High-order averaging schemes with error bounds for thermodynamical properties calculations by MD simulations
25
0
0
Texte intégral
(2) 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° 4875 Juillet 2003. ISSN 0249-6399. ISRN INRIA/RR--4875--FR+ENG. THÈME 4. apport de recherche.
(3)
(4)
(5) !#"$% $ $ '&( *) ,+- " . 0/* 12354 $
(6) 46 7"$1 !2312)31
(7) "1 !&8/:9<; =)31>"1 * ?@BADCFE8GIH$CKJKL MON @PGQHRKSTAULVE8GILWPXKYUYZG MO[
(8) \ ADYUAD]]^X_E \ GQ@BWPAUXK@ M ?
(9) @B`aGIH N GISTb M EaYZGIb$cX d X(ef@gADL MON @BhKchi@gAUC_dXijTSTYUY M^k GIl@gAUXKYnmob@BADHAUCKA pqsrt8u>va wy x{z|t*rBt8}s~uB- z1ii t8u1s ~uuBz t8zz| u { Ou 2Fu7o sI u
(10) ug#qsu##qsu¡ I¢$vK£K¤¥w§¦}sz|~|~uo¨©©iª>w§¨s«> ¬iuB ∗. †. ∗. ‡. ‡. ∗§. ∗. 0P·i®$u#¯B°¬i±-uB²s ³°i´ _µ q uauaz|ºQ Kz| » } ¸^1u8qsqsu8z|¬ {qs}s¶Dt8i u uBz|BoP~·iuz|#t*} ¬z|~| s¬z|¸¹ Fitf¸^}s~|~ su¬¸¹¶Uiz>t8uaqsutai~t8ug1s}s}~# - z|{ ( ta¸nzozz|BuB¼¶~ uBIzuguB¼½¿*¾z sqs¬6u01qsiuBt8us}¸¹t*z|}s ~ui~^ÀOuÁnÂIuiz|gta½ µ -u0·iu0 -qs·{uz 1uV {·iiut8u¬u K Q} 1tauVuB#-zu~O¸u1sq tau ~zugt8
(11) u_qQP-·ifu#qs¬-uÁÃÁ zqsq u u#ÄVqs1uBz|tau ugVÅÆ} ¸O#q5}s»#·qsuuB~|t81uz½ Fµ Ç.q uBu~| u1¼È¼ÀÁnuBu¼¬iz·izuuguiu*
(12) ºQsis}sP { z|t-q u * s} -z s·{¬z u8{¬it8}ssz ~|uBu1~|z sz
(13) ug
(14) z K_u¬##qs-{izi u uqsÄVua1z| u K# t8t8u1 su s#uBB½ ¸Oqsut8s}#-z| Fz ( uoug#q_¬z|·uB uBz|u B1}s#z| _qsuataK É#qsÊKuBËTtaÌÍ*ugÎ À±-uBÏ$¯´ 2ºI}s ~|uB }sÀs~|uB ¬i { z|t8q zuBÀ7ut #-z|z~OP·u#¬iu*1it8s}z| ÐÀqsz|¬q¶Z u>P·uB¬z| s¬. Õ
(15) ß#ðUâPÖ¹ÑÓÜæÔôõÒIÛîÕ>ßÔ7ô|ú$öTÕ>ÔOß#ÖÆÛÝÑ7ØPéü×ÜOØgâPÑTÒ á ì1ÜÙÜãPÚ¼ðÆÜæÛÝäDÜ.÷>á àUÞ$é^øgØgßãgàUù-ÜOáÝðDÚ¼ß#Ô7â-âgß#ÜÙÛÝâPܼÜâPØãPÜß#äOÜâPäÐý¼ã»åØÕ>Ú¼ÑÓâ1ß#ÖÆàDÑÓþ äÐÕ>ÿPÜæà7èPú$äÐÑTÑ$ãPç-ØgÜOß#Ö¹èPÞ$íTäUÜÔÐäDß#éÖõÜèPú6äÛÝáØ-ÜÔÐêOèKÚBØÜæÙà7¼óûënèP#Üßò1âPì1ø ÙÜÚ¼âBÔ7èPèPÜð¹ÜOâPà âPíTÜÛîä$ß#áÝÑTäUÜ$ÜãgåsÜß#÷äUØ-Ùù-ß#ÛðUØ-ß#ÑTâPÙá àDÜ é$ï$ÜäDÙß#ðÆàUÜäØ-ñ¼ñò1ó¼ó úÒIï${ú$ùaï$Ô Ö¹Þ^(ØgïnÖ¹Þ$ØgÔÐú$Öõâ-ú6ß#Û ÔÐBäDÜÜ$âPâPÜà7Üä Õ>Úã ÛÝÜä7ÞèPþ éðÆáÝûBèPÜäØKý¼Øgßì1ÜâBègÜ^ãgè $éâPéðUß#Û{ãPÜ ß#èPÛÝ Ü¼Ø ¼øgýò ÑÓÛîß#þß#ð¹àØgù-ðUß#âPÙÜ ∗. † ‡ §. . . . .
(16) . . . . 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.
(17) , $ $ 31 46 )0 31 n*31n)3 / ! /* a)= 3 2)3¼ z|ut8 ¯ z K
(18) »}s uB~| s¬Q´ ½
(19) nug}Qz ¸¹i z| iutf}s z|t*~ ug} }sz|~I-u zit8 u1uB {
(20) u}s K¸¹t i Btfz }sK~|}suBuB ~ uuVo~|u 1uB¼~| {u··iz|uugQ¬ ui Q}s 1u u ~| u0 {ug ~|t8t8}s-z~i} uB su0uV st8t8ug -~iBuBut8 s}s sQ~|ugizuVuB~~|uBuB- ¶ ·iuu#ut8~|suB~|uBnta K-} itu sB uBz K }s uB uB K}suBzTt*t8ºs~|uB Kuq Ku tai V u1{ÄV t8i1z zK }si½ug7½»n# uK}sufuo~ug Os#qsuB1u*uBz|uB^ug Kua }s»~|}ss ~~|B} uBnzuB }s } K}s 8z¬i~¬i}sz zu Kqs»t8~|u u#{qst8zs ~|uB ¼uB»zK }suÅU1t8itar1t8uBu ·i uu8~|~|1zztfÇ.}su~|~uz|#È1 ÐÀ sÀÓi! }. i s sºsu Q z uB OuguBo z|Bt8}s~uB}s> P·uguB}su s}su g1zz| s" u u>~ft szrBu>~as~}Q u1ÄVi1u>Iiz|ºs~|u½ # Î °g¯ Ì %³ $ ¯¿´ o{ t8z i} u0t8~ B}s~|zuÀOB~1}s~ ut8-uB s suBa-z|z K}suBBÀO#&q t u B~uB%· >I}s ~|u
(21) ~1} ~ u>t8-u suBBÀuBz|t-z| uuB}sgÀi&q irBt8u»uB¬i z K}su.
(22) ª.
(23)
(24)
(25) !"$#%&'( "
(26) *) +, -%.
(27) /102" 3+45
(28) &67
(29) . f "7 )^"1 ipzº iq
(30) u»u¸¹ }s·P iºs1Q~|uBuBz|O -zBug·iÀPnu¸¹^u¸u^qsau»uB¬ ssz|u·q uB¬i 0z|uBu»sBÀBq{ #i1z uo~ÓQ¸Ði$ uo1u{tu1uÄVs-12uBz|uqsu K K#uB#-t8À½Bz|½·½u È{} t8 *uzºQ~|u2K~I1z| su ¬ft8K}stasz}s~|zz|1ºsu i z|}s1t isPÅÆz·u# z¬i~uBuBT t;z¸Q:«iz|Àsºst8¨
(31) }<Du ¶ ½ qs¸7u æ _zt8t8z|uBK~ ug»oz|ÅU1 s}ss~|¸7z|uB K-qsu>~|zVis q i} u»¸z|uz| KQuBºs1z|u8ug~|ŹgÁ À ¬q zqsz|#z|qo~It8z| z|uKtÀ{Á z|s Kqsz·iuB
(32) u~|·{uotauBzti^uzqsu>q {}st8 (t*I«gºIi©u© uÀ ©i©¸©¸^ Vqstzz1¬i~|#quBȼ#ÀsÈO {}sz*t*q ºI~-u ·iupÁn~|¬i¸ÆuÀ{t8tz~|z|uB={Oz| s¬a¸Ótaqsuuqs t8ns}#u-1z|} u K¸$~|*q u}Q6M Pu · u#z ¬iiuB n auB.#MqQf~|~uB sta¬ z| s~u ¬fqsu =T ½ q uQ1uo z| s¬ z| *Á qsqsuz#1q0i {·qsuu zuB i~|2}ss~|z|º ºsz|z ~|zI u ¸¹u t8zu F? > ŹÁ qsAÇ z#@q ÀI uÇoIup> ÀQ ^z #p>uBÀÐ~ ¸$½B½½#¸7È > qsu»iqs iuBu*t82 {~a t8tau1zoq u $ u:tf«iÀ ºs-¨~u < iiz 0»t8zua 5qsuu
(33) ·iÇ» ~|p>}u»À7qQz| ( n p>ÀQ¸OÀO qs½½Buf½t8z~|i1~}sugi~1-}s1~iz|s Qz|foÈ KBÐ Q}s t8uB t uBz|B>
(34) z|>ÅÆqs_uztf>uB}sȬi~|i 0 u z| qsz uŹ_Iiiqsuz}sºs t8~|_qQ1 zii }s н.sÀTæ~|u qs u8q Á u
(35) z ~|q(uB1uBº -iqÐÀÐui¸2À{}Qqsqs#q uu zt8u
(36) uBu1t ¬iKug n5i1QiuBz| K ~|·i _us¬zug(t8>s: -ª ~-<ZugÁ½ » qs u q q uuBu u t*iº 1u8~uP · uBqs¬uauz|ta¸uVq Pu·iui#º ¬iuu*·P¸nºs~|u0qsuÁ iqsº uB u·Pqsuºs~|ufz|t*-·i}su~-»ziqs u ut8·{µz|s}qsuguB u 0 #qszu»t8tuu>-·Piz|·~B}suB~|z|~*¬ uB¸¹^i¸7t uqsu>uBqsuz|u1tsuBi-t,¼u^ zi sº{uVqsz|_u> 8s¡qsq uoiB~u»i~1 s}s¬> ~-i1z|zt8uii½$ up~|À{zqst8u»z {B}suB%½ t8Cs}s}suB~ zz q ¬*u~I {ta}s#t8iqsuuuit8Àz|uoBqsz|u~Q2 s#zuB-tfu }sug¼u~| iu us¼uB 1>uBz|1O¸I0}s¸.#z unQ s¸ z u* z ~|uu s¬~uB sqЬ½ qÐZo½z|æ 8qss-#ÁnuB1·zuB1 uiÀÁn¸^u21it8qK}Qta.i unºQ¸ÆuB~$z|u1¸¹¸^it q 1iu2taz t8ugi~s si~ºuBºQ1uB¸I·iu q - g·ÀÐuBz| (Vtq z¬i {q0_z } uB-¼zi VBÀÐB1}sz#¸¹{Vz| s¬ uB}sqs~u
(37) > uauºt8#uBz| s u }Q¸¹~|»~iqs sua¬zt8z|t*u} P~|·uBz| ¬uBBz|taÀ uiuB½ szuaqsu ZiV#qsz|æ~u >~|·-uÁqs*z| iºI su1uz1uu~|zuÀgqsÁnuBB8u 1Q}s #Ðz|s¬ s-- z 0·KQzB uO K¬iz~| >·i u sI 1uiu t8 uB is} }~ sqsz|uuÁ 1t8~ {u11·iKqsug¬ ½ u¸¹ Qæ17 uB¸¹i~s18 }sz6~-z|¬iz Ðz s·iÀ¬u Á2qsuFiu.11iz} t8 Ku^i1zPºs ·i}su#ua¬ui ½ ºIqsuF} ºQ-·i¸Tuq ut8ut8i. ºQuºQuBÁn~z|uBu1u¸
(38) aqsq u}s¬i1q } ~ Du QFuBE,t* "7ºs ~| "u Pu·uBi8¬uBuz| t8 qsz|u {}sÁ t8qsuz#qz|Bs~|~a-·K1z i taug } u }sszIt8uu {Pz·i}st8ut8#u8uB¬izuuÀ-F¸¹~}s Q #¼qs uBz|taqs uu¸Ó} Å suqszuouP {È }st81iuBt8uBz¼s}~ u~|qsu tau¬u#q7uB-½»^ugp¼qs¸Óizoq 5uBu»iB8z|t8~Án1}suB~~z|~- Fizia Ð~|ÀiVzDt8½ uz|u {½$· qs~|u8·#ugÅ Q t8suqsz u*uug#È^iaz K uBuBu~|¬#u -¸O ziqs u qsqsu8uz|z|t8t*uV}s~P-·zuBi ¬z|z| sta¬Vu t8u1qsuB B P½ µ uV qs-Á zz¸ qs u0z|z|ta}su~|P>·uBq ¬>u8qsz|au1uBit8s } }su u8º{ qsz|u 1 u1 z zHGBuz s sug¬_
(39) qsu ¸ T 1/T ¸¹t*} ~| 1 Z A(t) dt Å A(t) u sug.q u·-~|}su¸Óqsuiº u·Pºs~|u-^zt8u tÈÁ qsuuBi qsz*1it8Qi su K¸Tqsu0uB ug1uBuB* 1/T Á qsuu k z*#qsKuB 6Iizz|·uz| iu¬iuz¸ qsu zt8u»P·u#¬iuz|^u·-~|} -u Á zqVqsu¸¹t*}s~ Z A(t) f (t) dt Á qsuBu f (t) z^t8u ~uBz| s¬ ¸¹} ¼zi Á q iu»usugz| 0zs-·{z u z| x{uB½Q¨½ 8. 9. T. 0. k. T. k,T. 0. ÔÔ â
(40) Isò1ë1ñ#ó. k,T.
(41) v.
(42) -"& 44H
(43)
(44) "&67
(45) + "
(46) 44 + " 7 . Á q z|µ#uq8tVu*1ȼiiÀ sÁ ~uB qsz|Q #iq_ug B~7 q u>ºIuuqsÁ z.ugz uBq iuBq t*ufºs~|!uÇ @unt8q qsz|u~ uta s uzÁ { i 8{t8 u Kz|B} t8-~7z|BuBzi ~Ð u {t*Åõº qsuB~u tui½ ½ spq{qs-uÁ2zu uK·i~ } u g Àsz|K Qz| t8t8z~z|¸.Bt8ug¸I}szi~qs u z" uat8o~|sHz =it8}u#z|~|-~zi s z*qs z|BO 5~ q qsÁ ufz qsz
(47) zÇo p,q uqsuB uBi
(48) uu0ut* Á º u~qsuitfu½ ºsµ~uguiÀÓ 1¶Zu-·{1z z|}s u uB Kaq ~|6q
(49) uÁ2qstau8 K={}sz # -¬ #u1 ~: {v
(50) <z| s} ¬ #q u0 {u} K u ut8 zz|( ~.QuB¸¸¹iu}stt ss¾i i #uqн t8u ·uBai s¬
(51) q{KIqsuBuBn1i~~|uB1u z| V QuB zV¥sÀiz| ~} z s¬*z VQz1}s~.q - t8qsuBu# izi st8u z ~i sºQz|-·i u utu1*z| z| ¿z| Kx{uBug¬#½_ºs¨~|u½À{pÁ2qsuu0u»qsºsuB~u>z Kzz|¬iuVi} ~ 8uBt8ºsqs~|uBz|tq0-zqsu
(52) ~u~| utuB z|t z- s¬u {}s{t8uB¸Iz¸Q~Ðqszu1.suBt8z|st~|uB-u Bz a KºQuu ¬#uBº ~uz| >1utt8 qQ s-z|o fqs-Á z|~q u :Ý-¥ <Z½ }sµ ou uBsz|t-·{-z u
(53) u2zz| z|x uB uBBu ½ ªs½|«2q Ð ½ªs$½Ý¨ ¸ z¸TzÁn}s} #u uz|ÀPu2· uBB sÀÓoO¸¹»uº Á¿z
(54) ~u2z>¸s>qsqssu2-Á -· Fu2zzu Otx{z|¬uB} i½*uiªs} ½ ªsz|~| fÀÓÀP#Á iqsqs}suBuBt8uBuaz|1z|it8> K·{$z| u s1q{z| s~{¬»z$uguuB}st8~u^$»z| i>i (uqs¬i0~ ºs~ui}s½ uuC t z at8¸ }s %©0u2 uB~uBiÀg~uBzz1·~|uBuBz z|z| ¿ {·ªiug z¬Kz K-uua#z| ¼x{z uB¬_½7ªsq ½ vi}sqs¬quVu_st8}ss Q~|ua-u ¸ _Шu, ¶æz|¦i~u suBz| 5*Q u K}sº z|~u~Á2uu~|~OsI-·KuBz K u z| u½~DÀ7Á 7iqsz#~q iÀOz|Ánu -8taugI uB ~T taz z|¬0t8z| ={t8z| su¬ff~ºs¬~|uuV2#-uguBtýVpÁ #q i uz Izi uB» KºQzu~TÁnuuB suuB 6¬iq uB}suV¸Æ#-uugs
(55) uBuu8 KQ#nKuz·iº u~#u~ ºsºQ}s*z u #i *uqsuz·i.u Kz|t8.sz|~ Vu 1uist8t8 s~|z|unq a-Á .zz¸Qqqsqsu u u ¸Æu¬uB
(56) ^#¸Qq qsu i¼QuBzz1z~|^unzIzt8} u^ºI~u -·un¸Tqsqsuu s u~tuI½Z^z| iz.^uB sqsu-Á ¬ ¸^-Á qs uVI uB√ KzÀ ~DÀÓ }sqsuou0fq u8u K¸} 1i {z¶Z·iu¬#uq Qi1u*z| ¸n#qs u8zz|zt8i VuVsP·uB1uB¬nuB»ºQuÁn~|uB}su~ -u qs u»º{ÁnFfFº !zz Q
(57) ^~|-¸ÐÁnqsu u ~}s¬ºsuBÐ~|u
(58) q Á21/u *~|~ÓQqsTu u Ksz| ~|~Uu^½OIZz|z K.u~| suB¬iqs-ÀgÁ qs u2qQzt8-BuÀ Pz ¸·uq #u
(59) ¬iuuB sÐuB¬i {V·uB¸$¬iuqsÀgu
(60) ºs}Q. sz1~|u»-zÁ2 t ~|qs~uBu2 u ut*tf}Qºs#~q u Pqs·i¸7u
(61) u#qsuu Q¬ituuB½Àst*zºs-~|z|Ánu
(62) uBP·qs·u-ug#Á À 0z¬i¸7u½q u } qsuBu
(63) 2t8uBug·u #·P~ z~ s} zu»z~T¸1iqs uBu»zz|z|t8 2uP · u#1¬it8uB^sz|}suoa¬iKz|t8 uoPs· uBP¬z|t8u2¸¹2z|uB Vi#q ¸ 9<1%72"Ð" : 7)3#" & t8i7 u1ita }QuBo K}s *t z ) u ∈MºK z|z~ug tz| Fª fm½ @n½Ð#u1q H('z|,) ~u ) iºQu2z| q uBu 1z|ºQt8u z| ~ º{ szz foQ¸QKqszuz|( ' ut∈À u ÀQszu 3M. ×. i 3M. i. 3. 3. i. ' ). H( , ) =. M X i=1. ). 2 i. 2mi. +V(. '. 1, . . . ,. '. M ),. Ö¹Þ$ÔÐÖõú.
(64) ¥.
(65)
(66)
(67) !"$#%&'( "
(68) *) +, -%.
(69) /102" 3+45
(70) &67
(71) . Á ) q uu V ) |z
(72) qs uVuBz|Q Kug¼uBzi·i¼u~|z|½ pIqsu#u Kzta~Dz|À$~ z| ' { t8) z|B~Ðu s{#uB-t'z| Q>{¸¹z|('u ,. . . , ' z)¬z|· uB º{¸¹ ( ,..., ) H 1. 1. M. ' d dt. ∂H ' ) ( (t), ) (t)), ∂. =. ) d dt. M. = −. Å«gÈ. ∂H ' '∂ ( (t), ) (t)).. µ u_} s½7QÐKuÐu} ÐqQ-uB8B~|qs~Pu q K Qt8' z|B) ~ z7{ >uBz|t Å«PzÈa Kq T¸¹ }s 3M1z|
(73) z| {z·-¸ z|' ia¸¹}s) 1z|ÀP Á qsIu(u ' ,' ) ) À j) = { P · 1, ( ' (t), (u(t), (t)) z2 K.i.B.~Às},3M zz| i1 Vu»¸2qsu
(74) Å«guÈ1 sÀ{uBz|2¬i1i z|#s KugnIuBÁ (z·iu q, ºKug)VIuBqs1u ^ { z|t8t8uz|½ B$½ ¸µ u }s #u suÀIH( ) z|n(t)) s o u Ð ¸ s q B u | z { · z¶ , ) u
(75) º{ n Ë ' ). × } #q0q - ∀j ∈ [1, 3M ], I ( , Ë ) = I (' , ) )o ÅU¨È S( , ) = ( , ) ∈ qsu
(76) ~|u·uB~Ðu ¸$qsu
(77) z| {·Pz K2¸¹}s 1z| {I } 1i i#z| sz| s¬*qsu
(78) sq iu
(79) u>Qiz K (' , ) ) ½ ¬iu zz| {·-Á27uu1z|~|~ ¶ i (#u ' Às, uqs) u»)z| 1ºQu6} q u~s #z|-Ç szug@ ¼z|u~o QfuBt* ºsz~|uz|z|½ s 7¬
(80) À¸¹* Ántà uV~|u1qs}s usIA(z| siz'uz,) ~Iq )Q-iºIzuF K qsu ' i{º ) uBt8·-z|uBºst~~nuz| Á Oq ¸¹uitnu_q P~~Q·uz|#t83M i 0qsu
(81) ~|u·uB~Ðu S(' , ) ) ½^pq{} À{qsu
(82) u~|u·- K u utfºs~|u>P·iu#¬iu( ¸,qsu
(83) ) iº u·Pºs~|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. ' ). ' ). ÅƪiÈ. A( , ) dµ( , ) 0). ' ,). ' ) dµ( ,. ). Á q uu dµ(' , ) ) z2q u
(84) z {·-z| K2t8uBi} u> S(' , ) ) ½ qsui¾ uBt uÀ{Oq t8unz| qK{IqsuBuB1i~~|uB1u z|Z f QuB za¥À-i su2B fs-·iuA:Ý¥{À %-<qsu2¶æ~~|u u¬ z ' ) Źv{È 1 lim A( (t), (t)) dt = hAi, T Á oq u u (' (t), sP) z|(t)) t-z|z|n Vq u8q uuBP¼·u#f¬iu*¸ÐÅƪKq ȼu>Àszzu8t #Å«P È^ z s¬ ¸¹Kt #qVq 8uo} z| su>zzqs~Tu Q izz K u(' u
(85) ,}s) t ) ½ ' ,) , ÅU¥È 1 X hAi (δt, T ) = A N Á q u u s~(z|'u , i) F)@ {½
(86) zÅ «gȼÀTqsua K } taÁ qsuBuzu ~ÐuB¼½0¬i-zÁ2·iuu _·iuº{gÀ _qsz| Kufu¬i {·uBz|¬i _u #qsuouz|t8 uÅÆvKȸ.z|t8}sua#$uB: % < δt i s~|0-QuBu 1/T ÀKÁ2u
(87) s s u{IuB1T=f¸ÆNδtu {·uB¬iu u#-u»¸ÅD¥È^-Á hAi ½ S(. 0. 0). 0. 0. T. T →∞. 0. 0. N −1. Rie num. j. j=0. j. ÔÔ â
(88) Isò1ë1ñ#ó. N −1 j j=0. j. 0.
(89)
(90) -"& 44H
(91)
(92) "&67
(93) + "
(94) 44 + " 7 %. æ
(95) i uBTz|t8s-·u$qszZÐi{t8z.1i K·iu¬uB 1u$uiÀBÁ2uO} ¬¬ugÓ uBs~1uqsuO}s z ¸¹it8~|i¶ Á2uz|¬qKu z|tauP·iu#¬iu 1 A(' (t), ) (t)) dt º{ ~uBu zt8uP·uB¬uo¸7qsu>¸¹t T Z ' ) Å %iÈ A( (t), (t)) f (t) dt, Á q uu f i¼#7 ~uBz| s¬ ¸¹} ¼zi ¸qsunz¬i ~ t 7→ A(' (t), ) (t)) %½ Csi7z| # uÀ uB sz s¬ºK Z ' ) qsuzt8u
(96) P·uB¬u z| Vqsu>z KuB·-~ [t , t + T /2] ÅÆÁ z q t ∈ 2 A( (t), (t)) dt I (t ) = T ¼ È s À i su* F1i z uB qsuaP·uB¬u»¸ I (t ) -·iu t ÀQzD½ u½ hAi (T ) = 2 Z I (t )dt ½ [0, T /2] T 7u1 } u su>qsu>¸¹}s 1z| α ºK. 7→ 1/T z¸ t ∈ [0, T ], 0 qsuBÁ zu . α : t∈ pq u hAiÁ zq (Tz)u~¸½V ºQuu8uB¬iu au#z K~|~¿iÀ Å %Kuȼ sÀnÁ z|z sq ¬fº{ ≡ f qsÀOu Á ¶DqsuBq5u 1if K·i~|}z*z|qs (u1¸ni Kqs·iu8~|¸¹}}s Qz|¼ 5z| ¸ α f k Z Á z F q z # B u ~ ¸ Ó À 2 Á u u . s * u s q f u | z 8 t f u P · B u ¬ u ' ) ½µu α hAi (T ) = A( (t), (t)) f (t) dt B s-·u : ¥ <ZÀs}s u qsut8u»i}st8zi ºQ-·iuÀKq ÅD¤È 1 . hAi (T ) = hAi + O T pÁ q q{} z|#BqÀ.1¸¹ii8 K·iu {¬ug²O±P® 8Æ°qs±-²u
(97) ± uBË¡ QuBt*Kºsz~|z|u
(98) ·uP·z|uB iu¬¬iuu kÀ^Izuau z aQKz|½ ºs~u ugz|¬ ¿P·iu#¬z| s¬_#qsut8u 1/T pqsu z|1u1u»·iu#z| ¸ hAi (T ) z| ' ,) , ÅÆ£iÈ 1 X hAi (δt, T ) = C(k, N − 1, j)A N Á q uu (' , ) ) Á qszu.u q u {}st8uz|B~znuB¼qsu> {}s¬t*z|·ºIuBu f º{8¸ ¶D 8}sz| i ~uug¬i¸7z|Ii 8z#zqs·iuBuot8z| Ku uB¬¸Tu#z|t8 uu δt À T = kN δt Á q t8isu_}u }s t º{z ugs1}su # zi ÐC(k, ºÀsIz| Fu Nj}Q#−q_ 1, j)Á Á2q P0z|#q q ~~oºQqsuBu~z|io s¬5t8 sk}[0,#-Nz| F− z|1]o½¡~|tapKqsuBu_¸¹uB {u}sztf (ºQ1uBjt8 ,.. .z,ºIj u aqsut8s}#-z| ¸$q u>#-uB1i (' , ) ) ½ ºIz|uBt*8}sp~qs-¸uzi¸¹} 3t*ÀÓ={}si s
(99) ~--Á ÅÆ>~|£iu ~È ugz|¬iu»8À$¬iq uu sºIuBu8#qs~uBu5z QGBt8iu6z|uÅÆ £i È ¸ 6{ÅU¥iuBz (fÈ sz|qs u8a1u ~|z uhAiuu#t -}s(uiºI½ (δt,u$}Q¸nTu) =}sz# hAiuÀÐ {~|uB}s(δt, ~z| ¸¹fTto) ½n{- pzi8t8 Fzqsu i}s}s¬i ¬ºIuBuBz|- 8¸¹u Kt z| 7qs½ u»ug uBBÀK z1} ~|~|fi Vqsz|2Iz| K2ini 0t8u¬iu suB~ z}suBBÀ{Á z~|~Tug1uBz·iu 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. Ö¹Þ$ÔÐÖõú.
(100) ¤.
(101)
(102)
(103) !"$#%&'( "
(104) *) +, -%.
(105) /102" 3+45
(106) &67
(107) . q --·igæ _Àu3¸¹:Ý ¥sf<Ð zz qsu>z|¸¹ Ð Às~|1~z-¸.uiÁ À$z| sqs{¬aut8u·suB~|~iuB¼1zuBz|
(108) Fz|t8Ç.#qsuBuBta~|u u1>8u
(109) #¸$q iu t8 uu0 zr*{z| ta}Q ~u ug ¼zV01i ta }¸u>i qsuu> r#-=ug¼2iÈ1À.(ÁnŹuuB ~~ 0. 0. | hAi(k) num (δt, T ) − hAi | ≤ C. . 1 + δtr0 Tk. Å iÈ. . . ¸¹iit8u> K C Á qsz#q Kug s uIu T δt ½ z| 6z|t*}sµ qs~uV-uVz}sui Q uaTu~zqQz| s-uat
(110) q q uu
(111) (Iiq uu8z|Qºsii~|ºu>~{ºK sz0 siutaqszuº{~ ¸ÆF ¼uB qsPuVq F-z tf¸2}sqs~qsu-u·-z|uB 6~|}sug> ¸Á A zu8q' z|, sug) ¬I~|uauB1
(112) #u-1uBi1qsiu8u~i-½ u µ ~Oq z| uz|ta z|ut8 suu ¸ À }Qug.
(113) Ku2~|ot8uqs Àiqsu ·-~}sug A ' , ) u2 s^uB~|u {t8uÀi q u ui i s~| ugPO-^qsuo}s sz|·u#~#-u 1/√T ÅÆuu~|>q uu1sta ~u ¸Iqsu }sºs~|uÁ2u~|~sz| x{uB½$ª ½ v{ȼ½ q -t8p t8qsuo sz|¬zsiqK aK¸¹}s¸Iz|¸^qs~uBÅu$KÈ:ݼ¥-zÀit8<Z½Osn¬iu1-}s·qsuouuO Kq OÁ u
(114) uzÂIu1qÄVi1#z|u1.u# ¸Óz|~|V^uB¸$uB 8#}sq #qfu ~ssu uB^izi} #uBqÐu1½ u z| 8¸¹-Á2t* } ~| V ÅUuB£iÈ1zÀi¬it szP a¬»ºQuqsuBuB ~ z VuB ) 1n*3 V* 43 µ u
(115) s-Á }s 0 {}st8uz|B~Tu1st8s~|uB2q -qs-Á > ~ugqsu_q itasu> uB¸*È2q ¸uq u}suBt8z|t8zi u52Å }sKÈ * ¸¹uB8Á z Kqszu#q0¬#qsº u
(116) ~u uBz|tta-z|~u z| ºIu
(117) z|¬uit}s8i} ~z8¸¹s{z| s-¬·u ÅÆ-½ q qsu
(118) ugztuVÅ KÈ2~8qsi~ 2¸¹i s s¶Dz| Ku¬iºs~|u»ut¸$si¼z|B~Óz| iuuBB½ µxuBu½Qªs½|«goÈ1sÀuBq uu K uBqs}su~o>zu _ ~uBqsu8siºsu~|ut ¸^0ÅÆuuf~|~x{ugug¼z|½ ªs½Ý¨È1¸O½ z 2 uIqu u Ku»tqQtau
(119) iz| s Kz
(120) u¬ii1ºsz|~~|~u-À # ÅÆuuu ~|z5¸=u%~|©( f¸Æz|~~Ó~ugÁ azq }sz ºugq ¼u
(121) u qK {I(qsqsuBuuBÐuB s¸ } ¶æ¦iqsuB sug*u1Iz~Tu Kugz}s~D~½B½.C$æ z| x{ug~|~i½OÀOªsz| ½ ªsÀx{Á2ugu
(122) ½ ªs} ½ v À^0Á28u} u t tºQ Kz >z|z| ~uÁ uBt8}sqsºsz#t8q(z} qs u uVz| u {_z| _uB#tqsiuBt8}sºs 5z~|u#ÁnPuB~ ~.¸¹ Isq{u K~|z|z B¬~OÀsz|z| z t8¨uifq ½0u*½7ppuBqsqs uz>u>u1-sq K~|¶Zt8#- qsz u*iu (~.Qz>¸nu KuBqssuz|ug~7Á q uBqs Kii~-uuBsz·i·qQufuBu ¸~ z ÄVi11u } s~ guB½ u u *i1} u~|ft8s}u q uta { t8z| iQuBzug.z|^z 8qsuBuBuB. z|}s~~ zi ~.uC #{^Ot88qssuo~|zuB K¼uBz|=i¬au #-z| K¸7uou¬iqst8us z|~|{u1 6 uB st8#uBqszuBBt8Ài~Ðu ÀÓ Á2uauft qQPqs·Å-«Pu*Á¿È >}Qq u niqs}snuBu s sz ÂTiuu#qa Kz^z|~| ¬iu zq z t8z·i3u :î¤<Oqs¸ uo KzÂI} uBtauBuB i¶ qs ufu*# ¨ B Á2u ·iu u~|{uz º{0 Ç.u~uo~|¬iuB6zqst¸nÀqsÁ uVqsz#~|¬qizzÀ q it uB¸2~z|~|tau uºQu- ·iuÀs> Kt8' s~|uB1z) ¸ φ δt ( , )= ' ) B j. j. j. j. . . . . . . . . . . φVδtV. ÔÔ â
(123) Isò1ë1ñ#ó. (. n,. n). VV δt. n+1. n+1.
(124) £.
(125) -"& 44H
(126)
(127) "&67
(128) + "
(129) 44 + " 7 . 1zitaq t'Iiz| z1z|uB i2ug^¸qsu
(130) v B·i u~|t81zu~|À Ç.u~u ~¬iz qst Á z qz #uB~ ¸fq qsu>i uBn¸7qsu~|¬¶ . '. n+1 ,. '. n+1 ). (4). = φδt (. 1/3. 1. . ). '. ). = φVb1Vδt ◦ φVb0Vδt ◦ φVb1Vδt (. '. ). Á z q b = 1/(2 − 2 ) b = 1 − 2b ½ 1zitaq t'Iiz| z1z|uB i2ug^¸ qsu
(131) % B·i u~|t81zu~|À Ç.u~u ~¬iz qst Á z qz #uB~ ¸fq qsu>i uBn¸7qsu~|¬¶ (. ). (6). n,. 0. '. ). n). n,. n). 1. '. ). Á z q c = −1.17767998417887 À À c = 0.235573213359357 c = 0.784513610477560 ½ c = 1 − 2(c + c + c ) æ 6~|~qsu ¬} ugºIu~|-Á
(132) ÀÓÁnuas~|>uzqsuBqsu8 K} tauBz~7zt8u8P·uB¬u hAi (δt, T ) ¸n iº t8us·P}#ºs-~|uziA ~Ðfiqsuz2uq iu | tahAiu»¸¹i(δt, ~|~ÓQiTz K) −·hAiuz| i~~|fVF~|¸¹z|}s¬ s u1z|½ ¸ T ½ n u K}suB K~|À.qsu (. n+1 ,. n+1 ). = φδt (. n,. n). = φVc3Vδt ◦ φVc2Vδt ◦ φVc1Vδt ◦ φVc0Vδt ◦ φVc1Vδt φVc2Vδt ◦ φVc3Vδt (. 1. 0. 2. 1. 2. n,. n). 3. 3. (k) num. (k) num. .
(133) !" #$% &'(. æ qsu#q z| t8uBz1~iz| s Ðz|À Á2u
(134) 1 Qz u 8{uBt,¸ ·u> z|~ugnz| «B Á qsKu { tazz uB1z|ºQu º{ Á zqqsu>¸¹~|~-Á z| s¬as}s~z| >. H=. 5 X p2 i. 2. i=1. + ωi2. qi2 , 2. 7Bu1 »} ºQou
(135) ¸¹iº»z z u ºK1ua } ~Kzqsua~ÐP·iu~|#}s~|¬iu
(136) z|¸ A> = p ½
(137) pqs½ uau1si¼o·-~}suf¸Oqsuau utfºs~uaP·uB¬u = 0.5 7 1 u . } s º | z *u ) 0#q uB =Á qsuqsuBqsu
(138) q{{QqshAi g u g u 2 pqsq u ufz| Kt8u¬iz|~ sz|z _ V¸¹~}s¬i ¼z ziqs 6t ÅUz|»uBuo{t8sQs~|uBuB 1 zz½¥p½|«gqsÈ^u8uo su-~u K z|u z| uz 8½ sµ - u»·z|u» iu»u}s¬iºsug~|uºsz z~|tzuz-qsiuBuB n}sft8u> -z|·iz QuQu si ½ µ z|us} -·uuoqsqsuu ¸¹ zit*s}sq ~ KÅÆ£Kz sÈ$u
(139) Á iz q }s t8z ÂTuz|u K ^ÅÆ·-u~u*}sugx.uB½O¸ ¥s½ ¨iŠȼ½
(140) vKÈ1À¸¹n>¬z|·u Å Á zqfq u2i u¶ %~¬iz qst½$n~|uB~|VÅÆuu Cz|¬ ½2«gkÈ1À-zk Q1=uB1z| s¬ ¬iz·iuBoºQuδtuB.1δt {=·iu0.025 ¬u Q1uȼÀ½ ω1 = 1.0,. ω2 = 8.01256,. ω3 = 12.25245,. ω4 = 17.234,. ω5 = 20.98765.. 2 4. k. ,+. - /.0 21 231 '45
(141) . µ u
(142) suK}s 8qsuztf}s~-z| ¸$qsu >uBs~|u sºs~|ut½ µ u1 Qz u2qsu#taz|~ z| . ). H=. 2. 2. −. |. '1 | ,. Ö¹Þ$ÔÐÖõú.
(143)
(144)
(145)
(146) !"$#%&'( "
(147) *) +, -%.
(148) /102" 3+45
(149) &67
(150) (k). | hp24 inum − hp24 i |. s© ½|« © ½ © « ©s½ ©© « ©s½ ©©i©s« «u¶Z©K¥ «u¶Z© % «u¶Z©{¤ «u¶Z©i£ «u¶Z©. k k k. ¥. «B©. $z|¬}su*«. «g¥. ¨v«. ¨©. ¨i¥. T. >
(151)
(152) 2 i(k) (δt, T ) hp2 i
(153) !
(154) "# k $ ! % &' (") *+ ,
(155) ! ! ! C .- ( /01!2")! ! 3 "4657hp 1 48 (num 1
(156) 9 3"6 ( 4 4 % q1 = 4 $ p1 = −1.2 $ qi = 0 pi = 1 i = 2, . . . , 5. uB17u1zºs8z| s} ¬8¸¹ifª z| z|1u~u>z| } 6 -q uiP¼·iuz#·iu¬uni}s¸ ~|t*ºQ' uB i½5z|~U½ fz| q usu·{z|} fu1sta ~uiÀqsu uB uznt*º qs~u»uzP ·z uBz|~¬ÅÆua su¬K -zºI·iuVugÈ.1iuB stau ¬}u ~|Kz^zqsA~u»~| s>=h||t ' |i¸Ð| =qsuoz|3 sz+z2~ E={z Lu1zo/(4tai|t8Eu K|)}stÀÁ ½Oq puqsuu E L Kz|t*z}s~|~~--iziB _À{z|q t8"u utau6z|~z| δtz| =¸¹} 0.01 Ð À Á2 u8} ~|Kufzq ~7uai uz| K¶ %Vu¬i~¬iºs~|uz Àqstqsu½*z| i»u¬i¸¹i»z|qs u8q #qsuBt8t8u
(157) z|z|. ¼ z i | z ÅU{uBt8usx{~uBug¼½^z|¥À.½Ý¨ºsȼ}½ fÁ2uuVºs~|u0 suBz qsuf(s-·iu s* z|s-·u8qsu z| q Kz| su}st8zi ¸Æit8¼sµ ~z|znua} ¬uB su
(158) t8us¸z}À{qsu*u
(159) K qsu8 szP~|~|·u#z|ug ¬iuBz¼Á 2u zqsq u
(160) kºIuqs#¬u z| s s>¬i szuBz| ss s¬V~|¬8u¸¹2i¸$st q iºsu«*~|uz|tt*½ ¥}s½~--µzÁ2iu8u 0·z|uBu Bu1ÀiuBq }si-gugÀT> isÅÆ ukKuzC$# Qq8z1¬Q¸¹uB½O1¨}QȼuB½OBugÀQp^q iqsz| u ~|Iu ¬z| z|t*È1À{}s~-qsu
(161) zi~ (¬zuBt8 ugÀ{½ qsµ u>u0¸Æiuuuq uq uuf1q {-·igÀ7u¸¹¬iu QT1uf≥ÅUuB40uC$ÅÆz¬Qs½sªKȼP½ zt-u~|_ºIui vuB·~|}zi 0. 0. 2 0. 0. 0. . k. u } z| $o 1q Qqsu z#» u KBIu uÀI uBÁn ua1u>u* ¸7 q º u»~u*uB#nq Á } uB #z =qqq q ug IuBq 1nufug Tzt- uδtÅ iÈ z| z»i q uB^7½ Cs#iqsuB =qszÁ qssu} qsIuBi2uqsÀIuBÁnuu C (k) C (r ) 1. 2. 0. | hAi(k) num (δt, T ) − hAi | ≈. ÔÔ â
(162) Isò1ë1ñ#ó. C1 (k) + C2 (r0 )δtr0 . Tk.
(163) «g©.
(164) -"& 44H
(165)
(166) "&67
(167) + "
(168) 44 + " 7 h|. '. (k). |inum. «½ %. ª¥ «. k k k. «½Ý¥ «½ v «½ ª «½Ý¨ ¥. «B© «P¥ ¨© ¨¥ ª© ªi¥ vK©. C$z|¬}su»¨ >&")
(169) 3 ! 3 !" h| |i (δt, T ) 2% 31! % &")!/ 3 ! !
(170) 1
(171) 1 ! / $ ( 1
(172) ! "2( 4 5 3 1 !" (9 4 = (0.9, 0, 0) k = (0, 1.1, 0.5) T. (k) num. ~~$·-~|½ }sµ uBu
(173) 1¸ δtt8s}s u» sqsu µ {fu ~Q¬uo·P~} uB2 u1¸ uBTt8Àz|} s#u q0Cq (k)½qsp7u» ~ug z| sqs¬fz|BÀTuBÁnt!u*Án¸Ði=q u»Á uBz qF2·uBzCt(k)/T ºIuB} ~ u u uz| iuB QTuB |uBhAi ½uZu ¸^} qQ-uB¸¹iz|t -uz|»ÀQq q uÐÀQugqsBz ~|u K } u Kz zqsºI} }s~ ºIu u (δt, µ T ) − hAi | i ~ ¸ ½ u Q | z º{0© O½ % À 1uB$u Ï Ê z| s)^¬8Ê $z Ï Ê qsTu° $Ë t8Îu» TÁr P8²$¸¹Ï δtqs½u
(174) qsu ·-~}sug¸ k À=Á21u
(175) º#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.. Ƹi¼>Oq }s-snz|qsz|uo si¬~|1À z|~~-qsuazi·- O~|}suBOoqs¸uoCºQuB¬(k)z| s sKz| s}s¬z ={¸Iz|qs uougz|t*}su~-Á zziq VkŹu qs~|z»¬iz|uo.z|¸¹ Fi^~¬iuB¬uut8.u·- K~}sÁ ugOzqF¸ kqsȼu ½ ~i s¬¶ zzt8·iæu u _(i tuBu1 ~|~^ Á zÁ uq5zuqVuBt8
(176) Qz ugu
(177) ¼.qs{uf*·-qst~|u0}suB~|~suB·-t ¸ ~|}sCu(r¸ δt) ÀT½%ÁnCsu*iÀ. sk>uq u1uV ÀKu {Ti·uBzOz*~u~|0 ¬suiÀ- Pqsz|ut1i-utauBt ~}FCua¬z|(k)/T ·u¬ z|·º{uB C (r )δt µ Ê Æ° Ê ± Î Î ± Î ½ u. · | ~ s } g u ^ ¹ ¸ i Á s q z # q u I u » u C (r )δt C (r ) k T º{ 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¸¹u01i t qsuq suB¸ qsu 1E, "7
(178) "nuBz|t-u Å iÈ1½ 1. 2. 2. 0. 2. r0. 2. 2. 0. 0. 0. k. 1. r0. 0. 2. 0. 2. 0. Ö¹Þ$ÔÐÖõú.
(179) «i«.
(180)
(181)
(182) !"$#%&'( "
(183) *) +, -%.
(184) /102" 3+45
(185) &67
(186) | h|. s© ½|« © ½ © « ©s½ ©© « ©s½ ©©i©s« «u¶Z©K¥ «u¶Z© % «u¶Z©{¤ «u¶Z©i£ «u¶Z©. '. (k). |inum − h|. '. |i |. ª¥ «. k k k. ¥©. «B©i©. «g¥©. $z|¬}suVª. ¨©©. ¨i¥©. T. >
(187)
(188) 0( 1
(189) ! " ( 3 ' (") C (k) % !
(190)
(191) ! / 4 5 3 1h| !1|i"num (δt, T ) h| |i 9 4 k $ % = (0.9, 0, 0) = (0, 1.1, 0.5) ,. 1 &$ #
(192) $ &
(193)
(194) !
(195) 1 &
(196) ! . æ7 u s Qqsz ¶Z¦ig À uBÁ2nuQ} u Kz|V~U½OpqsuBu t'ot8¸ Mz~i= sz60 ¸qsz|u~ug z| _ut!ª uBz| KuB i¼z| s¬8qsi}s¬qa}s Q-u H=. M X i=1. ). 2 i. 2. +. M X X. V (|. i=1 j>i. '. j. −. '. |). i. Á zq V (z) = 4 − 4 V (z) = V (z)−V (z )−V (z )(z−z ) z ¸ z ≤ z À 0 qsuÁ z|u½ z z oqsu81}sÂ5È$# z|} }Q unz qs=u2t83.08z ztf½ µ}st u z|ptsq{s} ~|¬iB>uoÀTQºQ1uBi {z|qF· u Kz|qsnuazºIi Q}s: «uB<D i½.æz| s>z~$z ~Ó z zqsi uf zn¸¹z|Ź# Q1qs2uBu oztfu»uf}s}Q~-#qV Kz|z| *qQ {-}sºQ2P} qs»zu -[−3.2; 3.2] u Q z z¸Ð~Oqsu
(197) z uBzit, ÐÀTz| qs0.229 uB t µ qsutu8Pz· uÐ#z *¬iu~z ¸^K}sqsz u s={q z u1uz ½ *u{t8IIuq #uB-uB}s7uaz~|~ug2.50 ½ z q . } # 6 q ( z s z 8 u { qsz|uu8 1u>u*ºI su1Á2»uuBu s }s ¬iq_z|z| K~·-ugÀ z K¸¹} ¼zi È1½*¼pu qsuaz| *º suI·-u ºs ~|z u*f} ¥ u.u> sut8zz uz| _Źz| zo qsuaz|t8}suB~|B À LJ. 12. LJ. 6. LJ. c. 0 LJ. c. c. 3. A(. ÔÔ â
(198) Isò1ë1ñ#ó. '. 1, . . . ,. '. M. M) =. XX 2 | M (M − 1) i=1 j>i. '. j. −. '. i. |.. c. c.
(199) «P¨.
(200) -"& 44H
(201)
(202) "&67
(203) + "
(204) 44 + " 7 . µ
(205) u $oÁ2 u=Á ¬Kz z|q ÐÀqsÁ2u
(206) u·uBu~u q z -gÀKÇ.¸¹uBO~|u1~¬~u ¬iuB sz qs}st¬iqVÀ} z|t*z| s} ¬~|fz|z| 8t8uz|t8uBu À¸ δtqsu=~5.10 ½ qsu¸Æiu^qsu ¬ B u À T k {·uB ¬iu ufÅUuBu C$z|¬ ½.v{ȼ½ −3. '. '. (k). | hA( )inum − hA( )iestim |. ©s½ ©©i©s«. k k k. ª¥ «. «u¶Z©K¥ ¥©©i©© ¨©i©©i©© ªi¥©i©©i© ¥©i©©i©© i% ¥©©©i© C$z|¬}suv >
(207) ,. T. ! !
(208) "& ! ' ") (k) ( """ / hA( *
(209) )i1num ")
(210) (δt, ! T!)" * hA( /0 ' ()i estim 3" (
(211) ( ! ! " k4 $ % % % % . .
(212) 3 1 & # 0 45
(213) $ 1 ' % #. æ qsz|QBÀsÁ2u uB~ÓÁ z qqsu t8z~i sz| H=. p2y p2x + + V (qx , qy ), 2 2. Á q uu uB 1q zºs2z| s¬8qsuu¨ 1z zz|~ÓI~u»z|z K V (q> iÁn,}s8qºs~|¬i)u»~i=Á2º u(q~|~Ó~Ót8I−z| s1)uBzt K8z+~|~D(q½p-q +u u» qI- −u K1)z,~ VÀs-¬i{uBÁ nq z|#q +∞ - z| sz À 0) V (±1, 0) = 0 pqs su8u
(214) ºIuq sP ·{~|u>z|Q}si»z K¸n~|z|t8-u8u P ·iu-# (0,¬iuB1)Às- s¬iÁ ~qsz# q uBQVuB (0, >1)i _=1q ½ua(±1, QB ºQz1u
(215) ~|u*z Á uBz iqFz QuuB? I> uB¼o0qsu8 s ~|u*Iz| K»uB su¬ ŹqsuBufu K} ~7·- ~1}sÈ1ua½
(216) p¸nqsqsuBu8u uB szÂTuu¬u KEou¬i¸nzt8qsuBu x. y. 2 x. 2. y. 2 x. 2. 0. Ö¹Þ$ÔÐÖõú.
(217) «gª.
(218)
(219)
(220) !"$#%&'( "
(221) *) +, -%.
(222) /102" 3+45
(223) &67
(224) . «½2¸¹1i K·iEu¬ut*~|~uB ~|zt8q z 1 À } qsu*z ¬#-q ugu¼ ~i0uBz| s s¬
(225) ~|¸¹}su Qs¼~iz| ugVz zt8¬~|su
(226) º -·ugz 7½qspu»qs1iu
(227) K·iz|t8uuf¬uBP ·i1uu ##¬-ugu ¸¹it 1/T 1/T Åõ¸¹i ºsz# k Èz| qsu
(228) suf·{z|} 2u1sta ~ug B ¨½2¸¹ug~|E~ 渹t*uBu}Q~ # oq_q ~|u»¬iºQuz|q u 1 ÅÆqsPu 1Ei K·i≥u5¬uBȼÀT 1qsuou8¸7 qsu>z1~|z|u*t8zu>P·uBu s¬uBuB¬i^u1z|z
(229) z|t8q z-~
Documents relatifs