tistiq ues à partir de simulations num é riq ues de trajectoires de dynamiq ue molé culaire en temps long. Ces formules permettent d 'acc é lé rer la convergence ...
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
High-order averaging schemes with error bounds for thermodynamical properties calculations by MD simulations Eric Cancès — François Castella — Philippe Chartier — Erwan Faou — Claude Le Bris — Frédéric Legoll — Gabriel Turinici
N° ???? Juillet 2003 THÈME 4
ISSN 0249-6399
apport de recherche
!#"$% $ $ '&( *) ,+- " . 0/* 12354 $ 46 7"$1 !2312)31 "1 !&8/:9"1 * ?@BADCFE8GIH$CKJKL MON @PGQHRKSTAULVE8GILWPXKYUYZG MO[ \ ADYUAD]]^X_E \ GQ@BWPAUXK@ M ? @B`aGIH N GISTb M EaYZGIb$cX d X(ef@gADL MON @BhKchi@gAUC_dXijTSTYUY M^k GIl@gAUXKYnmob@BADHAUCKA pqsrt8u>va wy x{z|t*rBt8}s~uB- z1ii
t8u1s ~uuBz t8zz|
u { Ou 2Fu7o sI u ug#qsu##qsu¡
I¢.£P£P£P£wy¤i}sz|~~|u1o¥¦¦§ w¨¥©> ªug ∗
†
∗
‡
∗§
‡
∗
0«Pµi¬$u#B®ªi¯-uB°s ±®i²
_³ q uauaz|¸Q
Kz| ¹ } ¶^1u8qsqsu8z|ª
{qs}s´Dt8i u uBz|BoP~µiuz|#t*} ªz|~|
sªz|¶·
Fitf¶^}s~|~
suª¶·´Uiz>t8uaqsutai~t8ug1s}s}~# - z|{
(
ta¶nzozz|BuBº´~ uBIzuguBº»½*¼z
sqsª6u01qsiuBt8us}¶·t*z|}s
~ui~^¾Ou¿nÀIuiz|gta» ³ -u0µiu0 -qsµ{uz 1uV
{µiiut8uªu
K
Q} 1tauVuB#-zu~O¶u1sq tau ~zugt8 u_qQP-µifu#qsª-u¿Á¿ zqsq u u#ÂVqs1uBz|tau
ugVÃÄ} ¶O#q5}s¹#µqsuuB~|t81uz» F³ Å.q uBu~|
u1ºÆº¾¿nuBuºªizµizuuguiu* ¸Qsis}sP
{ z|t-q u * s} -z
sµ{ªz u8{ªit8}ssz ~|uBu1~|z
sz ug z
K_uª##qs-{izi
u uqsÂVua1z| u
K# t8t8u1
su
s#uBB» ¶Oqsut8s}#-z|
Fz
( uoug#q_ªz|µuB
uBz|u B1}s#z|
_qsuataK Ç#qsÈKuBÉTtaÊË*ugÌ ¾¯-uBÍ$² 2¸I}s
~|uB }s¾s~|uB ªi {
z|t8q zuB¾7ut #-z|z~OPµu#ªiu*1it8s}z|
ξqsz|ªq´Z u>PµuBªz|
sª
Ó Ý#îUàPÔ·ÏÑÚäÒòóÐIÙìÓ>ÝÒ7ò|ø$ôTÓ>ÒOÝ#ÔÄÙÛÏ7ÖPçúÕÚOÖgàPÏTÐ ß ê1Ú×ÚáPغîÄÚäÙÛâDÚ.õ>ß ÞUÜ$ç^ögÖgÝágÞU÷-ÚOßÛîDغÝ#Ò7à-àgÝ#Ú×ÙÛàPÚºÚàPÖáPÚÝ#âOÚàPâÎûºá¹ãÖÓ>غÏÑà1Ý#ÔÄÞDÏÑü âÎÓ>ýPÚäÞ7æPø$âÎÏTÏ$áPå-ÖgÚOÝ#Ô·æPÜ$ëTâUÚÒÎâDÝ#çÔóÚæPø6âÙÛßÖ-ÚÒÎèOæKØBÖ-Úä×Þ7þºùñºénæPÿ#ÚÝð1àPê1ö ×ÚغàBÒ7æPæPÚî·ÚOàPÞ àPëTÚÙìâ$Ý#ßÛÏTâUÚ$ÚágãsÚÝ#õâUÖ-×÷-Ý#ÙîUÖ-Ý#ÏTàP×ß ÞDÚ ç$í$ÚâD×Ý#îÄÞUÚâÖ-ïºïð1ñºñ øÐIí${ø$÷aí$Ò Ô·Ü^(ÖgínÔ·Ü$ÖgÒÎø$Ôóà-ø6Ý#Ù ÒÎBâDÚÚ$àPàPÚÞ7Úâ Ó>Øá ÙÛÚâ7ÜæPü çîÄßÛùBæPÚâÖKûºÖgÝê1ÚàBægÚ^ágæ $çàPçîUÝ#Ù{áP Ú Ý#æPÙÛ Úº Ö ºögûð1ÿÏÑÙìÝ#üÝ#î·ÞÖg÷-îUÝ#àP×Ú ∗
† ‡ §
Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30
, $ $ 31 46 )0 31 n*31n)3 / !
/* a)= 3 2)3º z|ut8 z K ¹}s uB~|
sªQ² » nug}Qz ¶·i z|
iutf}s z|t*~ ug} }sz|~I-u
zit8
u1uB
{ u}s
K¶·t i Btfz }sK~|}suBuB ~ uuVo~|u 1uBº~|
{uµµiz|uugQª ui
Q}s 1u u ~| u0 {ug
~|t8t8}s-z~i} uB
su0uV
st8t8ug -~iBuBut8
s}s
sQ~|ugizuVuB~~|uBuB-
´ µiuu#ut8~|suB~|uBnta
K-} itu
sB
uBz K }s uB uB
K}suBzTt*t8¸s~|uB
Kuq
Ku tai
V u1{ÂV
t8i1z zK }si»ug7»¹n# uK}sufuo~ug Os#qsuB1u*uBz|uB^ug
Kua }s¹~|}ss ~~|B} uBnzuB }s } K}s
8zªi~ªi}sz zu
Kqs¹t8~|u u#{qst8zs ~|uB ºuB¹zK }suÃU1t8itar1t8uBu µi uu8~|~|1zztfÅ.}su~|~uz|#Æ1
ξ
s¾Ñi! }
i
s
s¸su
Q
z uB OuguBo z|Bt8}s~uB}s> PµuguB}su
s}su g1zz|
s"
u u>~ft
szrBu>~as~}Q u1ÂVi1u>Iiz|¸s~|u» # Ì ®g Ê %± $ ½² o{
t8z i} u0t8~ B}s~|zu¾OB~1}s~ ut8-uB
s
suBa-z|z K}suBB¾O#&q t u B~uB%µ >I}s ~|u ~1} ~ u>t8-u
suBB¾uBz|t-z|
uuB}sg¾i&q irBt8u¹uBªi z K}su
§
!"$#%&'( " *) +, -%. /102" 3+45 &67
f "7 )^"1 ipz¸ iq
u¹u¶· }sµP
i¸s1Q~|uBuBz|O
-zBugµi¾Pnu¶·^u¶u^qsau¹uBª
ssz|uµq uBªi
0z|uBu¹sB¾Bq{ #i1z
uo~ÑQ¶Îi$
uo1u{tu1uÂVs-12uBz|uqsu
K
K#uB#-t8¾»Bz|»µ»u Æ{}
t8
*uz¸Q~|u2K~I1z|
su ªft8K}stasz}s~|zz|1¸su i z|}s1t isPÃÄzµu# zªi~uBuBT t;z¶Q:©iz|¾s¸st8¥ }~|zVis
q i} u¹¶z|uz|
KQuB¸s1z|u8ug~|÷g¿ ¾ ªq zqsz|#z|qo~It8z| z|uKt¾{¿ z|s
KqszµiuB u~|µ{uotauBzti^uzqsu>q
{}st8
(t*I©g¸Ii¦u¦ u¾ ¦i¦¶¦¶^ Vqstzz1ªi~|#quBƺ#¾sÆO
{}sz*t*q ¸I~-u µiup¿n~|ªi¶Äu¾{t8tz~|z|uB={Oz|
sªa¶Ñtaqsuuqs t8ns}#u-1z|}
u
K¶$~|*q u}Q6M Pu µ u#z
ªiiuB n auB.#MqQf~|~uB
staª z|
s~u ªfqsu =T » q uQ1uo z|
sª z|
*¿ qsqsuz#1q0i
{µqsuu
zuB
i~|2}ss~|z|¸
¸sz|z ~|zI u ¶·u
t8zu F? > ÷¿ qsAÅ z#@q ¾I uÅoIup>
¾Q ^z #p>uB¾Î~ ¶$»B»»#¶7Æ > qsu¹iqs
iuBu*t82 {~a
t8tau1zoq u
$ u:tf©i¾ ¸s-¥~u < iiz
0¹t8zua
5qsuu µiŹ
~|p>}u¹¾7qQz|
(
n p>¾Q¶O¾O
qs»»Buf»t8z~|i1~}sugi~1-}s1~iz|s
Qz|foÆ KBÎ
Q}s
t8uB t uBz|B> z|>ÃÄqs_uztf>uB}sƪi~|i
0 u z| qsz u÷_Iiiqsuz}s¸s t8~|_qQ1
zii }s
λ.s¾Tä~|u
qs u8q ¿ u z ~|q(uB1uB¸ -iqξÎui¶2¾{}Qqsqs#q uu zt8u uBu1t ªiKug n5i1QiuBz|
K
~|µi
_usªzug(t8>s: -§ ~--µPiz|µ~B}suB~|z|~*ª
uB¶·^i¶7t uqsu>uBqsuz|u1tsuBi-t,ºu^ zi
s¸{uVqsz|_u>
8s¡qsq uoiB~u¹i~1
s}sª> ~-i1z|zt8uii»$
up~|¾{zqst8u¹z
{B}suB%» t8Cs}s}suB~ zz
q ª*u~I
{ta}s#t8iqsuuuit8¾z|uoBqsz|u~Q2
s#zuB-tfu }sugºu~| iu usºuB 1>uBz|1O¶I0}s¶.#z
unQ
s¶ z
u* z ~|uu
sª~uB
sqΪ» qÎZo»z|ä
8qss-#¿nuB1µzuB1 ui¾¿n¶^u21it8qK}Qta.i
un¸Q¶ÄuB~$z|u1¶·¶^it
q 1iu2taz t8ugi~s
si~¸uB¸Q1uB¶Iµiu
q - gµ¾ÎuBz|
(Vtq zªi
{q0_z } uB-ºzi
VB¾ÎB1}sz#¶·{Vz|
sª
uB}sqs~u >
uau¸t8#uBz|
s u }Q¶·~|¹~iqs
suaªzt8z|t*u} P~|µuBz|
ªuBBz|ta¾ uiuB» szuaqsu ZiV#qsz|ä~u
>~|µ-u¿qs*z| i¸I
su1uz1uu~|zu¾gqs¿nuBB8u 1Q}s #Îz|sª
s-- z 0µKQzB uO
Kªiz~|
>µi u
sI 1uiu t8 uB
is} }~
sqsz|uu¿
1t8~
{u11µiKqsugª » u¶·
Qä17
uB¶·i~s18 }sz6~-z|ªiz
Îz
sµi¾ªu
¿2qsuFiu.11iz} t8
Ku^i1zP¸s µi}su#uaªui » ¸IqsuF}
¸Q-µi¶Tuq ut8ut8i.
¸Qu¸QuB¿n~z|uBu1u¶
aqsq u}sªi1q } ~
Du
QFuBE,t* "7¸s ~| "u PuµuBi8ªuBuz|
t8 qsz|u
{}s¿ t8qsuz#qz|Bs~|~a-µK1z i taug }
u }sszIt8uu
{Pzµi}st8ut8#u8uBªizuu¾-F¶·~}s
Q
#ºqs uBz|taqs
uu¶Ñ} Ã
suqszuouP
{Æ }st81iuBt8uBzºs}~ u~|qsu
tauªu#q7uB-»¹^ugpºqs¶Ñizoq 5uBu¹iB8z|t8~¿n1}suB~~z|~-
Fizia
Î~|¾iVzDt8» uz|u
{»$µ qs~|u8µ#ugà Q
t8suqsz u*uug#Æ^iaz
K uBuBu~|ª#u -¶O ziqs
u qsqsu8uz|z|t8t*uV}s~P-µzuBi
ªz|z|
staªVu t8u1qsuB B P» ³ uV qs-¿ zz¶
qs u0z|z|ta}su~|P>µuBq ª>u8qsz|au1uBit8s } }su u8¸{ qsz|u 1
u1 z zHGBuz
s
sugª_ qsu ¶ T 1/T
¶·t*} ~| 1 Z A(t) dt à A(t) u
sug.q uµ-~|}su¶Ñqsui¸ uµP¸s~|u-^zt8u tÆ¿ qsuuBi qsz*1it8Qi
su
K¶Tqsu0uB ug1uBuB* 1/T ¿ qsuu k z*#qsKuB
6Iizz|µuz|
iuªiuz¶ qsu zt8u¹Pµu#ªiuz|^uµ-~|} -u ¿ zqVqsu¶·t*}s~ Z A(t) f (t) dt ¿ qsuBu f (t) z^t8u ~uBz|
sª ¶·}
ºzi
¿ q iu¹usugz|
0zs-µ{z u z|
x{uB»Q¥» 8
9
T
0
k
T
k,T
0
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
k,T
v
-"& 44H "&67 + " 44 + " 7
¿ q z|³#uq8tVu*1ƺii¾
s¿ ~uB qsz|Q #iq_ug
B~7 q
u>¸Iuuqs¿ z.ugz uBq
iuBq t*uf¸s~|!uÅ @unt8q qsz|u~ uta
s uz¿
{
i
8{t8
u Kz|B} t8-~7z|BuBzi
~Î
u {t*Ãó¸ qsuB~u tui» » spq{qs-u¿2zu uKµi~ } u g ¾sz|K
Qz|
t8t8z~z|¶.Bt8ug¶I}szi~qs
u z" uat8o~|sHz =it8}u#z|~|-~zi
s
z*qs
z|BO
5~ q qs¿ ufz qsz zÅo
p,q uqsuB
uBi uu0ut*
¿ ¸ u~qsuitfu»
¸s³~ugui¾Ñ 1´Zu-µ{1z z|}s
u uB
Kaq ~|6q u¿2qstau8 K={}sz
#
-ª #u1
~: {v z
Kzz|ªiuVi}
~ 8uBt8¸sqs~|uBz|tq0-zqsu ~u~| utuB
z|t z-
sªu
{}s{t8uB¶Iz¶Q~Îqszu1.suBt8z|st~|uB-u
Bz
a
K¸Quu ª#uB¸ ~uz|
>1utt8 qQ
s-z|o
fqs-¿ z|~q u :&#-Qiz
K (* , , ) » ªiu zz|
{µ-¿27uu1z|~|~ ´
i (#u *
¾s, uqs, u¹)z|
1¸Qu6} q u~s
#z|-Å
szug@ ºz|u~o
QfuBt*
¸sz~|uz|z|»
s
7ª ¾¶·*
¿ntÁ uV~|u1qs}s usIA(z|
siz*uz,, ~Iq )Q-i¸IzuF
K qsu
* i{¸ ,
uBt8µ-z|uB¸st~~nuz|
¿ Oq ¶·uitnu_q P~~Qµuz|#t83M i
0qsu ~|uµuB~Îu S(* , , ) »^pq{} ¾{qsu u~|uµ-
K u
utf¸s~|u>Pµiu#ªiu( ¶,qsu ) i¸ uµP¸s~|u A uB uB j
j
j
3M
3M
j
j
j 1≤j≤3M
0
0
0
0
0
0
hAi =
Z
* , S( , Z0
* ,
* ,
ÃħiÆ
A( , ) dµ( , ) 0)
* ,,
* , dµ( ,
)
¿ q uu dµ(* , , ) z2q u z
{µ-z|
K2t8uBi} u>
S(* , , ) » qsui¼
uBt u¾{Oq t8unz| qK{IqsuBuB1i~~|uB1u z|Z
f QuB
z#¾-i
su2B
fs-µiuA:&#{¾ (-¾szzu8t #éP
Æ^ z
sª ¶·Kt #qVq 8uo} z|
su>zzqs~Tu Q izz
K u(* u ,}s, t ) » * ,, , Ã #Æ 1 X hAi (δt, T ) = A N ¿ q u u s~(z|*u , i,
F)@ {» zà ©gƺ¾Tqsua
K } ta¿ qsuBuzu ~ÎuBº»0ªi-z¿2µiuu
_µiu¸{g¾
_qsz|
Kufuªi
{µuBz|ªi
_u
#qsuouz|t8
uÃÄvKƶ.z|t8}sua#$uB: ( < δt i
s~|0-QuBu 1/T ¾K¿2u
s
s u{IuB1T=f¶ÄNδtu
{µuBªiu
u#-u¹¶Ã #Æ^-¿ hAi » S(
0
0)
0
0
T
T →∞
0
0
N −1
Rie num
j
j=0
j
N −1 j j=0
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
j
0
-"& 44H "&67 + " 44 + " 7 (
ä
i uBTz|t8s-µu$qszZÎi{t8z.1i
KµiuªuB
1u$ui¾B¿2uO} ªªugÑ uBs~1uqsuO}s
z ¶·it8~|i´ ¿2uz|ªqKu z|tauPµiu#ªiu 1 A(* (t), , (t)) dt ¸{ ~uBu zt8uPµuBªuo¶7qsu>¶·t T Z * , Ã (iÆ A( (t), (t)) f (t) dt, ¿ q uu f iº#7 ~uBz|
sª ¶·}
ºzi
¶qsunzªi
~ t 7→ A(* (t), , (t)) %» Csi7z|
#
u¾ uB
sz
sª¸K Z * , qsuzt8u PµuBªu z|
Vqsu>z
KuBµ-~ [t , t + T /2] ÃÄ¿ z q t ∈ 2 A( (t), (t)) dt I (t ) = T º Æ s ¾ i
su*
F1i
z uB qsuaPµuBªu¹¶ I (t ) -µiu t ¾QzD» u» hAi (T ) = 2 Z I (t )dt » [0, T /2] T 7u1 } u
su>qsu>¶·}s
1z|
α ¸K
7→ 1/T z¶ t ∈ [0, T ], 0 qsuB¿ zu . α : t∈ pq u
hAi¿ zq (Tz)u~¶»V
¸Quu8uBªiu
au#z
K~|~½i¾ à (Kuƺ
s¾n¿ z|z
sq ªf¸{ ≡ f qs¾Ou ¿ ´DqsuBq5u 1if
Kµi~|}z*z|qs
(u1¶ni
Kqsµiu8~|¶·}}s
Qz|º
5z|
¶ α f k Z ¿ z F q z # B u ~ ¶ Ñ ¾ 2 ¿ u u
s * u s q f u | z 8 t f u P µ B u ª u * , »³u α hAi (T ) = A( (t), (t)) f (t) dt B
s-µu : #
{}sªt*z|µ¸IuBu
f ¸{8¶ ´D
8}sz|
i ~uugªi¶7z|Ii
8z#zqsµiuBuot8z|
Ku uBª¶Tu#z|t8 uu δt ¾ T = kN δt ¿ q t8isu_}u }s t ¸{z ugs1}su # zi
ÎC(k, ¸¾sIz|
Fu Nj}Q#−q_
1, j)¿ ¿2q P0z|#q q ~~o¸QqsuBu~z|io
sª5t8 sk}[0,#-Nz|
F− z|1]o»¡~|tapKqsuBu_¶·uB
{u}sztf
(¸Q1uBjt8 ,..
.z,¸Ij
u aqsut8s}#-z|
¶$q u>#-uB1i (* , , ) » ¸Iz|uBt*8}sp~qs-¶uzi¶·}
3t*¾Ñ={}si
s ~--¿ Ã>~|iu ~Æ ugz|ªiu¹8¾$ªiq uu
s¸IuBu8#qs~uBu5z QGBt8iu6z|uÃ
i Æ ¶
6{Ã #iuBz
(fÆ
sz|qs
u8a1u ~|z uhAiuu#t -}s(ui¸I» (δt,u'}Q¶nTu) =}sz#
hAiu¾Î
{~|uB}s(δt, ~z|
¶·fTto) »n{-
pzi8t8
Fzqsu i}s}sªi ª¸IuBuBz|-
8¶·u
Kt z|
7qs» u¹ug uBB¾K z1} ~|~|fi
Vqsz|2Iz|
K2ini
0t8uªiu
suB~ z}suBB¾{¿ z~|~Tug1uBzµiu T
0
T
T
0
T
t1 +T /2
1
1
1
1
1
t1
1
1
T /2
(2)
1
1
1
1
0
[0,T ]
[0,T ]
(2)
T
2,T
2,T
k,T
[0,T /2]
T
(k)
[0,T /k]
k,T
0
(k)
k
k
(k)
k(N −1)
(k) num
j
k
j
j=0
j
kN j j=0
1
j
k
kN j j=0
(k=1) num
Rie num
Ô·Ü$ÒÎÔóø
!"$#%&'( " *) +, -%. /102" 3+45 &67
q --µigä
_¾u3¶·:& #sfz|¶·
Î
¾s~|1~z-¶.ui¿ ¾$z|
sqs{ªaut8uµsuB~|~iuBº1zuBz| Fz|t8Å.#qsuBuBta~|u u1>8u #¶$q iu t8 uu0 zr*{z| ta}Q ~u ug ºzV01i
ta }¶u>i qsuu> r#-=ugº2iÆ1¾.(¿n÷uuB
~~ 0
0
| hAi(k) num (δt, T ) − hAi | ≤ C
1 + δtr0 Tk
à iÆ
¶·iit8u>
K C ¿ qsz#q Kug
s uIu
T δt » z|
6z|t*}s³ qs~uV-uVz}sui
Q
uaTu~zqQz|
s-uat q q uu (Iiq uu8z|Q¸sii~|¸u>~{¸K
sz0
siutaqszu¸{~ ¶ÄF ºuB qsPuVq F-z tf¶2}sqs~qsu-uµ-z|uB
6~|}sug> ¶¿ A
zu8q* z|,
sug, ªI~|uauB1 #u-1uBi1qsiu8u~i-» u ³ ~Oq z|
uz|ta
z|ut8
suu ¶ ¾ }Qug.
Ku2~|ot8uqs ¾iqsu µ-~}sug A * , , u2
s^uB~|u
{t8u¾i
q u ui i
s~| ugPO-^qsuo}s
sz|µu#~#-u 1/√T ÃÄuu~|>q uu1sta ~u ¶Iqsu }s¸s~|u¿2u~|~sz|
x{uB»$§ » v{ƺ» q -t8p t8qsuo
sz|ªzsiqK
aK¶·}s¶Iz|¶^qs~uBÃu$KÆ:&º#-z¾it8 ~ugqsu_q itasu>
uB¶*Æ2q ¶uq u}suBt8z|t8zi
u52Ã }sKÆ
* ¶·uB8¿ z
Kqszu#q0ª#qs¸ u ~u uBz|tta-z|~u
z|
¸Iu z|ªuit}s8i} ~z8¶·s{z|
s-ªµu ÃÄ-» q qsu ugztuVà KÆ2~8qsi~ 2¶·i
s
s´Dz|
Kuªi¸s~|u¹ut¶$siºz|B~Ñz|
iuuBB» ³xuBu»Q§s»|©goÆ1s¾uBq uu
K
uBqs}su~o>zu
_ ~uBqsu8si¸su~|ut ¶^0ÃÄuuf~|~x{ugugºz|»
§s»Û¥Æ1¶O» z
2 uIqu
u
Ku¹tqQtau iz|
s
Kz uªii1¸sz|~~|~u-¾ #
ÃÄuuu ~|z5¶=u(~|¦( f¶Äz|~~Ñ~ug¿ azq }sz
¸ugq ºu u qK {I(qsqsuBuuBÎuB
s¶
} ´ä¤iqsuB
sug*u1Iz~Tu
Kugz}s~D~»B».C$ä
z|
x{ug~|~i»O¾O§sz|
» §s¾x{¿2ugu » §s} » v ¾^0¿28u} u t t¸Q
Kz
>z|z|
~u¿ uBt8}sqs¸sz#t8q(z} qs u uVz| u {_z|
_uB#tqsiuBt8}s¸s
5z~|u#¿nPuB~
~.¶· Isq{u
K~|z|z
Bª~O¾sz|z|
z
t8¥uifq »0u*»7ppuBqsqs
uz>u>u1-sq K~|´Zt8#- qsz u*iu
(~.Qz>¶nu
KuBqssuz|ug~7¿ q uBqs
Kii~-uuBszµiµqQufuBu ¶~ z ÂVi11u }
s~ guB» u u *i1} u~|ft8s}u q uta {
t8z| iQuBzug.z|^z
8qsuBuBuB. z|}s~~ zi ~.uC #{^Ot88qssuo~|zuB
KºuBz|=iªau #-z|
K¶7uouªiqst8us z|~|{u1
6
uB
st8#uBqszuBBt8¾i~Îu
¾Ñ ¿2uauft qQPqsµÃ-©Pu*¿½Æ >}Qq u niqs}snuBu s sz ÀTiuu#qa
Kz^z|~|
ªiu
zq z t8zµi3u : Kt8* s~|uB1z,
¶ φ δt ( , )= * , B j
j
j
j
φVδtV
(
n,
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
n)
VV δt
n+1
n+1
-"& 44H "&67 + " 44 + " 7
1zitaq t'Iiz|
z1z|uB
i2ug^¶qsu v Bµi
u~|t81zu~|¾ Å.u~u ~ªiz qst ¿ z qz #uB~ ¶fq qsu>i uBn¶7qsu~|ª´
*
n+1 ,
*
n+1 )
(4)
= φδt (
1/3
1
,
*
,
= φVb1Vδt ◦ φVb0Vδt ◦ φVb1Vδt (
*
,
¿ z q b = 1/(2 − 2 )
b = 1 − 2b » 1zitaq t'Iiz|
z1z|uB
i2ug^¶$qsu ( Bµi
u~|t81zu~|¾ Å.u~u ~ªiz qst ¿ z qz #uB~ ¶fq qsu>i uBn¶7qsu~|ª´ (
,
(6)
n,
0
*
,
n)
n,
n)
1
*
,
¿ z q c = −1.17767998417887 ¾ ¾
c = 0.235573213359357 c = 0.784513610477560 » c = 1 − 2(c + c + c ) ä
6~|~qsu ª} ug¸Iu~|-¿ ¾Ñ¿nuas~|>uzqsuBqsu8
K} tauBz~7zt8u8PµuBªu hAi (δt, T ) ¶n
i¸ t8usµP}#¸s-~|uziA
~Îfiqsuz2uq iu | tahAiu¹¶·i(δt, ~|~ÑQiTz
K) −µhAiuz| i~~|fVF~|¶·z|}sª
s
u1z|»
¶ T » n
u K}suB
K~|¾.qsu (
n+1 ,
n+1 )
= φδt (
n,
n)
= φVc3Vδt ◦ φVc2Vδt ◦ φVc1Vδt ◦ φVc0Vδt ◦ φVc1Vδt φVc2Vδt ◦ φVc3Vδt (
1
0
2
1
2
n,
n)
3
3
(k) num
(k) num
!" #$% &'(
ä
qsu%q z| t8uBz1~iz|
s
Îz|¾
¿2u 1
Qz u 8{uBt,¶ µu> z|~ugnz|
©B ¿ qsKu {
tazz uB1z|¸Qu ¸{ ¿ zqqsu>¶·~|~-¿ z|
sªas}s~z|
>
H=
5 X p2 i
2
i=1
+ ωi2
qi2 , 2
7Bu1
¹} ¸Qou ¶·i¸¹z
z
u
¸K1ua
} ~Kzqsua~ÎPµiu~|#}s~|ªiu z|¶
A> = p » pqs» uau1siºoµ-~}suf¶Oqsuau
utf¸s~uaPµuBªu = 0.5 7 1 u
} s ¸ | z *u ) 0#q uB =¿ qsuqsuBqsu q{{QqshAi g u g u 2 pqsq u ufz|
Kt8uªiz|~
sz|z
_
V¶·~}sªi
ºz ziqs
6t ÃUz|¹uBuo{t8sQs~|uBuB
1 zz»#p»|©gqsÆ^u8uo
su-~u K z|u z| uz 8» s³
- u¹µz|u¹
iu¹u}sªi¸sug~|u¸sz
z~|tzuz-qsiuBuB n}sft8u> -z|µiz
QuQu
si
» ³ z|us} -µuuoqsqsuu ¶· zit*s}sq ~
KÃ Kz
sÆ$u ¿ iz q }s t8z ÀTuz|u
K
^Ãĵ-u~u*}sugx.uB» ¶ #s» ¥ià ƺ» vKÆ1¾¶·n>ªz|µu
Ã
¿ zqfq u2i u´ (~ªiz qst»$n~|uB~|VÃÄuu Cz|ª »2©gkÆ1¾-zk
Q1=uB1z|
sª ªizµiuBo¸QuδtuB.1δt
{=µiu0.025 ªu
Q1uƺ¾» ω1 = 1.0,
ω2 = 8.01256,
ω3 = 12.25245,
ω4 = 17.234,
ω5 = 20.98765.
2 4
k
,+
- /.0 21 231 '45
³ u
suK}s
8qsuztf}s~-z|
¶$qsu >uBs~|u s¸s~|ut» ³ u1
Qz u2qsu%taz|~
z|
,
H=
2
2
−
|
*1 | ,
Ô·Ü$ÒÎÔóø
!"$#%&'( " *) +, -%. /102" 3+45 &67 (k)
| hp24 inum − hp24 i |
s¦ »|© ¦ » ¦ © ¦s» ¦¦ © ¦s» ¦¦i¦s© ©u´Z¦ # ©u´Z¦ ( ©u´Z¦ ©u´Z¦ ©u´Z¦
k k k
¥v©
©B¦ #
$z|ª}su*©
©#
¥¦
¥#
T
> 2 i(k) (δt, T ) hp2 i ! "# k $ ! % &' (") *+ , ! ! ! C .- ( /01!2")! ! 3 "4657hp 1 48 (num 1 9 3"6 ( 4 4 % q1 = 4 $ p1 = −1.2 $ qi = 0 pi = 1 i = 2, . . . , 5
uB17u1z¸s8z|
s} ª8¶·if§ z|
z|1u~u>z|
} 6
-q uiPºµiuz#µiuªuni}s¶ ~|t*¸Q* uB
i»5z|~U» fz|
q usuµ{z|} fu1sta ~ui¾qsu uB
uznt*¸ qs~u¹uzP
µz uBz|~ªÃÄua
suªK
-z¸IµiuVugÆ.1iuB
stau ª}u
~|Kz^zqsA~u¹~|
s>=h||t * |i¶Î| =qsuoz|3
sz+z2~ E={z
Lu1zo/(4tai|t8Eu
K|)}st¾¿ »Oq puqsuu E L Kz|t*z}s~|~~--iziB
_¾{z|q t8"u utau6z|~z|
δtz|
=¶·} 0.01 Î ¾
¿2
u8} ~|Kufzq ~7uai
uz|
K´ (Vuªi~ªi¸s~|uz ¾qstqsu»*z|
i¹uªi¶·i¹z|qs
u8q #qsuBt8t8u
z|z|
º z i
| z ÃU{uBt8usx{~uBugº» z|#¾.»Û¥¸sƺ}» f¿2uuV¸s~|u0
suBz qsuf(s-µiu
s* z|s-µu8qsu z| q
Kz|
su}st8zi
¶Äit8ºs³ ~z|znua} ªuB
su t8us¶z}¾{qsu*u
K qsu8
szP~|~|µu#z|ug
ªiuBzº¿ 2u zqsq u k¸Iuqs#ªu z|
s
s>ªi
szuBz|
ss
sªV~|ª8u¶·2i¶$st q i¸su©*~|uz|t t*» #}s»~--³z¿2iu8u
0µz|uBu
Bu1¾iuBq }si-gug¾T> isÃÄ ukKuzC$#
Qq8z1ªQ¶·uB»O1¥}QƺuB»OBug¾Qp^q iqsz|
u ~|Iu
ªz| z|t*Æ1¾{}s~-qsu zi~
(ªzuBt8 ug¾{» qs³ u>u0¶Äiuuuq uq uuf1q
{-µig¾7u¶·ªiu
QT1uf≥ÃUuB40uC$ÃÄzªQs»s§KƺP» zt-u~|_¸Iui
vuBµ~|}zi
0
0
2 0
0
0
k
u } z| 'o
1q
Qqsu z#¹ u
KBIu
u¾I uB¿n
ua1u>u*
¶7 q ¸ u¹~u*uB#nq ¿ } uB #z =qqq q ug IuBq 1nufug Tzt-
uδtà iÆ z|
z¹i q uB^7» Cs#iqsuB =qsz¿ qssu} qsIuBi2uqs¾IuB¿nuu C (k) C (r ) 1
2
0
| hAi(k) num (δt, T ) − hAi | ≈
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
C1 (k) + C2 (r0 )δtr0 . Tk
©g¦
-"& 44H "&67 + " 44 + " 7 h|
*
(k)
|inum
©» (
§# ©
k k k
©»&# ©» v ©» § ©»Û¥
©B¦ © # ¥¦ ¥ # §¦ § # vK¦ #
C$z|ª}su¹¥ >&") 3 ! 3 !" h| |i (δt, T ) 2% 31! % &")!/ 3 ! ! 1 1 ! / $ ( 1 ! "2( 4 5 3 1 !" (9 4 = (0.9, 0, 0) k = (0, 1.1, 0.5) T
(k) num
~~$µ-~|» }s³ uBu 1¶ δtt8s}s
u¹
sqsu ³ {fu ~QªuoµP~} uB2 u1¶ uBTt8¾z|}
s#u q0Cq (k)»qsp7u¹ ~ug z|
sqsªfz|B¾TuB¿nt!u*¿n¶Îi=q u¹¿ uBz qF2µuBzCt(k)/T ¸IuB}
~ u u uz|
iuB QTuB
|uBhAi »uZu ¶^} qQ-uB¶·iz|t -uz|¹¾Qq q uξQugqsBz ~|u K } u
Kz zqs¸I} }s~
¸Iu u (δt, ³ T ) − hAi | i
~ ¶ » u Q
| z
¸{0¦ O» ( ¾ 1uB$u Í È z|
s,^ª8È $z
Í È qsTu® $É t8Ìu¹ T¿r P8°$¶·Í δtqs»u qsu µ-~}sug¶ k ¾=¿21u ¸#z|
k 1
1
k
(k) num
k
0
C1 (k = 1) ≈ 0.6, C1 (k = 2) ≈ 14.4, C1 (k = 3) ≈ 212, C1 (k = 4) ≈ 8270, C1 (k = 5) ≈ 250, 000.
Ķiº>Oq }s-snz|qsz|uo
siª~|1¾ z|~~-qsuaziµ-
O~|}suBOoqs¶uoC¸QuBª(k)z|
s
sKz|
s}sªz ={¶Iz|qs
uougz|t*}su~-¿ zziq
Vk÷u qs~|z¹ªiz|uo.z|¶·
Fi^~ªiuBªuut8.uµ-
K~}s¿ ugOzqF¶ kqsƺu » ~i
sª´ zzt8µiäu u
_(i tuBu1 ~|~^ ¿ z¿ uq5zuqVuBt8 Qz
ugu º.qs{uf*µ-qst~|u0}suB~|~suBµ-t ¶ ~|}sCu(r¶ δt) ¾T»%¿nCsu*i¾. sk>uq u1uV ¾Ku
{TiµuBzOz*~u~|0 ªsui¾-
Pqsz|ut1i-utauBt ~}FCuaªz|(k)/T µuª
z|µ¸{uB
C (r )δt ³ È Ä® È ¯ Ì Ì ¯ Ì » u
µ | ~ s } g u ^ · ¶ i ¿ s q z # q u I u
¹ u C (r )δt C (r ) k T ¸{ C (r = 6) = 0.163, C (r = 4) = 1.09, C (r = 2) = 0.622. p s q g u u {
s } 8 t uz|B~^uQuBz|t8u
K*qsuBu¶·u01i
t qsuq
suB¶ qsu 1E, "7 "nuBz|t-u à iÆ1» 1
2
2
0
2
r0
2
2
0
0
0
k
1
r0
0
2
0
2
0
Ô·Ü$ÒÎÔóø
©i©
!"$#%&'( " *) +, -%. /102" 3+45 &67 *
| h|
s¦ »|© ¦ » ¦ © ¦s» ¦¦ © ¦s» ¦¦i¦s© ©u´Z¦ # ©u´Z¦ ( ©u´Z¦ ©u´Z¦ ©u´Z¦
(k)
|inum − h|
*
|i |
§# ©
k k k
¦
©B¦i¦
#
© #¦
$z|ª}suV§
¥¦¦
¥ #¦
T
> 0( 1 ! " ( 3 ' (") C (k) % ! ! / 4 5 3 1h| !1|i"num (δt, T ) h| |i 9 4 k $ % = (0.9, 0, 0) = (0, 1.1, 0.5) ,
1 &$ # $ & ! 1 & !
ä7
u
s
Qqsz ´Z¤ig
¾ uB¿2nuQ} u
Kz|V~U»OpqsuBu t'ot8¶ Mz~i=
sz60
¶qsz|u~ug z|
_ut!§ uBz|
KuB iºz|
sª8qsi}sªqa}s
Q-u H=
M X i=1
,
2 i
2
+
M X X
V (|
i=1 j>i
*
j
−
*
|)
i
¿ zq V (z) = 4 − 4
V (z) = V (z)−V (z )−V (z )(z−z ) z ¶ z ≤ z ¾ 0 qsu¿ z|u» z z oqsu81}sÀ5Æ$#
z|} }Q unz qs=u2t83.08z
ztf» ³}st u z|ptsq{s} ~|ªiB>uo¾TQ¸Q1uBi
{z|qFµ u
Kz|qsnuaz¸Ii
Q}s:
©uBz~$z
~Ñ z
zqsi uf
zn¶·z|÷#
Q1qs2uBu oztfu¹uf}s}Q~-#qV
Kz|z|
*qQ
{-}s¸Q2P} qs¹zu -[−3.2; 3.2] u Q
z z¶Î~Oqsu
z uBzit,
ξTz| qs0.229 uB
t ³ qsutu8Pzµ
uÎ#z
*ªiu~z ¶^K}sqsz u s={q z
u1uz » *u{t8IIuq #uB-uB}s7uaz~|~ug2.50 » z q
} # 6 q (
z s
z 8 u { qsz|uu8
1u>u*¸I
su1¿2¹uuBu
s
}s ªiq_z|z|
K~µ-ug¾ z
K¶·}
ºzi
Æ1»*ºpu qsuaz|
*¸ suIµ-u
¸s ~|z u* } #
u.u>
sut8zz
uz|
_÷z|
zo qsuaz|t8}suB~|B
¾ LJ
12
LJ
6
LJ
c
0 LJ
c
c
3
A(
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
*
1, . . . ,
*
M
M) =
XX 2 | M (M − 1) i=1 j>i
*
j
−
*
i
|.
c
c
©P¥
-"& 44H "&67 + " 44 + " 7
³ u 'o¿2
u=¿ ªKz z|q
ξqs¿2u uµuBu~u q z -g¾KÅ.¶·uBO~|u1~ª~u ªiuB
sz qs}stªiqV¾} z|t*z|
s} ª~|fz|z|
8t8uz|t8uBu ¾¶ δtqsu=~5.10 » qsu¶Äiu^qsu ª B u ¾ T k
{µuB ªiu
ufÃUuBu C$z|ª ».v{ƺ» −3
*
*
(k)
| hA( )inum − hA( )iestim |
¦s» ¦¦i¦s©
k k k
§# ©
©u´Z¦ # ¦¦i¦¦ ¥¦i¦¦i¦¦ § #¦i¦¦i¦ # ¦i¦¦i¦¦ #
C$z|ª}suv > ,
¦¦¦i¦
(#
T
! ! "& ! ' ") (k) ( """ / hA( * )i1num ") (δt, ! T!)" * hA( /0 ' ()i estim 3" ( ( ! ! " k4 $ % % % %
3 1 & # 0 45 $ 1 ' % %
ä
qsz|QB¾s¿2u uB~Ñ¿ z qqsu t8z~i
sz|
H=
p2y p2x + + V (qx , qy ), 2 2
¿ q uu
uB 1q z¸s2z|
sª8qsuu¥ 1z zz|~ÑI~u¹z|z
K
V (q> i¿n,}s8q¸s~|ªi)u¹~i=¿2¸ u(q~|~Ñ~Ñt8I−z|
s1)uBzt
K8z+~|~D(q»p-q +u u¹ qI- −u
K1)z,~ V¾s-ªi{uB¿ nq z|#q +∞ - z|
sz ¾ 0) V (±1, 0) = 0
pqs
su8u ¸Iuq sP µ{~|u>z|Q}si¹z
K¶n~|z|t8-u8u P µiu-# (0,ªiuB1)¾s-
sªi¿ ~qsz# q uBQVuB
(0, >1)i
_=1q »ua(±1, QB
¸Qz1u ~|u*z ¿ uBz
iqFz QuuB? I> uBºo0qsu8 s ~|u*Iz|
K¹uB
suª ÷qsuBufu K} ~7µ- ~1}sÆ1ua» p¶nqsqsuBu8u uB
szÀTuuªu
KEouªi¶nzt8qsuBu x
y
2 x
2
y
2 x
2
0
Ô·Ü$ÒÎÔóø
©g§
!"$#%&'( " *) +, -%. /102" 3+45 &67
©»2¶·1i
KµiEuªut*~|~uB ~|zt8q z
1
¾ } qsu*z
ª#-q uguº ~i0uBz|
s
sª ~|¶·}su
Qsº~iz|
ugVz
zt8ª~|su ¸ -µugz
7»qspu¹qs1iu
Kµiz|t8uufªuBP
µi1uu ##ª-ugu ¶·it 1/T 1/T Ãó¶·i
¸sz# k Æz|
qsu sufµ{z|} 2u1sta ~ug B ¥»2¶·ug~|E~ 䶷t*uBu}Q~ # oq_q ~|u¹ªi¸Quz|q u
1 ÃÄqsPu 1Ei
Kµi≥u5ªuBƺ¾T
1qsuou8¶7 qsu>z1~|z|u*t8zu>PµuBu
sªuBuBªi^u1z|z z|t8q z-~ 2z8 Kqsugu>¹
s
su ¸ uµu z|
qsu # Bu B §s»2¶·
E
¹u ~| z Kug~|u ¹~|
sªi u¸Qq uB
µiua1 ¾-
Kqs_u
{µuuBt ªiu
ugu~|~ ¶n ¶·uqsuB~|u8 nzt8qsu usPµuBuBu
ªugun ¶·¶ ¿2 ztf }sz|~-z|
z|ºO
(¸Qz|t8z
uB 1i¶O
KµiqsuufªuB
t81ufuoz
uq ou s¶OtuBµ{zªii
s} z2}¿n u8÷iiugu µiuÃÄ
u$u ©g¦sC$¾ z¦ªQ¦i» ¦a#Æ1z|» taug¹~i
sªuB#Æ iqsiu*¿ q z|#q_~|uB ¦s» ¦¦i§ hq i © ¦s» ¦¦K¥ k § k # ¦s» ¦¦ © k ¦ ´ä¦s» ¦¦ © ´ä¦s» ¦¦K¥ ´ä¦s» ¦¦i§ ´ä¦s» ¦¦v ´ä¦s» ¦¦ # ´ä¦s» ¦¦ ( ´ä¦s» ¦¦ ¦ ¥-u ¹¦ ( viu ¹¦ ( (u ¹¦ ( u ¹¦ ( ©u ¹¦ 0
k,T
k
0
0
0
(k) x num
C$z|ª}su# > ! 30
T
"6 &1 !0( 1 ! ! 8 ( (4 - !") (k) (9 T !/0 *" "4 $ ( ) ! 1 hq x i num !/0(δt, *" T ) % $ δt = 0.05 2.108
qszt8u
u #1K¾}s}
z ~=K~|~|¾^*}si*
7 »$iuB
sä
Vuªi{BuBqs¹z
Vz|ut8u0s¶·
~u¹iztu1z|s
(C$tazzªQ
K » ~ui0( ¾i¾qsqsufu¹qsu~ qsuB>z|¸ z|
~nuz
7¶ÑI¾Ñuzqs
(
uo ¿ ¿nqs fz|_#q6¸ ~|z
>zª
QQnuBz|t8
u u =K
z|
s-¿ i
0
sqsu0u
¸ i~|z|
qs
Ϊu ¾ t8Qug#
Ft8uu1z >5z Guz|t88u¿2´ä
6¸I-uauug_z~| =-uBµz|#tq -z|
u t8¶· tu~ : _ ¸Qu¿nqsu uBu
6{t8q t8u8u1¿n_¶
{qsµuuBQªiu
u
Ku*z|#~-Æ1¾ugz| 1uiI
t8u q -ug*uB
qs1uu¸Q¶nu~|µuuB
uB ~7=6¿2zu
~|~ ´äqsuu t8us }#¸ -iz|z|
BѶ z
FPµiqsu#z¹ªiuB¹u¸{¾I(qs¸sua}u
{µ¶·uB#1ªiu0u
Fu 'zuftfz|}so~|
s0z|~|
*
ªz|u¹qsz|
u √ 1/T zt8ugi¾ ¸I}su1u¹u8g¾ q ¸su }si}Q¶Ä~|~|z ´ä -¿ #q
z 1/T
¾ zz|
Q~ug¸Quz¿n
uBuqs
u#qsu
s¸ ªiu>iz|
¶$B» PiuBz|¸s~|uz|t*}s~-zi
p#-uB1i q (t) ©»&# q (t) © ¦s»&# ¦ ´ä¦s»&# ´#© ´#©»&# ¦ ¥¦ vi¦ (¦ ¦ ©g¦¦ z|t*}s~-zi
0zt8u t (" ,1 ! ( 3/0 ! ! 3 4 - !") C$z|ª}su ( > 3 ! ( 4 % $ ( ) ! 1 !/0 *" x
x
x
%
$
qx (t)
t
δt = 0.01
¶^q u8¥ }s¸ ~ua¿nuB~~.
_u¸Qt u } Q
# 7ut8¹u1 u1} } ¹z5 Guiu
» Z¸{oq zz| q uguBuBu>1Vz|
sª
F~|#uB¸{qs _ug> =0 q z-o} qsz|uf
_z|{iu
_
suBq ªiuau1uBz ªi }s z|¶Äiz 1u H(q ,q ,p ,p ) = E x
qx px
q p = ± 1 + E0 cos(θ), p 2E0 sin(θ) sin(φ) cos(ψ), =
qy py
= =
y
x
y
0
p E (sin(θ) cos(φ) − cos(θ)), p 0 2E0 sin(θ) sin(φ) sin(ψ),
éB¦iÆ √ ¿ z¶ zq (θ, φ,»ψ)Zz|∈n[0,q uθu1¶·]×uI[0,iπ]z|¸s×~|u[0,a2π]t8¾¿ sqs}uBuou
{θ}st8u=zπz~|¶~|*Eqs≤u¹1ÅA@
u
θutf=¸sarccos(−1/ E ) | ~ o u P µ B u ª u ¶ {
E >1 ÃÄuBª}s~ºÆ^¸ uµ-¸s~|uo¿ zqaqsz|ªqB1}s#»$Zug Z é©gÆ A (θ, φ, ψ) w(θ, φ) dθ dφ dψ, hAi(E ) = M
M
0
M
0
0
0
[0,θM ]×[0,π]×[0,2π]
Ô·Ü$ÒÎÔóø
©#
!"$#%&'( " *) +, -%. /102" 3+45 &67
¦s» ¦s©
|h
PN
j j=1 qx iic
| /N
p
755/N p 1.96 755/N
¦s» ¦¦s© ©Bu ¹¦
©u ¹¦ à ¦s» ¦s©gÆ N δt
C$z|ª}su > " !/ 3 % * 1
©u ¹¦
% " /01 1 ! 3 3 ( 4 - !j=1 ") 9 ) ! 1 % $ ( ! 3 . 1 !$ 1" 4 N 1 X j qx N
N
% ( % !
δt = 0.01 h·iic
1 1 "
¿ q uu w(θ, φ) = C qsin(φ)psin(θ) zqsu8z
{µ-z
Ks¸ ¸sz|~z u
z¶Oqsu {
t8z~ Tt1~|t8zHGBzs
s}sªVu
¶·it
K##ƺ-»äug
Fºpz¸sug~|ÑuV¿ ©z ¾Iq ¿nµ-uf1zit8} Ñz|
su>z z|qs~izuB
zuBt*z|¸s
Q~|u » Pµiu#ut ªiu¿
z(θ,q*φ,uµiuψ)#~PÃ Czt81z|+u^PµiqsuufE#
sªcos(θ) g u ä¸I
u>ªu1uB
ssu~#z|
s~D¾iu qsuugzu¹qsuBz|t8¸Ku 8Pµqsuuo#¶ÄªiºuBOq -u zq ÀIuouBzuB
K{nuB
s¶·ut,ªuzqsu>}su
¶Äiu1tfu¹¸sq ~|u¹2Pz|µi
{uµP#ªzui»7
Kpqs}sz¸Q uzÀI>uBÃóuBqs
z|1u>tP
¶Ä¸Iizu t8º ug}s qQuo¿n-.u fBq qsu
u>-u1ÀTzu t u»
z.uuBªi¶ f z|nuB¸s}
O iz
f
{µ-µiuz|
K~i^
sªqQ-zt8¿nuu z ~ug
s¾-tf2} t#q8
Q~iª
suªuBaz q u
f
Kzq ¶·suƺ¾z|t*2}s¸{~8-ziqs
u ¿2qsuu2#C qsz|t8i{tiuPu oµitsu##
{ªi10uBzÑz|
s-µiz~iuQzi~$z
KuB1µu
#¶szµ{~ z|z uiz¿
ÀI¾-uBuB
*
Ku^¿,z|
sPzzi¹~s
~|
qsµufu.z}sz|qsz
Q¶Äz»u C}sz|un#¶Oz|7g1i¾
>$1z|i
at8
K¹suB}
su^u
Kªqsu u22t8uB¾I~|
*t8µ-u1~| }sqs u^z|
sª ¶ ¾ ( , ) E C aqsu>~~7sqsug¸ u*¸szz|
s~zz z~u1
i
z zz|w(θ, ó Ã 8
{ ¾ 2 ¿ o u Q } u s q u 1 B u
1 u Z ´ u B u 1 | z
t8uu1¾sq
: qs- qs)z
t8 u1qsqsu {z|
u~ aªu{µ{ z -z|s
sσPztz|
V(E¶)qsu ¶uqs
uBuu tf ¸s-~|u>#sP»Oµi u#ugªi}su~» u 2
0
i
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
0
i i
0
M C+M D
0
©(
-"& 44H "&67 + " 44 + " 7
o¸ uµP¸s~|u A '
qx qx2 qx4 qy qy2 p2x p2x V (qx , qy )
p$ !¸ "~u5 © > ! /
¦s¦sä´ ¦s»» i¥» vivvi(v§# ¦ ¦s» ¦»Û¥ {©# #s# © (i¥ ¦s¦s»Û»Û¥-¥-vvvv ii¥¥ ¦s»Û¥ # #s©
hAi(E0 = 0.5)
*
ä´ ¦s¦s» »¦§i ¦ ©© ¦s¦s¦s» ¦»»|i¥©v{i¥¥ # ¥s# © ¦s¦s»»|§©v§ v(s© # ¦s»Û¥ # {©
(k=5)
hAinum (
1,
,
*
,
Z´ ¦s¦ » » ## © # -¦ # ¦ ¦s©» ¦»» ¦§v viv K¦¥¥ Pvi¥# ¦s¦s»|» §i© (¥ # vPv ¦s»Û¥ #§ © #
(k=5)
hAinum (
1)
2,
3,
*
ä´ ¦ ¦s» » i¥# § ( # ( ¦s¦s¦ »|» ¦»©©Bvi## §© ( vi¥# ( ¦s¦ ¦ » ¦»» §¥ # #{©¥
(k=5)
hAinum (
2)
1 ' ( ! "0( 1") / 1 3 (9 !/0 ' ( ,! ") ! 3 4 % 1") / 1 3 ' ( " ( % / 1 ' 8 / 4 !/0 % " / ' 8 / !% 3 9 !" ( $ $ ( $ ( , ) = (−1, 0.5; 0.5, −0.5) 1 1 (9 ( ) " 1! 2 , 12 ) = $ , ) = (−1.05, 0.549; 0.3606 . . . , 0) ( (−0.9, 0.772 . . . ; 0, −0.5) 3 3 4 - !/0 " !" 4 ,")!/ 3 ! . ! ("# !% 4 δt = 0.005 T = 5000 $ % % % % % #3/0 * , !1 "1 1 9 4 k
©»|© ©» ¦ @^
uBp t*z¸st8~|uuPPµµuBuBªªuaua÷÷
s
siitt~|~|z5z5GGuu ÆÆ ©» ¦ ( ©» ¦v ©» ¦i¥ © ¦s» ¦s» ( ¦s» v ¦ ¦ »© ¦ » ¥ ¦s» § ¦s» v ¦ » # ¦s» ( ¦s»
$ z|ª}su
") / ! C > / ( !") ' !/0 ' ( ) ( " % ( ) " 3 ( 1 1 ! !" 4 ! E0 $ 2 4
hqy i(E0 )
E0
' ( / / 2 A =1 qy$ 3 1 $ ! % ! 9 ( 95% ! " ' / ( ! $ ' % ") / ! % %
Ô·Ü$ÒÎÔóø
,
3)
©
!"$#%&'( " *) +, -%. /102" 3+45 &67
√ ¿ qsq uup#u-qsunMut8uB¶z
*qsµP
{uVµ~uB}
{u}sªit*uµ(E ¸
Iuu**)z¶ z
{
µz uBz|ªi~nuBI1i
√ q zu^z|¾u
Q¿ Bqs»uBut*Z¶ u¸s~|u^PµuB
ªu hAi z| iqs-
iu0$u2u
σ ~ 6t8 u1uµ{qsz (E- ziz|)/*
(} ¶ Mu ¾¾ σqsuV1(E )uªuB
1uf1
Q#
K A ¿
z q6uBQugº>u q u1st8 i¸ u ¸ σz~|z z
C$(E z|uª
» )/z > M¶·iw(θ, V » p i {
µ φ) σ (E ) I}sz|¶Ä
Kσ u¾su>(Eqsu*} uB)u
ut*¸ ªi~uua
suBPµu#u>ªi_u hAi#-z|¹ug1ºita zug}28u ª
z|µ ¿ uB
z z|ta
{q(}sutfPªi¸QµiuuBug#-ªiuB¶uBBI»Bz|
i1}s##¶0qsz¶.u*1qsiua
K
KuauB
s2uª~ x{
0 uBµ{z|z|
¦s&» # ¦s» v # ¦s» v ¦s» § # ¦s» § ¦s»Û¥ # ¦s»Û¥ ¦s»|© # ¦s»|© 6 ¹_¡t8u1qs ¦s» ¦ # ( t8u1qs ¦ ¦ ¦s»Û¥ ¦ » v ¦s» ( ¦ » © 0
M C+M D
MC
0
MC
0
M C+M D
MC
0
0
0
E0
C$z|ª}su > / ! ( % ") / ! ' hq i(E ) % ! ! 3 *" (9 1 3 ! ' ! !/0 ' " ( 31 8 ( 991 !1" ") !8 !! /0 !' ! σ ' (E ) !"% 1 "6 9 / ! 3 9 9 3" " ( 9 % 6 % ")! 0 " 91 ( /0 % 4 (" ' 3!1 4 $ 1 3 ! % σ (E ) % % %
M C+M D
MC
2 y
0
0
0
p z q{z} iz|B
ta¾K2u8P
V1µut8qs#u>sªi}s1u»8u>
Q,#
0ª
KuB{2
uB u
st*u¸ sª~su¹PP}sµzut¶Ä#1ªiuiuz|¾¾i
F¿n
u¹¶na
qs} uV} u¹uBª
uBªiuBug#t*q0^¸s~|8¶ÎuV#PqsqsµuB{uBt,iuoª8ufu1z|µ>iuBt8qss~u} t8zu ÀTuB*u
6u#
K-µ-2ugz|~
sº}szuiz ¶~ qsuBu¹z|t8uPµu#ªiuBB» n31)7 ä
t8is }u.u¹*f1z|it8t8u¹sP}µiuu#u
ªiu utf¶Î¸sqs~uouBPuoµui#¸ ªiuuB$µP¸s¶Ñ~|iuB¸ n-uBµµ-un¸s~
sugu8¾i÷i
s2uouµiu
#}Q~óÆu_uB~|uBº1} ~|f^¶{
{t8
z|Bt8^z
~
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
©
-"& 44H "&67 + " 44 + " 7
iªI{uuBuu t´D}s» z¼
Q
qss z|z uB0.i1¾Tq
{-uµi¿uuªª u
Q 1zuqsu86u}s
t8qQu-tf0z¸si6~|
Îua¾-P#-µiqsuz|ug#Oºªz|t8ui» 5u³ Pµ¶>ufuBqQu P ªµ} unu*1u
{ugµ~uBuBuB
s
iªiªuBu Bq ¾-VÃU
{
}sqst8uuqK#}Q-z|Bug
sº~ut8iu u1*z|
sqs~ªuB
s ~ªugq t8Qut8#qsst8 }u1#-¾Ku¿n#ziuo»
q P~ µuu ÀI iugºÆauB
iz|u
uuq u 8u ii#Oqs¸IzuBµ}su
6Oªz|q µ-uB^
½tsPauB¸Qz|uoz|}
λ u ~i
sª z¿ t8z 5z Gq uqsqszu¹
{z|t*}st8}s~u-zzi
~ ¿ q u ug
¹»qs{pu^sqszQu8~.t8us
Ku1z|qs¸s~~|uu
st¿2uBufzªi
5qQPµ¶{iua~qsugsuO1}sug~uB
KuBot u {q
zi
(Ît8zqsuBµz>¹uBz >~-I¸ qsuu0 u1z
Q{Îsug1~|oi#
-uB z|Q
(ii
~µi ¶2uz|
sqsªnq u0z|> ¿ z qsz}s~|u8uz
Qs÷ºzq 7 t8 {uVu1#uB¹
s¸su ~u z|t8t8usqs- µ u> qQq Pu*µuosq i8¸Qufu }Qi1u u ¾u¶·s ~iz|
z|
Î1¾ u¹¸s}q iu>qsuf}sªiªt8uBs}u # -z|
zi
u1¶Ä¶.B» q : u¾Î©Bz|t8¦s¾Îuf©P"© µi ~$qsnz} >qssµQzuBiosqsqsuB¶.uo~n¶·q } z~^I} Oª¶·ªqsiug^} ziz|ªt*
qB}s¾Î~q -nufq z
s1ªiz
K z|z|#
(1º} 7}sªiz¸siz
#
Kq » ºzi
u uB#qsuB#q u ni
1uB u¹^ @nxpx x{ ' º» 446 äug
*ztqsz uVà KÆBº¾-¾I¿2u
_ u zz~Kzq i
sun
q{{~7Qqs} ugtaugs7
z|u
u z|u
suou su ? -µ> uOqsqsu*un
KuB} taz|t8uBzug~72ÃÄ#vKqsÆ7ut8
u } Ã Æ1u » po z
K-uBµiªu.#-qsuu qsu_
uz iu}Q2-*z|
Q{uBt ¶>t8¶ z|
qsiz|}s~~ug Oz|¸I
uF§i*K¾t8 suB~|uB11z|z¸Qu » ¸{f0¿nqsuu q Pµit8uz|~ t8u
s
Kzz|
su * z 0, z|
»O7x{u1ug »¡* ¥¾, ¿nu H( , ) ( , ) ¸Iu>q u z
szz~Î
z ziM
λ
0
0
! & 0 &
1 &
³ &u_ }s s) I4Hi¾uqQ-q -z82Vqs+u P8taq z|~*ÃÄ
uz|u
Hu¶» z|:0(- zi
ξH
qQ-z Vz( AE,45& 4 Ãqs©PÆouB÷u*qsu1u z| u
s3Mzz|
z
{µ-¶z
0
Kz|
{µP¶·}s
z1
Kz|n
¶·}s
Iº(z* i,
, z)¾ juB=~1,~|u . .z.
, 3M ¾Q¥¶Æ"B¿2qsu u {u
s
t8u¹z¸{ ~*u, t { x B u O » S( , ) qsu ~|uµiu~Îu1¶7q u z
{µ-z|
K2¶·}
ºzi
{I } ÃUº¶»ÃD¥ÆÆ B qsugu z|
{µ-z|
i# I u>z|
z
{µi~}sz|
ξ{zD» ui»Oz¶·qsu1i
zz|
j
j 1≤j≤3M
* ,
3M
j
3M
*
* ,
,
* ,
,
¿ ¿ qsz uqu ug∇I* uB1z| qsauªiqsu t8 ziuB
KtauB¿
Kz}sqtuBµPIuBzº.¸ ~ug B q u QKzz|
µ-z¸s~|uB.
∀( , ) ∈
×
* ,
*
* ,
z|qsuªi zuB
K
, ∀j1 , j2 , ∇ Ij1 ( , )·∇ Ij2 ( , ) = ∇ Ij1 ( , )·∇ Ij2 ( , ), ∇
,
Ô·Ü$ÒÎÔóø
©
!"$#%&'( " *) +, -%. /102" 3+45 &67
z
0qsuBqsufznu1
szuz|#ªoq{¸I0
s}suz|qsª{q{¸I ¾I}sÃÄzqsÆO{ q uo~|u¶.µiuq ~Tufuz|
szz* ~1,
z nz zit8
(* 12, ,
) 1} i#
sq
sugqQº-u B ¾I¶·:|i©g¹¥~|z
uB~|Vz
uIu
u
KB 0
0
j
,+
'15 ! &$ " 1
7¸I u1}
u ¸I uiQqsuBu
u1 } /2π » ¼
K
uz
z
sqsª uF¸I-* µu, q{{Qqsugug¾2z * z|, Ii¾$z¸s~|¸Iu } : (
¾ qsuBu0u1z|a_
i C > 0
uQi
suq
K
Kγz|
su≥0}st8} #q ´ ° C , \ {0}, | α · ω( ) | ≥ ∀α ∈ |α| q{¿ {q Iuu q ¿nuBu>z|$
st8uBu |
α$|q =αqsu +K.} ..
K+z α α·ω(
° )α·ω(
aª° uO) 1=~|iαu^ω (GB° u)o+i
s.~|. . z +¶ |αα | ªiω{uB(°¹z)&
»n
pszq z| » qsu1u ¶ÄB¼ »
z|: #{su¾ qQ( » »>» » »>» »>» »¹» »>» » »>» »>» »>» » »¹» » § § »» §¥uzs1~|~|uuBsz|
a¸s~|ut }
» »>u » λ>uB»
s
» »¹ » ´ä¤i»>»
sug»>^» I» »>u
K» »>z» ~o»¹»¹» » »>»>» » » » »>»>» » »>»>» » »>»>» » » » »¹»¹» » »» § » v 'o
u¹Qz1~|u¹z|
}s¸ ~u ¿2u~|~ÑQuB
iz|~F»>» » »>» »>» »¹» »>» » »>» »>» »>» » »¹» » Ì ± $ $ Ì $ «#s»©, ,^È o$Í }st8zi
q %u t8z|~
sz
¶·}s
1z|
> » »>» »¹» »>» » »>» »>» »>» » »¹» » #s» ¥ » »>» » »¹» »>» »>» H» »>» »>» »¹» »>» » »>» »>» »>» » »¹» »
(
ÒÒ à I ÿgûöºþ#ð1ñºè1ï#é
©i©P¥©
©
©
Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399
