +,/-!3%1 ,0 /-Q!3" !3/+ -::+#3/ /9-/ 2+.#/!+3 &'R .%-12 /+ 4+..+5!3"
$%.-/!+3 ,%/5%%3 G -31 -78.!/#1% +4 /9% 5-6%2 %.%:/$!: ;.%1) G = 2 ln (E/E0 ) -31 /9#2
/9% 1!2/$!,#/!+3 +4 /9% 5-6%2 -78.!/#1%2 29+#.1 +,%02 /9% .+"C3+$7-. 1!2/$!,#/!+3)
P ln
E
E0
[2 ln (E/E0 ) − hGi]2
exp −
=p
2σ 2 (G)
2πσ(G)
1
!
@&'(KA
B9%$%4+$% /9% STB 8$%1!:/2 - 8-$-,+.!: 8$+;.% 4+$ ln P (ln E) 59%3 !/ 8.+//%1 -2 4#3:/!+3
+4 ln E ' B9!2 /9%+$%/!:-. 8$%1!:/!+3 !2 !3 /9% "++1 -"$%%7%3/ 5!/9 +,2%$6-/!+32 +4
/9% U-"7#!$ 5-6%2 -78.!/#1%2 !3 /9% V-$/9W2 4+$%29+:Q G<-!$32 -31 ?+,!32+3J (NNNM
-31 8+.-$ :-8 $%"!+3 G<-!$32 -31 X%3!%//!J KLL(M' +$ !32/-3:%J !"#$% &'( 8$+6!1%2
/9% %6!1%3:% +4 /9% .+"C3+$7-. +,2%$6%1 !32!1% /9% S-/#$3W2 4+$%29+:Q ,0 /9% <-22!3!
28-:%:$-4/ G4$+7 *HI- %/ -.'J KL(&M' E3% :-3 2%% /9-/ 1%6!-/!+3 +4 /9% 1!2/$!,#/!+3 4$+7
/9% .+"C3+$7-. 4+$ -78.!/#1%2 > 0.25mV /m -$% $-/9%$ .-$"%' B9% 8+5%$ .+5 %D8+3%3/
1!2/$!,#/!+3 -/ /9% 9!"9 -78.!/#1%2 5-2 %D8.-!3%1 ,0 -""$%"-/%1 1!2/$!,#/!+3J -2 $%2#./ +4 - :+7,!3-/!+3 +4 7-30 .+"C3+$7-. 1!2/$!,#/!+32 4+$ 28-/!-..0 .!7!/%1 $%"!+32
G>+29#!Y%3 %/ -.'J KLL(M'
((Z
!"#$% &'() *+%,$%-!./0 1!2-$!3#-!,4 P (G) 5!-+ /41 5!-+,#- -+% 6$%2%4.% ,7 / -+$%%85/9%
1%./: -+$%2+,01 /2 "!9%4 3: %;#/-!,42 &'< /41 &'=>? 2,0!1 /41 1/2+%1 .#$9%2 $%26%.-!9%0:
@7$,A B,3!42,4 %- /0'? =<<CD'
*+% EF* 5/2 $%G4%1 -, -/H% !4-, /..,#4- -+% 4,480!4%/$ 5/9%85/9% !4-%$/.-!,4
@I!4 %- /0'? =<JK? B,3!42,4 %- /0'? =<<CD' L7 -+% +!"+%2- 5/9% 0%9%02 2/-#$/-% 9!/ -+$%%8
5/9% 1%./: M,$ 9!/ 2,A% ,-+%$ 4,480!4%/$ 6$,.%22N -+% 1!2-$!3#-!,4 ,7 G 5!00 7/00 $/6!10:
3%:,41 -+% 6,!4- Gc .,$$%26,41!4" -, -+% -+% -+$%2+,01 A/"4!-#1% ,7 -+% %0%.-$!. G%01
Ec 7,$ 1%./:' O:4/A!./00:? -+% %4%$": 1%42!-: ,7 -+% 5/9%2 -+/- $%/.+%1 -+% -+$%2+,01
$/6!10: 1%.$%/2%2 /2 6$,1#.- 5/9%2 /$% "%4%$/-%1? 7/00!4" -, /4 /2:A6-,-!. 0%9%0 -+/- !2
%P6,4%4-!/00: 1%.$%/2!4" 7#4.-!,4 ,7 -+% %P-%4- -, 5+!.+ -+% -+$%2+,01 5/2 %P.%%1%1'
*+% /#-+,$2 /66$,P!A/-%1 -+% -+$%2+,01 /2 /4 /32,$3!4" 3,#41/$: /41 2,09%1 -+% -+%
1!Q#2!,4 6$,30%A !4 G 7,$ -+% +/078!4-%$9/0 G < Gc !4 ,$1%$ -, ./0.#0/-% P (G)' R7-%$
4,$A/0!S/-!,4 -, #4!- -+% 6$,3/3!0!-: 7#4.-!,4 P (G) 5/2 ,3-/!4%1 /2 7,00,52)
Gc − hGi −1
√
×
P (G) =
2σ(G)
(2Gc − hGi − G)2
(G − hGi)2
− exp −
exp −
2σ 2 (G)
2σ 2 (G)
p
2πσ 2 (G)Erf
M&'=CN
7,$ G < Gc /41 P (G) = 0 7,$ G > Gc ' !"#$% &'( 2+,52 -+% 1!2-$!3#-!,42 &'< /41 &'=>
/41 !00#2-$/-%2 -+% %Q%.- ,7 -+% -+$%%85/9% 1%./: 6$,.%22'
!" #$%$&'$&(%) *+,*-+$&-' ,. $/- '0%))1%0*)&$23- 4%5602&+
7%8-' &5 $/- *)%'0% 7&$/ 3-5'&$9 :2($2%$&,5'!
B%.%4-0:? -+% A,1%0 -+/- 1%2.$!3%2 -+% 0!4%/$ !4-%$/.-!,4 ,7 I/4"A#!$ 5/9% 6/.H%-2
5!-+ -+% %0%.-$,4 3%/A /41 -/H%2 !4-, /..,#4- -+% /4"#0/$ 1!Q#2!,4 ,7 -+% 5/9% 9%.-,$
==T
!" #$ #%" &'(" )*'##"+,-. $- #%" )/'001'/20,#! " "-),#3 4!*#!'#,$-)5 ') &"00 ') '
)!22+")),$- $6 #%" ,-)#'7,0,#3 *'!)" 73 #%" +"/$('0 $6 #%" &'(" 6+$/ #%" +")$-'-*"
&,#% 2'+#,*0") !+,-. *+$)),-. "-),#3 4!*#!'#,$-) &,#% 0'+." '/20,#! ") &') 2+$2$)"
73 8+')-$)"0)9,9% "# '0: ;<==>?: @%" '!#%$+) !)" #%" A"'+)$- #"*%-,B!" #$ *0')),63 #%"
,)#+,7!#,$-) $6 #%" &'(" )2"*#+'0 "-"+.3 "-),#3 $7#',-" 6+$/ #%" -!/"+,*'0 /$ "0,-.:
@%"3 '+.!" #%'# !- "+ *"+#',- *,+*!/)#'-*") #%" "C"*#,(" -!/7"+ $6 #%" $6 #%" +".,$-)
&%"+" #%" &'(") *'- .+$& "C"*#,("03 ,) -$# ("+3 0'+."5 '- #%!) #%" *"-#+'0 0,/,# #%"$+"/
6',0) #$ 7" #+!": D$+ #%,) *')") #%" ,)#+,7!#,$- $6 #%" 0$.'+,#%/ $6 #%" &'(" "-"+.3
"-),#3 *$++")2$- ) +'#%"+ #$ A"'+)$- #32" EF ,)#+,7!#,$-) +'#%"+ #%'- -$+/'0: E#%" 6$00$&,-. )"*#,$- &" 7+,"43 ")*+,7" /',- , "') '- +")!0#) $6 #%" $+,.,-'0 2'2"+
;8+')-$)"0)9,9% "# '0:5 <==>?:
@%" /$ "0 *$-), "+) #$ "C"*#) +"0'#" #$ #%" "-),#3 ,-%$/$."-",#3 $- #%" ,-#"+'*1
#,$- $6 #%" "0"*#+$- 7"'/ &,#% G'-./!,+ &'(")5 -'/"03H #%" '-.!0'+ ,C!),$- $6 #%"
&'(") '- +"/$('0 #%" &'(" 6+$/ #%" +")$-'-*" &,#% #%" 7"'/: I"2"- ,-. $- #%",+
/'.-,#! " #%" 4!*#!'#,$-) '+" ,-($0(" ,- $-" $6 #%")" 2+$*"))"): @%" J+)# 2+$*")) ,)
*'!)" 73 ' )/'001'/20,#! " 4!*#!'#,$-)5 &,#% '/20,#! ")5 ∆n5 #%'# )'#,)63 *$- ,#,$-)H
2
∆n/N0 ≪ 3kT /(mωpe
),
KL:MNO
&%"+" T ,) "0"*#+$- #"/2"+'#!+" '- k ,) P$0#Q/'-- *$-)#'-#: @%" )"*$- "C"*#
,) 2+$(, " 73 #%" 0'+."1'/20,#! " 4!*#!'#,$-)5 &,#% '/20,#! ")5 ∆N 5 #%'# )'#,)63
,-"B!'0,#3H
2
∆N/N0 ≤ 3kT /(mωpe
),
KL:MLO
E- $#%"+ &$+ ) #%" &'(" ("*#$+ *'- 7" *%'-." ),.-,J*'-#03 !+,-. *+$)),-. $6 )!*%
,-%$/$."-",#,") 7!# #%" ,-%$/$."-",#,") '+" -$# 0'+." "-$!.% #$ +"4"*# #%" 2'*9"#
7'*9&'+ :
E6 #%" #32,*'0 &'(" -!/7"+) q '- Q5 *$++")2$- ,-. #$ #%" 4!*#!'#,$-) $6 #%" J+)#
'- #%" )"*$- #32"5 +")2"*#,("035 '+" +"0'#,("03 0'+." ') *$/2'+" &,#% &'("0"-.#% $6
#%" G'-./!,+ &'(")H
Q < Γ/vg ≪ q ≪ k
KL:MRO
&%"+" Γ ,) #%" .+$&#% +'#"5 vg ,) #%" .+$!2 ("0$*,#3 $6 #%" &'(" 2'*9"#5 '- k ,) '
#32,*'0 &'(" -!/7"+ $6 G'-./!,+ &'(")5 #%" &'(") )2"*#+'0 "-"+.3 "-),#35 Ws 5 *'7" 2"+6$+/" ') '("+'." ,- #,/" '- )2'*" $- #%" *%'+'*#"+,)#,* ,-#"+('0) ∆t '- ∆r
#%'# )'#,)63 #%" *$- ,#,$-)H
1/(qvg ) ≪ ∆t ≪ 1/Γ, 1/(Qvg )
KL:M>O
1/q ≪ ∆r ≪ vg /Γ, 1/(Q)
KL:MSO
'-
MMS
!" #$%& '(&)* #$) )+!,-#%!. !/ #$) (+)"(0)1 2(+) &3)'#"(, ).)"04 1).&%#4 '(. 5) 1)&'"%5)
54 #$) !66)"78,(.6 )9-(#%!.&* 2$)") #$) 1%:-&%!. 3"!')&& 1-) #! #$) &;(,,7(;3,%#-1)
<-'#-(#%!.& %& '!.&%1)")1 (& ( &#!'$(&#%' 3"!')&&* 2$%,) #$) %.#)"('#%!. !/ #$) 2(+)
3('6)# 2%#$ )('$ ,("0)7(;3,%#-1) 1).&%#4 1%&#-"5(.')& %& '!.&%1)")1 %. 1)#)";%.%&#%'
(33"!('$= >& %# 2(& &$!2. %. ?@"(&.!&),&6%6$ )# (,=* ABBCD #$) )9-(#%!.& '(. 5) 2"%##).
(& /!,,!2&E
∂Ws
+ (v~g · ∇) Ws = ∆DWs + ΓWs .
∂t
FG=HIJ
Z
W0
~
Ws (t, ~r) = √ exp −iωt + i kd~r ,
kn
FG=ABJ
K# %& 2!"#$ .!#%.0 #$(# 2$). #$) ):)'# !/ #$) 1%:-&%!. %& .)0,%0%5,) (.1 &!;) %.7
$!;!0).)%#%)& (") ,("0) ).!-0$ #! 3"!+%1) ")<)'#%!. !/ #$) 2(+) 3('6)# 5('62("1* #$)
(;3,%#-1) !/ #$) 2(+)& '(. (,&! '$(.0) 1-) #! #$) +("%(#%!.& !/ #$) 0"!-3 +),!'%#4=
L-'$ ( 3%'#-") '!"")&3!.1& #! ( M@N &!,-#%!. #$(# '(. 5) 2"%##). (&E
2$)") ~k %& ( '!;3,)O 2(+) +)'#!" 1)&'"%5%.0 +("%(#%!.& %. 5!#$ #$) 2(+) 3$(&) (.1
(;3,%#-1)* (.1 kn %& ( '!;3!.).# !/ #$) 2(+) +)'#!" 3)"3).1%'-,(" #! #$) &-"/(') !/
#$) %.$!;!0).)%#4=
P$) &%;3,%Q)1 &#(#%!.("4 &#(#) 3"!5,); 2(& '!.&%1)")1 54 @"(&.!&),&6%6$ )# (,=
?ABBCD* 2$). #$) R(.0;-%" 2(+)& ).#)" #$) $(,/ &3(') x > 0 !''-3%)1 54 -.&#(5,)
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D sin (θ)
∂Ws
∂θ
+ ΓWs ,
FG=AHJ
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∆ ln Ws
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∆x
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δ2
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&*9- -+-/78 1-+4(.8:
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Ws (x, θ) *+1 30/ .'- .0.*6
Ws (x, θ) sin θdθ,
!"#;%
0
(3 .'- 1-5-+1-+)(-4 03
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θ
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(+4.*+)- *4 30660&4 ?@/*44+04-64A(A' -. *66 #BBCDE
#
%
θ 2
1
Γ(x, θ)
.
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Γmax
2 ∆θ
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4=*66 /*.(04 03
Γmax /D%:
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θ:
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Γmax /D
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# # %
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,
Γmax
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f or
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AF=BGD
2!"$" Z = nres V ; V %, ' &)-15" )//13%"# 64 2'&".3'$(%/-" ,4,("5; ~rp %, 3),%(%)* )+
(!" 3'$(%/-" p; ǫk %, (!" 3-',5' #%"-"/($%/ +1*/(%)* '*# ǫk = ∂ǫk /∂ωk = !" "-"/($)*, '$"
5)&%*7 %* (!" 2'&"H, "-"/($%/ >"-# '//)$#%*7 () (!" I"2()* "01'(%)*,E
′
X
∂ v~p
e
e
~kψk exp i(~k · ~rp − ωk t),
+
v~p × B~0 = Re
∂t
mc
m
AF=BCD
∂Hw−p
dP~p
=−
dt
∂~rp
AF=KLD
∂Hw−p
d~rp
=
= v~p ,
dt
∂ P~p
AF=KMD
k
2!"$" B~0 %, 6'/<7$)1*# 5'7*"(%/ >"-# /!),"* () 6" #%$"/("# '-)*7 (!" '9%, z ; c %, '
,3""# )+ -%7!(; '*# v~p %, ' &"-)/%(4 )+ (!" 3'$(%/-"= !" "01'(%)* F=BC /'* 6" 3$","*("#
', (!" J'5%-()*%'* "01'(%)*, ',
'*#
~ rp )/c %, (!" 7"*"$'-%?"# 3'$(%/-" 5)5"*(; A(~
~ rp ) = (B~0 × ~rp )/2
2!"$" Pp = mv~p − eA(~
%, (!" &"/()$ 3)("*(%'-; '*# Hw−p %, (!" J'5%-()*%'* )+ (!" 2'&".3'$(%/-" ,4,("5= @)$
(!" /'," )+ '6,"*/" )+ 2'&".2'&" %*("$'/(%)*; (!" J'5%-()*%'* %, 7%&"* ',E
MBN
Hw−p =
!
X
~ rp ))2
(P~p + eA(~
ψl exp i(~k · ~rp − ωp k)
− eRe
2m
k
~ 2
kψ
X
k
′
+V
ωk ǫ k
16π
Z
X
p
!"#$%
k
&'()( *&+ ,)-* *().- )(/)(-(0* *'( 120(*23 /4)*235( (0()67 408 *'( &49(:/4)*235( 20*():
43*2+0 (0()67; )(-/(3*29(57< *'( 54-* *(). 3+))(-/+08- *+ *'( &49( (0()67
(=/)(--(8 *')+>6' *'( 0+).45 &49( 4./52*>8(-
&'()(
α=k
q
X
k
X
ωk |Ck |2 = V
′
(V ǫk )/(16π)"
Ck = αψk exp −i(ωk t)
Wk = V
X
k
k
′
ωk ǫ k
~ 2
kψk 16π
4-
P
,
k
ωk |Ck |2
!"##%
?* 2- &+)*' 0+*206 *'4* (@>4*2+0 !"$A 8(-3)2B206 *(./+)45
3'406(- 20 &49(C- /+*(0*245 340 B( +B*420(8 20 *'( D)4.(&+)1 +D E4.25*+0240 8(-3)2/*2+0
4- D+55+&-F
Z
∂Hw−p
∂Ck
e ∂ψk∗ X
= −iωk Ck + i
exp i(ωk t − ~k · ~rp )
= −i
∂t
∂Ck∗
2 ∂Ck∗ p
!"#G%
H+ 3+0-28() *'( 0+0:520(4) (I(3*- )(54*(8 *+ &49(:&49( 20*()43*2+0; 40 4//)+/)24*(
E4.25*+0240; 8(-3)2B206 20*()43*2+0 B(*&((0 &49(-; 3+>58 B( 488(8 *+ *'(
Hw−p "
*'( 34-( +D *')((:&49( )(-+0403( /)+3(-- &2*' .4*3'206 408 )(-+0403( 3+082*2+0-
~k ′ + ~k ′′
408
ωk = ωk′ + ωk′′
*'( E4.25*+0240 340 B( &)2**(0 4- D+55+&-F
1
Hw−w = √
V
&'()(
Vk,k′ ,k′′
J+)
~k =
X
′
k,k ,k
Vk,k′ ,k′′ Ck∗ Ck′ Ck′′ + c.c
!"#!%
′′
2- *'( &49(:&49( 20*()43*2+0 .4*)2= (5(.(0*" H'>-; *'( *+*45 E4.25*+0240
+D *'( -7-*(.;
Htot = Hw−p + Hw−w ;
Htot =
q
′
′
′
Z
X
p
340 B( &)2**(0 20 D+55+&206 D+).F
!
X
mvp2
eψk exp i(~k~rp )
− Re
2
k
~ 2
X kψk X kk ′ k ′′
V
+
8π
(16π)3/2
k
k,k′ ,k′′
!
!"#K%
ǫk′ ǫk′′ ǫ∗k Vkk′ k′′ ψk∗ ψk′ ψk′′ exp i(ωk − ωk′ − ωk′′ ) + c.c
H'( (@>4*2+0 !"#G 8(-3)2B206 *'( *(./+)45 (9+5>*2+0 +D *'( 0+).45 &49( 4./52*>8(-
Ck
D+) *'( 34-( +D *')((:&49( /)+3(-- *41(- D+55+&206 D+).F
L$!
∂ (Hw−p + Hw−w )
∂Ck
= −i
∂t
∂Ck∗
!"#$%
&'( )*+,-./0,*0 1(234,5.,/0 6*2(1 /0 .'( (78*.,/02 !"#$ *01 !"#$ *--/92 1(:(-/5,0;
.'( 08+(4,3*- 3/1( .'*. ,0:/-:(2 2<+5-(3.,3 +(.'/1"
=4/+ .'( 08+(4,3*- 5/,0. />
:,(9? .'( 82( /> 283' +(.'/1 54/:,1(2 2(:(4*- *1:*0.*;(2 3/+5*4(1 ./ .'( @,0(.,3
A-*2/:BC/,22/0 *554/*3'? 0*+(-<? ,. *--/92 ./ 3*-38-*.( .'( 1<0*+,32 /> 9*:(2 *01
5*4.,3-(2 184,0; :(4< -/0; .,+( 5(4,/12? ./ 2.81< .'( ,0.(4*3.,/0 /> 2+*-- 4(-*.,:( 1(02,.<
5*4.,3-( 1,2.4,68.,/02 9,.' 9*:(2 *01 ./ 3/--(3. -*4;( 08+6(42 /> 5*4.,3-(2 .4*D(3./4,(2
.'*. 3*0 6( *0*-<E(1 >84.'(4 82,0; 2.*.,2.,3*- *-;/4,.'+2" F/4(/:(4? ,. 3*0 6( 82(1
./ ,0:(2.,;*.( 5'<2,3*- 54/3(22(2 3/00(3.(1 9,.' .'( 1<0*+,32 /> ,01,:,18*- 5*4.,3-(2
*2? >/4 ,02.*03(? .'( 2-/9 ;4/9.' /> 9*:( (0(4;< 1(02,.< 9,.' .,+( ,0 .'( 2*.84*.,/0
2.*;( /> .'( 9*:(B5*4.,3-( ,0.(4*3.,/0%" G/+5*4(1 ./ C*4.,3-(BH0BG(-- CHG% 3/1(2? .'(
08+6(4 /> 5*4.,3-(2 82(1 ,0 .'( 2,+8-*.,/02 ,2 14*2.,3*--< 4(183(1? 2/ .'*. -/0; .,+(
3/+58.*.,/02 ,0 #I +*;0(.,E(1 5-*2+*2 3*0 6( 3*44,(1 /8." F/4(/:(4? *2 .'( 2<2.(+
/> (78*.,/02 ,0:/-:,0; 9*:(B5*4.,3-( *2 9(-- *2 9*:(B9*:( ,0.(4*3.,/02 .(4+2 '*:( *
)*+,-./0,*0 >/4+? 2<+5-(3.,3 ,0.(;4*./42 ;,:( .'( 5/22,6,-,.< ./ 82( -*4;(4 .,+( 2.(52
*01 ./ *33(-(4*.( 2.4/0;-< .'( 3*-38-*.,/02"
!"!#
$%&'( %) *(+,-+, ./01 '20'34+( &'4,/05 /41%-%6'4'/0/',
&'( *554/54,*.( +/1(- 1(234,6,0; .'( ,0.(4*3.,/0 /> .'( *0 (-(3.4/0 6(*+ 9,.' J*0;B
+8,4 9*:(2 5*3@(. ,0 * 5-*2+* 9,.' 1(02,.< K83.8*.,/02 9*2 1(4,:(1 >4/+ .'( L*B
@'*4/:M2 (78*.,/02" &'( (78*.,/02 >/4 .'( (-(3.4,3 N(-1
:*4,*.,/02
δn
~
3λ2
∂E
~
~ = ωp δn E,
+ D ωp ∇(∇ · E)
∂t
2
2 N0

%+"
*01 -/9 >4(78(03< 1(02,.<
3*0 6( 94,..(0 ,0 >/--/9,0; >/4+O
i
9'(4(
~
E
2
~
E !"#P%

∂vi
∂Φ

 δn
=∇
= −∇ c2s
+
.
∂t
∂t
N0 16πmi N0
~ r − ωt)
~ = Re P E
~
E
k k exp i(k · ~
)3 ("3! %4 56$%6-% )3!1
mi
!
!"#Q%
! "#"$%& $ '"#( )* %+" ,-." /-$0"%1
)3 7-!!1 -3(
Φ
δn
("!$& 2"!
! %+" +4(&)(43-7 $ 568 /)%"3% -#9
:+" & ;+% +-3( ! (" %"&7 )* "<6-% )3 =9>? ("!$& 2"! %+" 3)3@# 3"-& "A"$% )* ("3! %4
/"&%6&2-% )3 %+-% $-6!"! &"*&-$% )3 )* %+" B-3;76 & ,-."!
&"; )39
:+" #-!% %+"&7
3%) #), ("3! %4 /#-!7-
3 "<6-% )3 =9>C $)&&"!/)3(! %) %+" /)3("&)7)% ." *)&$" %+-%
"8/"#! /-&% $#"! )6% )* &"; )3! )* #-&;"& ,-." "3"&;4 ("3! %49 D3 %+"
$)3% 36 %4 "<6-% )3 *)&
1D
;")7"%&4 %+"
)3! &"-(!E
∂vi
∂2Φ
∂ δn
=−
=− 2
∂t N0
∂z
∂z
JKL
F=9GHI
!"#" z $% & '((#)$*&+" &,(*- +!" .&-*"+$' /",) ,$*"0 12 .&3$*- 4%" (5 "64&+$(* 7089:
"64&+$(* 70;< (*" '&* )"#$=">
∂2
∂2
− c2s 2
2
∂t
∂z
δn
|E|2
∂2
= 2
,
N0
∂z 16πmi N0
?708@A
B64&+$(*% 70;C &*) 708@ '&* D" )"#$=") 5#(. +!" E&.$,+(*$&*
δn |E|2 3λ2D ∂E 2
+
H=
dz
2N0 8π
16π ∂z L
# %
!
N0 mi 2 δn 2
∂Φ 2
+
cs
= const,
+
2
N0
∂z
Z
?708FA
!"#" +!" $*+"-#&+$(* (* z $% G"#5(#.") (="# +!" ,"*-+! (5 +!" %2%+". L0 H+ $% (#+!
*(+$*- +!&+ +!" %2%+". !&% (+!"# +$." $*=&#$&*+%: +!" +(+&, *4.D"# (5 64&*+& (5 !$-!I
5#"64"*'2 /",)
Z
|E|2
I1 =
dz = const
?708;A
8π
L
&*) +!" +(+&, .(."*+4. P
δn ∂ ∂Φ
i
∂E ∗
∗ ∂E
P=
dz N0 mi
?7088A
+
−E
E
N0 ∂t ∂z
16πωp
∂z
∂z
L
p
p
E/ 8πωp &*) E ∗ / 8πωp : &% ",, &% δn/N0 &*) −N0 mi Φ &#" '&*(*$'&,,2 '(*I
Z
J!"*
K4-&+") =&#$&D,"% (5 !&.$,+(*$&* H0 J!4% "64&+$(*% 70;C &*) 70;< '(4,) D" #"'(="#")
D2 .&3$*- 4%" (5 E&.$,+(* "64&+$(*%>
1 ∂E
δH
= −i ∗ ,
8πωp ∂t
δE
N0 mi
∂Φ
δH
=− ∂t
δ δn
?7087A
N0
!"#" δ $% 54*'+$(*&, )"#$=&+$="0
J!" $*+"#&'+$(* (5 +!" D"&. ","'+#(*% $+! +!" L&*-.4$# &="% $* &* $*!(.(-"*"(4%
G,&%.& '&* D" )"%'#$D") D2 & +(+&, E&.$,+(*$&* Ht = H+Hw−p : !"#" Hw−p $% )"/*")
D2 70;;0 J!4%: +!" .(+$(* (5 +!" D"&. ","'+#(*% $% )"%'#$D") D2 +!" E&.$,+(* "64&+$(*%
70;9 &*) 70;@0 J!" +(+&, E&.$,+(*$&* (5 +!" %2%+".: Ht : #"&)% MN#&O+ "+ &,0: F9@;P>
Ht = H + Lnb
X Ek
1 X mvp2
− eRe
i Jk∗
Z p 2
k
k
!
= const
?708QA
!"#" nb $% )"*%$+2 (5 +!" D"&.: Z = Lnb $% +!" *4.D"# (5 +!" ","'+#(*% $* +!" D"&.:
&*)
Jk =
1 X
exp i(ωp t − kzp ).
Z p
?708RA
SGG,2$*- +!" "64&+$(*% 7087 +( +!" E&.$,+(*$&* Ht ?708QA: +!" "64&+$(* (5 L&*-.4$#
+4#D4,"*'" )#$="* D2 &* ","'+#(* D"&. &% )"#$=") $* MN#&O+ "+ &,0: F9@;P &% 5(,,( %>
@FR
i
X ωp
δn
∂E 3λ2D ∂ 2 E
+
ωp 2 − ωp
E = 4πienb
Jk exp (ikz)
∂t
2
∂z
2N0
k
!"#$%
k
&' ()*
k
+,-.* ()* */0-(12' !"#$ 3*-4+5
i
7)*3*
ωp
4πeωp nb
∂
3
(e)
Ek = ωp k 2 λ2D Ek +
− γk
(ρE)k + i
Jk
∂t
2
2
k
(e)
= −Imǫek /(∂Reǫek /∂ωk )
ρ = δn/N0 8 γk
*<*.(32'1. ,-3( 2; ()* 41*<*.(31. .2'+(-'(
ǫk 8
!"#6%
1+ - 4-9,1': ;-.(23 -'4
ǫ2k
1+ ()*
7)1.) :1=*+ ()* ,2++1>1<1(? (2 1'.<04* ()*
4-9,1': *@*.(+ 2; A-':9013 7-=*+ 40* (2 ()*13 A-'4-0 3*+2'-'.* 71() ()* ()*39-<
*<*.(32'+" B/0-(12' !"#6 .-' >* 0+*4 (2 2>(-1' - '2'<1'*-3 41+,*3+12' 3*<-(12' 2; A-':C
9013 7-=*+ -+ ;2<<27+5
3
δn
(e)
ωk ≃ ωp + ωp k 2 λ2D + ωp
+ iγk
2
2N0
&'(3240.1': ()* 419*'+12'<*++ ,<-+9- =*<2.1(?
u = vi /cs
!"!D%
8 ()* E2031*3 (3-'+;239+
2; */0-(12' !"F$8 G**,1': ()* +*.2'4 234*3 (*39+8 -'4 */0-(12' !"F6 .-' >* 731((*' -+
;2<<27+8
∂
ρk = ikcs (uk + (ρu)k ),
∂

!"!H%
|E|2

∂uk
(i)
k 
− 2γk uk = ikcs ρk +
.
∂t
16πmi N0 c2s
!"!I%
7)*3* 4-9,1': *@*.( 40* (2 ()* 12'+ 1+ 1'(3240.*4 ()320:) ()* 4-9,1': 3-(* 3-(*
(i)
(i)
(i)
γk = −Imǫk /(∂Reǫk /∂ωk "
B/0-(12'+ !"#$8 !"!H8 -'4 !"!I (2:*()*3 71() */0-(12'+
!"I6 !"FD8 -'4 !"FH ;239 ()* .29,<*(* +*( 2; */0-(12'+ 2; ()* 924*<" &' ()* +0>+*.(12'
,3*+*'(1': ()* '09*31.-< +190<-(12'J+ 3*+0<(+8 7* -3* 0+1': '239-<1K*4 =-<0*+ 4*+.31>*4
1' 4*(-1< 1' ()* L,,*'41M N"H O+** -<+2 P3-@( *( -<"8 IDHFQ"
!"!#
$%&'()*+, -)&%,+.)/0-1 ('-%,.-
R*3* 7* ,3*+*'( 3*+0<(+ 2; ()* +190<-(12'+ ;23 - 3-':* 2; ,-3-9*(*3+ 3*<*=-'( (2 +2<-3
(?,* &&& *<*.(32' >*-9+ -'4 ,<-+9- 2>+*3=*4 -(
10eV 8 N0 = 5 ·
106 8
1
λD = 15m8 ωp /(2π) = 20kHz 8
LS OB3:0' *( -<"8 H66$Q5
-'4
vsw = 450km/s"
Te =
L( T3+( 7*
.2'+14*3 - .-+* 2; +9-<< 4*'+1(? U0.(0-(12'+" E1:03* !"V +)27+ ()* +,-(1-< ,32T<* 2;
()* *'*3:? 4*'+1(?
U0.(0-(12'+
2; (19*"
δn/N0
|E|2
+0,*3,2+*4 (2 ()* *<*.(31. T*<4 *'=*<2,*
-+ 7*<< -+ ()* *<*.(32' 41+(31>0(12' ;0'.(12' ;23 ()* ()3** 929*'(+
W)* <*=*< 2; ()* 4*'+1(? U0.(0-(12'+ 1+ .)2+*' (2 >*
>*-9 )-+ >*-9 =*<2.1(?8
nb = 5 ·
10−5 N0 "
Re(E) -'4 ()* 4*'+1(?
vb = 14vt 8
>*-9 ()*39-< =*<2.1(?
0.001
-'4 ()* 431=*'
∆vb = vt 8
-'4 4*'+1(?
W)* *<*.(32' 41+(31>0(12' ;0'.(12' +)27+ >*)-=123 =*3? .<2+* (2
,3*41.(*4 >? XA ()*23? ;23 ()* )292:*'*20+ ,<-+9-" L( ()* >*:1''1':8 ()* 3*<-M-(12'
HI$
−4
0.5
0
−5
δn/N 0 ,|E|2
x 10
5
0
−3
x 10
2000
4000
6000
8000
10000
0.3
0.2
0.1
0
−5
tωp= 10
0.4
f(v)
Re(E)
5
0
10
0
2000
4000
6000
8000
15
20
10000
0.1
Re(E)
0.5
0
δn/N 0 ,|E|2
5
0
−3
x 10
2000
4000
6000
8000
10000
tω =10
p
0.3
0.2
0.1
0
0
−5
4
0.4
f(v)
−0.1
0
2000
4000
6000
8000
5
10
15
20
15
20
10000
0.5
0
δn/N 0 ,|E|2
−0.2
0
2000
4000
6000
8000
10000
0.02
tωp=3⋅ 10
0.3
0.2
0.1
0
0
−0.02
4
0.4
f(v)
Re(E)
0.2
0
2000
4000
6000
8000
10000
5
10
v
z / λD
!"#$% &'() *%+,-. /01+,!2. +1/ 34% 5+6% 7+28%3-. %69:#3!91 +3 34$%% ,9,%13. 9;
3!,%)tωp = 10<104 < +1/ 3 · 104 =;$9, #77%$ 39 :95%$ 7+1%:.>' ?4% :%;3 7+1%:.) .7+3!+:
7$9@:% 9; 34% %:%23$!2 @%:/ %16%:97% Re(E) =#77%$ 7+1%:.> +1/ 34% 5+6% %1%$"0 /%1.!30
|E|2 =.4951 5!34 $%/>< .#7%$79.%/ 39 /%1.!30 A#23#+3!91. δn/N0 =.4951 5!34 B:#%>
=:95%$ 7+1%:.>' C!"43 7+1%:.) %:%23$91 6%:92!30 /!.3$!B#3!91 ;#123!91 f (v) =.4951 5!34
B:#%>' D+.4%/ 2#$6%. 29$$%.791/ 39" 34% !1!3!+:
%:%23$91 /!.3$!B#3!91 ;#123!91' E+!1
1/2
2
7+$+,%3%$.) vb = 14vt < ∆vb = vt < hδn i /N0 = 0.001< L = 10000λD < nb /N0 =
5 · 10−5 '
$%.#:3. !1 34% 7:+3%+# ;9$,+3!91 !1 34% ,9.3 #1.3+B:% $%"!91 9; 34% 6%:92!30 .7+2%<
1+,%:0 !1 34% $+1"% 9; 6%:92!3!%. vb − 3∆vb < v < vb =.%% ;9$ !1.3+12% $!"43 ,!//:% 7:93
!1 !"#$% &'(>' F#B.%G#%13:0< 34% :%;3 B9$/%$ 9; 34% /!.3$!B#3!91 ;#123!91 ,96%. 395+$/
:95%$ 6%:92!3!%.< #13!: $%+24%. 34% $%"!91 922#7!%/ B0 7:+.,+ 9; 34% .9:+$ 5!1/' ?4%
1#,B%$ 9; 34% %1%$"%3!2 %:%23$91. 5!34 6%:92!3!%. :+$"%$ 34+1 vb + 3∆vb +3 34% %1/ 9; 34%
$%:+H+3!91 !. 6%$0 :95' ?4% +,7:!3#/%. 9; 34% I+1",#!$ 5+6%. !12$%+.% 5!34 3!,%' J.
91% 2+1 193!2%< 34% |E|2 !. ,9.3:0 #1!;9$,:0 /!.3$!B#3%/ !1 .7+2%< 495%6%$ .9,% 7%+8.
2+1 B% .%%1 !1 34% 6!2!1!30 9; 34% /%1.!30 "$+/!%13.' ?4!. 7%+8. 5!34 34% ,+"1!3#/%
.%6%$+: 3!,%. +B96% 34% ,%+1 2+1 !1/!2+3% 34+3 %6%1 5%+8 /%1.!30 A#23#+3!91. ,+0
:%+/ 39 9B.%$6%/ 2:#,7!1" 74%19,%1+'
P
!"#$% &'K .495. 34% %69:#3!91 9; 34% I+1",#!$ 5+6% %1%$"0 /%1.!30 W = k Ek2
=:%;3 7+1%:> +1/ 34% 393+: %1%$"0 9; 34% B%+, %:%23$91. .#7%$79.%/ 5!34 34% %1%$"0 9;
34% %1%$"%3!2 7+$3!2:%. =$!"43 7+1%:>' ?4% 5+6% %1%$"0 /%1.!30 ,919391!2+::0 !12$%+.%.
5!34 3!,%' J3 + ,9,%13 9; 3!,% 9; +77$9H!,+3%:0 tωp = 4 · 104 34% 5+6% %1%$"0 /%1.!30
LMN
−3
x 10
1
1.8
0.9
1.6
0.8
1.4
0.7
1.2
0.6
Et , Ea
W
2
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0
0.1
2
4
ωp t
6
0
8
4
x 10
2
4
ωp t
6
8
4
x 10
P
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tωp = 4 · 104 ' G3% 6!7,$!<#,!2/ 2+ ,3% %0%;,$!; H%068 P (E)8 !7 7324/ !/ !"#$% &'J >0%+,
-./%0?' G2 2<,.!/8 P (E)8 4!,3 #7%6 Kb = 128 <!/7 7-.;%6 <%,4%%/ −1.25 ./6 1.25' D,
!7 42$,3 /2,!/" ,3., ,3% 4.1% %0%;,$!; H%06 !7 /2$=.0!K%6 .;;2$6!/" ,2 |E|2 /(4πN0 Te )'
G2 ;0.77!+5 P (E) 4% ;.0;#0.,%6 ;2$$%7-2/6!/" L%$72/F7 β -.$.=%,%$7' G3% 2<,.!/%6
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C, ,3% /%E, 7,%- 4% 7%0%;,%6 +$2= .00 ,3% 6.,. 2/05 ,3% 1.0#%7 2+ ,3% %0%;,$!; H%068 Ep 8
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ln (Ep2 ) !7 7324/ ., ,3% $!"3, -./%0 !/ !"#$% &'J' D, 7%%=7 ,3., 2<,.!/%6 6!7,$!<#,!2/ $%M
7%=<0%7 ,3% 6!7,$!<#,!2/ -$%6!;,%6 <5 NOG >+2$ !/7,./;% 7%% !"#$% &'P?' G3% 02".$!,3=
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,3% 6!7,$!<#,!2/ <5 ,3% 02"M/2$=.0 6!7,$!<#,!2/ 4!,3 -.$.=%,%$7 hln (Ep2 )i = −8 ./6
σ 2 = 0.66 !7 -$%7%/,%6' C 6%1!.,!2/ +$2= ,3% 02"M/2$=.0 /2$=.0 6!7,$!<#,!2/ +2$ ,3%
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QRS
0.03
0.025
−2
10
ln P (ln (Ep2 ))
P(E)
0.02
0.015
0.01
−3
10
0.005
0
−0.1
−4
−0.05
0
0.05
0.1
10
−12
−10
−8
−6
−4
ln (Ep2 )
E
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4+ ,5% 0.$"% 9."/!,#8% -.$, 4+ ,5% 8!2,$!3#,!4/ :!,5 %B-4/%/,!.00C 8%6$%.2!/" +#/6,!4/
P (ln(Ep2 )) = exp (−a ln Ep2 − b)D :5%$% a = −7.9 ./8 b = 42.8 !2 254:/ :!,5 "$%%/'
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TVT
−4
0.5
0
−5
δn/N 0 ,10 |E|2
x 10
0
2000
4000
6000
8000
10000
0.1
0.3
0.2
0.1
0
−0.1
tωp= 10
0.4
f(v)
Re(E)
5
0
10
0
2000
4000
6000
8000
15
20
10000
0.5
0
δn/N 0 ,10 |E|2
−0.1
0
2000
4000
6000
8000
10000
0.1
tω =3⋅ 10
p
0.3
0.2
0.1
0
0
−0.1
4
0.4
f(v)
Re(E)
0.1
0
2000
4000
6000
8000
0
10
20
30
20
30
10000
0.05
Re(E)
0.5
0
δn/N 0 ,10 |E|2
0
2000
4000
6000
8000
10000
0.1
tωp=7⋅ 10
0.3
0.2
0.1
0
0
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4
0.4
f(v)
−0.05
0
2000
4000
6000
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!"#$% &'() *%+,-. /01+,!2. +1/ 34% 5+6% 7+28%3-. %69:#3!91 +3 34$%% ,9,%13. 9;
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B:#%>' D+.4%/ 2#$6%. 29$$%.791/ 39"34% !1!3!+:
%:%23$91 /!.3$!B#3!91 ;#123!91' E+!1
1/2
2
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5 · 10−5 '
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−3
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1
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W
1
0.5
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0.4
0.4
0.3
0.3
0.2
0.2
0.1
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0.1
2
4
ωp t
6
0
8
4
x 10
2
4
ωp t
6
8
4
x 10
P
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0.07
0.06
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10
ln P (ln (Ep2 ))
P(E)
0.05
0.04
0.03
0.02
−3
10
0.01
0
−0.06 −0.04 −0.02
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ln (Ep2 )
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p
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exp (−iωi t + iki (x)x)
ki (x)
<&'&M=
:5%$% Wi (t) !2 ,5% :.;% %/%$"A 8%/2!,A 43,.!/%8 +$49 ,5% 2!9#0.,!4/2I ki (x) !2 ,5% :.;%
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(MV
0.1
0.4
E
0
tωp= 10
δn/N0 ,5 · 10 −3 |E|2
0
2000
4000
6000
8000
10000
f(v)
0.3
−0.1
0.2
0.05
0.1
0
−0.05
0
10
0
2000
4000
6000
8000
15
20
10000
0.4
0
δn/N0 ,5 · 10 −3 |E|2
tωp=3⋅ 104
0.3
−5
0
2000
4000
6000
8000
10000
f(v)
Re(E)
5
0.2
0.05
0.1
0
0
−0.05
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
10
20
10000
0.4
0
δn/N0 ,5 · 10 −3 |E|2
tωp=7⋅ 104
0.3
−5
0
2000
4000
6000
8000
10000
f(v)
Re(E)
5
0.2
0.1
0.1
0.05
0
−0.05
1000
0
2000
3000
4000
5000 6000
z/λD
7000
8000
9000
10000
0
10
v
20
30
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5 · 10−5 '
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P
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0.05
0.6
0.5
0.04
0.4
0.03
0.3
0.02
0.2
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0
0.1
0
2
4
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time [ω p−1 ]
0
8
4
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0
2
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6
time [ωp−1 ]
8
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P(E)
0.06
0.05
0.04
0.03
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10
0.02
0.01
0
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0.2
0.4
0.6
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−8
−6
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