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Clustering model for complex fluids
- 3. iju N ( )NjiUp m H ji ij i i ,,2,1, 2 1 2 =+ = ∑∑ < )1(
- 4. ∑( ) 2 1−NN ijr ijij rrr −= iju ∫ −= ωβ dH hN TVQ NN )exp( ! 1 ),( 3 ∫ ∑ ∫ ∑< = −−= ji N i NNi ijNN drdp m p u hN TVQ 1 33 2 3 }) 2 exp(){exp( ! 1 ),( ββ ( ) rdpdup mhN TVQ NN ji ij i iNN 332 3 2 1 exp ! 1 , − −= ∑∑∫ < ββ )2( ∫∏∫ ∑ == −=−= N i i i N i Ni dp m p dp m p I 3 1 2 1 3 2 ) 2 exp() 2 exp( ββ
- 5. ∫ ∫ ∞ − =−= 0 223 2 )4()) 2 exp(( 2 Nm p N dppedp m p I πβ β 23 )2( N mkTI π= ( ) ( )TVZ N dru N TVQ NN N ji ijNN , ! 1 exp ! 1 , 3 3 3 λ β λ = −= ∑∫ < )3( ( ) rNd ji ij u erNd ji ij uTVNZ 33exp, ∫ ∏ < − =∫ ∑ < −= β β )4(
- 6. mkT h π λ 2 = ( )0iju =),( TVZN ijf (0) ( , ) N NZ V T V= (0) 3 ( , ) ! N N N V Q V T N λ =and )5( .1−= − iju ij ef β )6(
- 7. ( )0iju = ijf( )TVZN , ijf iju
- 9. 88 N 8=N ( )TVZN , [[ ]]1 2 6 8 3 5 7 4 ( ) ( ) [ ] Nklijij N ji ijN rdrdfff rdrdfTVZ 3 1 3 3 1 3 1 1, ∫ ∑ ∑ ∫∏ +++= += < )7( .8 3 1 3 6714128 3 1 3 6834 rdrdffftandrdrdfft BA ∫∫ == )8( ]][[1 2 6 8 3 5 7 4 )9(
- 10. ijfijrr [[ [[ [[[[ ]]]]]]]]3 5 8 42 1 [[ ]]76≡ 2 ]][[ 5 ]][[ 7[[ ]][[ ]]3 4 6 8 ]][[1 ]][[≡ ∫ ∫ ∫ ∫ ∫ ∫= 8 3 6 3 684 3 3 3 347 3 52 3 1 3 3 rdrdfrdrdfrdrdrdrdtA )10( ∫ ∫ ∫ ∫ ∫= 7 3 6 3 674 3 2 3 1 3 14128 3 5 3 3 3 rdrdfrdrdrdffrdrdrdtB )11(
- 11. NNNN11NN At Bt 8Z ( ,.., , )λ β α 3 3 1( ... ) ... Nf f f d r d rα β λ∫ )12(
- 12. NN 55 l l cluster− l 5 cluster− 3 4 5 1 2 ∫≡ 5 3 1 3 3425151412 .... rdrdfffff )14( ( , )NZ V T = )13(N 1
- 13. l l cluster− ( 1,2)l ≠l cluster− 3 cluster− l cluster− 2 3 1 2 3 1 2 3 1 2 3 1 and )15( )16(×= − Vl TVb ll )1(3 ! 1 ),( λ l cluster−
- 14. 1122(( 22 )(),(lim TbTVb l V l = ∞→ ( )TVbl , l1r ( )1−lijf r 1r V V
- 15. ∫ +++= 3 3 2 3 1 3 2313122313231213126 )( 6 1 rdrdrdfffffffff Vλ [ ]∫∫ ∫∫+≈ 13 3 12 3 23131213 33 13126 12 3 6 1 rdrdfffVrdrdffV Vλ V b 63 6 1 λ = [ 2 3 1 2 3 1 2 3 1 2 3 1 ]++ + V 1 1 =b 1 ][ ∫= 1 31 rd V 1= )17( ∫∫+= 13 3 12 3 2313126 2 2 6 1 2 rdrdfffb λ )19( V2 1 32 λ =b ][ 1 2 ∫∫ ∫≈= 12 3 1232 3 1 3 123 2 1 V2 1 rdfrdrdf λλ )18( ∫ ∫ ∞ ∞ − −== 0 0 2 )( 3 2 3 )1( 2 )( 2 drredrrrf kT ru λ π λ π
- 16. N 1mclaster−1 2mclaster−2 3mclaster−3 { }lm ( ) { }∑= / , lN mSTVZ N { }lmSclasterl − ),( TVZN Nlm N l l =∑=1 Nml ,.....,1,0= )20( )21(
- 17. 77 Nlm N l l =∑=1 { }lm { }lS m l l m∑ { }lm 11)NN 22
- 18. ( ) ( ) ( ) . ! ! !2!1 ! 21 ∏ = l mmm l l NN )22( { }lS m ∏l ( clasterl − lm ) )23(
- 19. 22 s{ms{mLL{{ ( ).!/1∏l lm [∏l ( clasterl − ! ) l m m l ] )24( 3( 1) ( ! ) / !lml l l l bl V mλ − ∏ )25( )22()25(
- 20. ( ) ( ) . ! 1 !, 31 3 ∑ ∏ = l l m l l m N N m V bNTVZ λ λ )26( ( ) ,333 1 Nlmm l l ll λλλ == ∑ ∏ )27( ( ) ( ) . ! 1 , 1 3 ′ = ∏∑ = N L l m l m N m V bTVQ l l λ ∑}{ lmNlm N l l =∑=1 )28(
- 21. NN00 ( ) ,11 ∏== ∑ l mlllmN zzz ( , )NQ V T 1 1 N l lm N = =∑ ∑ ∞ = =℘ 0 ),(),,( N N N TVQzTVz )26(}{ lm }{ lm )29( )30(
- 22. ( ) . 11 ,exp ! 1 ! 1 ,, 1 3 1 3 1 0 3 0,, 1 3 1 21 ∑ ∏ ∏ ∑ ∑ ∏ ∞ = ∞ = ∞ = ∞ = ∞ = ∞ = =℘ = = =℘ l l l l l l l m l m l l mm l l m l l zbIn V V zb m V zb m V zbTVz l l λ λ λ λ )31( )32(
- 23. zz ∞→Vif ∑ ∞ =∞→ = ℘≡ 1 3 1 ln 1 lim l l l V zb VKT P λ ∑ ∞ =∞→ = ∂ ℘∂ ≡ 1 3 1 l l l V zb z Ln V z V N Lim λ )33( )34(
- 24. zz343435353535zz3434Vv N = ( )la T lalb ( )3 vλ ( ) 13 1 − ∞ = = ∑ l l l v Ta KT Pv λ )1(
- 25. ( )∑ ∑∑∑ ∞ = −∞ = ∞ = ∞ = = 1 1 111 l l n n nl l l e l l e zbnTazblzb )2( ,111 ≡= ba )3( ( ) ( )∫ ∞ − −−=/−= 0 2/ 322 .1 2 drreba KTru λ π )4( , 3 1 24 0 0 13 3 12 3 23131262 2 23 ∫∫ ∞ ∞ −=−= rdrdfffbba λ )5( ,31820 432 3 24 =−+−= bbbba )6(
- 26. l-clasterl-claster l-clusterl-cluster l-clusterl-clusterLL33--clustercluster 1−lβ 3a ( ) ( ) × − = −− Vl ll 131 !1 1 λ β 2 3 1 )2( 1 1 ≥ − −= − l l l a ll β )7( )8(
- 27. = = ∫∫ ∫ ∞ ∞ 0 0 13 3 12 3 13126 3 3 2 3 1 3 23131262 2 1 2 1 rdrdffV V rdrdrdfff V λ λ β ∫∫ ∞ ∞ −= 0 0 13 3 12 3 23131263 2 1 rdrdfff V a λ )2( 1 1 ≥ − −= − l l l a ll β )7( . 23 3 2 β−=a (9)
- 28. 9.49.4 ))1010(( ))1111(( 1010 ))1212(( ))1313(( lb 1−lβ∞→V { } ∑ ∏ = − =lm l k k k l m l l b 1 1 2 ! )(1 β { }km ,...2,1,01 1 1 =−=∑ − = k l k k mlkm { } ∏ ∑ ∑ − +− ∑−= − − l l ll l m m l m lb l ml l l k ! )( )!1( )!2( )1( 1/ 1β { }lm ∑= =−=− l l ll mlml 2 ,....2,1,01)1(
- 29. ))11(( ))22(( 11 =a 2a )(ru )(ru ( ) ( )∫ ∞ − −−=/−= 0 2/ 322 .1 2 drreba KTru λ π −= 612 )()(4)( rr ru σσ ε
- 30. ))33(( ))44(( 3344 +∞=)(ru 0rr < σ6 0 2=rε− σ<rσ>>r 0r 6 00 )/()( rruru −=
- 31. 334411 ))55(( ))66(( ))77(( 10 << KT u .exp1 2 0 0 0 2 6 02 32 −+= ∫ ∫ ∞r o drr r r KT u drra λ π .1 3 2~ 0 3 3 0 2 −− KT ur a λ π ∫ ∞ −+= 0 2600 33 3 0 2 ))(( 2 3 2 r drr r r kT ur a λ π λ π xex +≈1 ...)()()( 3 21 1 3 1 ++== − ∞ = ∑ v aa v Ta kT pv l l l λλ ))(1( 3 2 1 3 0 3 3 0 vkT ur kT pv λ λ π −+=
- 32. ))88(( aabb ))99(( ))1010(( ))1111(( 1 3 0 3 0 2 0 3 0 ) 3 2 1() 3 2 1() 3 2 ( − −≈+=+ v r v kT v r v kT v ur p πππ 0 30 3 0 0 3 0 4}) 2 ( 3 4 {4 3 2 3 2 v rr b ur a =≡= = π π π ( ) ,~ 2 KTbv v a P −− +
- 33. aabb l>2l>2 ))1616) () (1515(( ))1717(( +∞= = )( 0)( ru ru 0rr < 0rr > −= = 1)( 0)( rf rf 0rr < 0rr > 3 0 3 3 0 2 4 3 2 λ ν λ π == r a ( ) ( )∫ ∞ − −−=/−= 0 2/ 322 .1 2 drreba KTru λ π ∫ += 0 0 2 3 0 2 r drr λ π
- 34. 9.2.59.2.5 ))1818(( 1122 33 ))11(-(- ))1919(( 33 012 rr < 13 r 13 r 23r12r0rr > , 3 1 24 0 0 13 3 12 3 23131262 2 23 ∫∫ ∞ ∞ −=−= rdrdfffbba λ { } , 3 1 12 3 0 13 3 63 0 12 rdrda r r ∫= ∫′= λ 2 1 3 12r 13r 23r 0r
- 35. 013 rr <023 rr < 1s2s0 r ( ){ } ( )[ ] ∫ − −−=∫′ 2 12 2 0 2/ 0 12 2/122 013 3 .22 rr dyyryrrd π 0ry 12r120 rr − dy 120 rr − 1s2s .
- 36. 1919( )( ) ))2020(( . 164 3 3 4 3 1212 2 03 013 3 +−=∫′ rrr rrd π { } , 3 1 12 3 0 13 3 63 0 12 rdrda r r ∫= ∫′= λ . 8 5 18 5 2 26 6 0 2 3 a r a == λ π
- 37. ““BoltzmanBoltzman” -1899” -1899MajumdarMajumdar “-1929“-1929 Ree_ HooverRee_ Hoover “-1964“-1964 Ree –HooverRee –Hoover 3 2 3 2 1 4 28695.0 1120 4 2(tan1377273 2 3 8960 1283 aaa = −+ += − π π 5 26 4 25 )004.00386.0( )003.01103.0( aa aa ±= ±= 6 27 0127.0 aa =
- 38. ))11(( ))22(( 11N V NZ0Z0Q 1 3 1 QZ λ=V ∞→V ( ) ( ) ( ) . ! , ,,, 3 0 0 N N N NN N z N TVZ zTVQTVz =≡℘ ∑ ∑ ∞ = ∞ = λ ),( TVQN ( ) ( ) N N N N TVZ TVQ 3 ! , , λ ≡
- 39. + + +==℘≡ ∑ ∞ = 2 3 2 1 3 1 3 0 !2!1 1ln 1 )( ! ),( ln 1 ln 1 λλλ zZzZ V Z N TVZ VVKT P N N N (3) (4) ∑ ∞ = ∞→ =℘= 1 3 1 )ln 1 (lim l l lV zb VkT p λ 3:Z { } [ ]...}) !1 () !2 (2{ !2 1 !1 1 ...) !2!1 1ln( 1 22 3 1 6 2 3 12 3 2 3 1 +−+=+++ z ZZ z Z V z Z z Z V λλλλλ ...)( 11 2 213 1 3 ++== ∑ ∞ = zbzbzb l l l λλ
- 40. ))55(( ))66(( ))77(( ))88(( l z 1l > i iz V≡1l > 1 1a = ,1 1 11 ≡= Z V b ( ), !2 1 2 1232 ZZ V b −= λ ( )3 112363 23 !3 1 ZZZZ V b +−= λ ( ).61234 !4 1 4 1 2 12 2 213494 ZZZZZZZ V b −++−= λ
- 41. 66889.1.169.1.16 l-clusterl-cluster 11 22 l cluster− lb lZ lb ∞→V V lZ ×= − Vl TVb ll )1(3 ! 1 ),( λ
- 42. RushbrookeRushbrooke 9.1.299.1.29( )( ) ))1212(( ))1313(( ×= − )1(3 ! 1 ll l b λ l VlZ ( ) ( ) . ! 1 , 1 3 ′ = ∏∑ = N L l m l m N m V bTVQ l l λ { } ∑∏= = l l m M l l m lM M m bV MZ 1 3 3 ! )/( ! λ λ ,...2,1,0 1 ==∑= l M l l mMlm { }lm
- 44. ))1717(( 1515 ))1818(( ))1919(( Z<<1Z<<1 1717191920201616 ))2121(( v nx 3 3 λ λ == ∑ ∞ = = 1 )( l l l zblzx { })(exp)( xxxz ϕ−= 0→x x ( ) ( ) . 1 1 11 1 3 0 1 33 0 3 + −= −= ′−= ∑ ∫ ∑∫ ∞ = ∞ = k k k x k k k x x k kx dxxkxdxx x KT xP β λ β λ ϕ λλ
- 45. 17172121 ))2222(( ))2323(( x(zx(z(( ))2424(( ))2525(( β ∑ ∞ = + −= 1 1 1 k k k x k k KT Pv β ( ).1 1 11 > − −= − l l l a lβ lb ( ) ( )xfxxz /= ( ) ( ){ } 0 1 1 1 ! = − −∞ = = ∑ ξ ξ ξ j j j j j f d d j z zx )!1( −j 1−j ξ { } j f )(ξ0=ξ
- 46. ))2626(( 1818 ))2727(( ))2828(( ∏ ∞ =1 )exp( k k kj ξβ ( ) ( ){ } ( ),expexpexp 11 ∏∑ ∞ = ∞ = = == k k k k k k xxxxf ββϕ ( ) ( ) ×−= ∑ ∞ = !1 !1 j j z zx j j 0=ξ 1−j ξ ∑ ∞ = = 1 )( l l l zblzx ( ) ( ) ( ) ∏ ∏ ∑ − = ∞ = ≥ − ∑′= ×= 1 1 2 1 0 1 2 , ! 1 ! 1 j k k m k m k m km k m kj j m j j m j inoftcoefficienthe j b k k k k k β ξ β ξ { }km 1 1 1 −=∑ − = jkm j k k ,..2,1,0=km
- 47. BethBethUhlenbeckUhlenbecku(ru(r(( 9.4.99.4.9 ))11(( ( ), !2 1 2 1232 ZZ V b −= λ ( ). 2 1 2 12322 ZZ V ab −=−= λ
- 48. ))22(( 00 ))33(( 9.4.29.4.2 ))44(( )0( 11 ZVZ ≡≡ ( ) ( ) ( ) ( ) ( ), 2 1 20 1 0 23 0 2 0 2 ZZ V ab −=−= λ ( ) ( ) ( )0 223 0 22 2 1 ZZ V bb −=− λ ( ) ( ).,!, 3 TVQNTVZ N N N λ≡ ( ) ( ) ( ) ( ) ( ). 0 22 ˆˆ 3 0 22 3 0 22 HH eeTr V QQ V bb ββλλ −− −≡−=−
- 49. ))55(( ))66(( ))77(( ))88(( PP2m2m ( ) ( ),,,ˆ 21212 rrErrH ααα ψψ = ( ) ( ). 2 ˆ 12 2 2 2 1 2 2 ru m H +∇+∇−= ( ) ( ) ( ) ( ) ( ), 1 , /. 2/1 re V rRrR n RPi nj j ψψψψα == ( ) . 22 2 n j m P E εα += ε
- 50. jjnnPP ))99(( ))1010(( 44 ))1111(( m 2 1 ε ( ) ( ) ( ), 2 1 2 2 2 rrru m nnnr ψεψ = +∇ − ( ) ,13 2 =∫ rdrnψ ∑ − − − ∑ − = ∑ − − − =− n nene j mjP e V n E e E e V bb )0(4/23 )0(3 22 )0( βεβεβλ α β αβλ α
- 51. ))1212(( ( )( ) 3 2/1 8 λ V = ( ) 3 2 3 2 1 2 0 4/ 3 4/ )2(8 4~ 22 h mkTV dPPe h V e mPmP j j π πββ =− ∫∑ ∞ −− mkT h π λ 2 = 3 1 λ
- 52. 1111 ))1313(( ))1414(( ))1515(( )0( nε ( ) ( ) { }.8 0 2/10 22 ∑ −− −=− n nn eebb βεβε ( ) ( ),/ 2 1 2 222 0 pk m k m P n == =ε )()0( kg nε ( ),/ 22 pk m k n ==ε
- 53. 1313 ))1616(( ))1717(( )(kg ( ) ( ) ( ) ( ){ }dkkgkgeebb B mkB 0 0 /2/12/10 22 22 88 −+=− ∑ ∫ ∞ −− ββε ∑ )(rχ ),( ϕθY ( ) ( ) ( ).,, ϕθ χ ψ ml kl klmklm Y r r Ar =
- 54. ))1818(( rr ))1919(( ))2020(( )()( rr ψψ =− )()( rr ψψ −=−l ( ) ,00 =Rklχ 0R ( )rklχ ( ) ( ) , 2 sin +−∝ k l krr lkl η π χ ( ) ,2,1,0, 2 ==+− nnk l kR lo πη π
- 55. u(ru(r((kknnn+1n+1 ))2121(( )(klη l )(kglk∆ ( ) ,π η =∆ + k dk kd R l o ππη π η π nnk l kRkk l Rkk ll −+= +−− ∆++−∆+ )1()( 2 )( 2 )( 00 π ηη =∆ ∆ −∆+ + k k kkk R ll )()( 0 dk kd l )(η
- 56. ))2222(( kkmmkk ))2323((l )12( +l l)12( +l { }lll −− ),...,1(, )(kg ( ) ( ) ( ) ( ) .12 1 ∂ ∂ ++∑′= ′ ∑= k k Rlkgkg l o l l l η π ( ) ( ) , 1212 ∂ ∂ + + = ∆ + = k k R l k l kg l ol η π =∆kkgl )(
- 57. ))2424(( ))2525(( 1616( )(( )( ))2626(( u(ru(r(( ∑ / ,...4,2,0=l ,...5,3,1=l )(klη ( ) ( ) ( ).120 + ′ ∑= l R kg l o π ( ) ( ) ( ) ( ) ( ).12 10 k k lkgkg l l ∂ ∂ + ′ ∑=− η π ( ) ( ) ( ) ( ){ }dkkgkgeebb B mkB 0 0 /2/12/10 22 22 88 −+=− ∑ ∫ ∞ −− ββε ( ) ( ) ( ) dk k k elebb lmk lB B ∂ ∂ +∑′+=− ∫∑ ∞ −− η π ββε 0 / 2/1 2/10 22 22 12 8 8 )(klη
- 58. ))2727(( 5.5.255.5.25( )(( )( ))2828(( )0( 22 bb −2b )0( 2b ( ) ( ) 2/5 0 2 0 2 2 1 ±=−= ab ( ) ( ) ( ) ( ) ( ) ( ) −≡−= 20 1 0 2 3 20 1 0 23 0 2 2 1 2 1 QQ V ZZ V b λ λ )0( 2Q ( ) . 2 1 2 1 2 1 2 1 2/5 2 3 1 32/5 2 3 3 0 2 ±= − ± = λλλ λ VVV V b ± = V V TVQ 3 2/3 2 3 )0( 2 2 1 1 2 1 ),( λ λ
- 59. 9.1.189.1.185.5.285.5.28( )(( )( ))2929(( )]/2exp(1ln[ 32 3 λπ rkTu −±−= ( ) ( ) ( ) . 2 12 1 2 2/5 0 2/2 3 2 0 / 3 0 2 22 ±=±= −= ∫ ∫ ∞ − ∞ − drre drreb r KTrus λπ λ π λ π )(ru
- 60. ))3030(( ))3131(( o o rrfor rrforru >= <+∞= 0 )( )( )( tan)( 1 ol ol l krn krj k − =η )(klη ( )rχ0rr = )(xjl)(xnl
- 61. ))3232(( ))3333(( ))3434(( { } 00 1 0 )tan(tan)( krkrk −=−= − η ,.... cos3sin)3( )(, cossin )(, sin )( 3 2 2210 x xxxx xj x xxx xj x x xj −− = − == ( ) ,... sin3cos)3( , sincos )(, cos )( 3 2 2210 x xxxx xn x xxx xn x x xn +− −= + −=−= { })(tan )tan(1 )tan( tan)( 0 1 0 00 001 1 krkr krkr krkr k −− −−= + − −=η .... 5 )( 3 )( 5 0 3 0 −+= krkr [ ] [ ] − −−= −+ −− −= −− 2 0 01 02 000 2 0001 2 )(3 )(3 tan )(3/)tan()(31 )(3/)(3)tan( tan)( kr kr kr krkrkr krkrkr kη ... 45 )( 3 0 +−= kr
- 62. 11 22 2626 ))3535(( l0)0( =lη dkkkelbb l k l )()12( 8 0 2 22/1 )0( 22 22 η π λ β ∫∑ ∞ − + ′ =− 2,0=l1=l )(.... 3 10 2 5 0 21 0)0( 22 Bose rr bb − − −=− λ π λ )(....186 5 02 2 0 Fermi rr − + −= λ π λ π
- 63. ))11(( ( ). 2 ˆ 2 1 2 ij ji i N i N ru m h H <= ∑+∇∑−=
- 64. ))22(( 11......NN ∑ −− =≡ α ββ αEH N eeTrTVQ N )(),( ˆ { }∑ ∫ − = α α β α ψψ V NH rdNeN N 3ˆ* ),....,1(),....,1( αψ Nrr ...1
- 65. ))33(( ))44(( NW ∧ { }∑ − ′′>≡′′< α α β α ψψλ ),...,1(),...,1(!,...,1ˆ,...,1 *ˆ3 NeNNNWN NHN N { } αβ α αα ψψλ EN eNNN − ∑ ′′= ),...,1(),...,1(! *3 NW ∧ ),...,1( NWN { } αβ α αα ψψλ EN N eNNNNW − ∑= ),...,1(),...,1(!),...,1( *3
- 66. 22 ))55(( 9.1.39.1.39.4.29.4.2 ))9.1.39.1.3(( ))9.4.29.4.2(( )( NWTr ∧ )( ! 1 ),...,1( ! 1 ),( 3 3 3 NN N V NNN WTr N rdNW N TVQ λλ == ∫ NZrdNW N N 3 ),...,1( ( ) ( ){ }NNN drrdrrrr ++ ,...,,,...... 111 ( ) ( )3 3 3 1 1 , exp , ! ! N N ijN N N i j Q V T u d r Z V T N N β λ λ< = − = ∑∫ ( ) ( ).,!, 3 TVQNTVZ N N N λ≡
- 67. 33 ))66(( 5.3.165.3.16 66 ))77(( rr ,′ }∑ >=′< −−′ P mprpirpi ee V e V W 2//).(/).(3 1 2 11 11 1ˆ1 β λ { }mprrip e h pVd V 2//).( 3 33 2 11 βλ −−′ +∞ ∞− ∫∫∫= 22 11 /)( λπ rr e −′− = ∞→T0→λ rr ′, 11 rr −′ 11ˆ1 1 >=< W ( ) ( ){ } 1 2 exp .kU r V i k r − =
- 68. 55 ))88(( ψ ),...,1( N N W ),...,1( N 3 3 31 1 1 ),( λλ V rdTVQ V == ∫ ),...,1( NWN{ }αψ
- 69. AABBAABB 11λλ 22řř ))99(( AABB 1122AABB Nrr ,....,1 )()(),...,( 1 BBAANN rWrWrrW ≈ ji rr , ijr ijr NWAW BW AB rr ijr λ? ijr r%?
- 70. ))1010(( 1111(( 2=N v ∞→12r 1)2()1()2,1( 112 ≡→ WWW ∞→12r )2,1(2w )2,1(2w )2()1( 11 ww )2()1( 11 ww)2,1(2U ( ) 02,12 →U ( ) ( ) ( ) ( )212,12,1 1122 WWWU −≡
- 71. ))1212(( ))1313(( ))1414(( )2,1(2Uijf lUˆ >′>=<′< 1ˆ11ˆ1 11 UW >′′<+>′><′>=<′′< 2,1ˆ2,12ˆ21ˆ12,1ˆ2,1 2112 UUUW >′><′><′>=<′′′< 3ˆ32ˆ21ˆ13,2,1ˆ3,2,1 1113 UUUW >′′><′<+ 3,2ˆ3,21ˆ1 21 UU >′′><′<+ 3,1ˆ3,12ˆ2 21 UU >′′><′<+ 2,1ˆ2,13ˆ3 21 UU >′′′<+ 3,2,1ˆ3,2,1 3U
- 72. ))1515(( ))1616(( lUˆl [ ][ ] ′ = ∑∑ P fatorsmfactorsmm N UUUU l NW ....)()....()().....( 2211 ),...,1( 21 )( ∑{ }lm ,....2,1,0; 1 ==∑= l N l l mNlm
- 73. 1717(( ))1818(( ))1919(( ∑p { }lm,1ˆ11ˆ1 11 WU ′=′ ,2ˆ21ˆ12,1ˆ2,12,1ˆ2,1 1122 WWWU ′′−′′=′′ 3,2,1ˆ3,2,13,2,1ˆ3,2,1 33 WU ′′′=′′′ 3,2ˆ3,21ˆ1 21 WW ′′′− 3,1ˆ3,12ˆ2 21 WW ′′′− 2,1ˆ2,13ˆ3 21 WW ′′′− 1 1 1 ˆ ˆ ˆ2 1 1 2 2 3 3 ,W W W′ ′ ′+
- 74. 11 ))2020(( 2218181919∑l lm ( ) ( ) ,!11 1 −∑− −∑ ll m mll W ( )lUl ,....,1 NWˆ ( )l,.....1 12 ,,....., WWWl
- 75. ))2121(( ))2222(( 1616--11--99 ( )l,.....1 ( ) rriflU ijl ~,0~,...,1 λ>>− lb ( ) ( ) ( )∫− = ;1,...,1 ! 1 , 3 13 rdU Vl TVb l lll λ lU ijr ( ) ( )3 1 1 , ! l l b V T l Vλ − = × l cluster−
- 76. 2121 551515 ))55(( ))1515(( ( )TVbl , V∞→V ( )Tbl ( )lUl ,....,1 )( ! 1 ),...,1( ! 1 ),( 3 3 3 NN N V NNN WTr N rdNW N TVQ λλ == ∫ [ ][ ] ′ = ∑∑ P fatorsmfactorsmm N UUUU l NW ....)()....()().....( 2211 ),...,1( 21 )(
- 77. ))2323(( ))2424(( ( ).!/1∏l lm ( ) ( ) ( ) . ! ! !2!1 ! 21 ∏ = l mmm l l NN ( ) ( ) [ ][ ] ∑∑′= ∫ ......... ! 1 , 2211 3 3 UUUUrd N TNQ p m N NN lλ ( ) ( ) ( ) [ ] { } [ ] ( ) [ ] { } [ ] ( ){ } 1 2 1 2 3 1 1 2 23 1 2 3 1 23 1 3 3 1 1 ! , ... ... ... ! 1! 2! ... ! !... 1 ! ... ! ! ! 1 ! ! ! ! l l l l l l N N m mN m N lm m m mlN m l l N ml lmlN m l l N Q V T d r U U U U N m m N d r U U N l m N d rU N l m ι ι ι λ λ λ = = = ∑ = = ∑ ∫ ∑ ∏ ∫ ∑ ∏ ∫
- 78. ))2525(( ))2626(( lU ( ) ( ) ( )( ) ∏∑′= − = !/ 1 , 13 1 1 3 l ml N lm NN mVbTVQ l l λ λ ( ) ; ! 1 31 1 ∏∑′= − l mN km m V b l l λ ( ) .333 Nlmml l lll λλλ ==∏ ∑ ( ) ( ) ( )∫− = ;1,...,1 ! 1 , 3 13 rdU Vl TVb l lll λ
- 79. ( ) ,11 ∏== ∑ l mlllmN zzz ( ) . 11 ,exp ! 1 ! 1 ,, 1 3 1 3 1 0 3 0,, 1 3 1 21 ∑ ∏ ∏ ∑ ∑ ∏ ∞ = ∞ = ∞ = ∞ = ∞ = ∞ = =℘ = = =℘ l l l l l l l m l m l l mm l l m l l zbIn V V zb m V zb m V zbTVz l l λ λ λ λ ( ) ( )∑ ∞ = =℘ 0 ,,, N N N TVQzTVz
- 80. l l l V zbLn VKT P Lim ∑ ∞ =∞→ = ℘= 1 3 11 λ 3 1 1 l l V l N z Ln lb z V V zLim λ ∞ →∞ = ∂ ℘ = = ÷ ∂ ∑
- 81. 25252525--11--99 ))2727(( 55--99 lb . 111 1 3 1 3 l l l l l l zblandzb KT P ∞ = ∞ = ∑=∑= λυλ lUnW( )ln ≤ n( )ln ≤ lb2=l
- 82. 1.71.71.81.8(( 2727--55--99 lb ( ) ( ) ,1 2/510 −− ±= l lb 2≥l ( )0 lb
- 84. ))9.6.19.6.1(( ))9.6.29.6.2(( ( ) 2 2 1 1 ˆ . 2 N N ij i i j h H u r m = < = − ∑∇ + ∑ ( ) ( ) ( ){ }ˆ* 31 1 , 1,..., 1,..., . ! ! i NE H N N i i i i V Q V T e N e N d r N N β β ψ ψ− − = ∑ = ∑ ∫
- 85. 9.6.3a9.6.3a 9.6.4a9.6.4a ((9.6.5a9.6.5a)) NWˆ ( ) ( ){ }3 *ˆ1,..., 1,..., 1,..., 1,..., ,iEN N i i i N W N N N e β λ ψ ψ − ′ ′ ′ ′= ∑ ( ) ( ) ( ){ }3 * 1,..., 1,..., 1,..., .iEN N i i i W N N N e β λ ψ ψ − = ∑ ( ) ( ) ( )∫ == .ˆ ! 1 ,...,1 ! 1 , 3 3 3 NN N NNN WTr N rdNW N TVQ λλ U
- 86. 9.6.129.6.129.6.169.6.169.6.179.6.179.6.209.6.20 ))9.6.279.6.27(( 9.6.229.6.22aa)) WU , 111 1 3 1 3 l l l l l l zblandzb KT P ∞ = ∞ = ∑=∑= λυλ ( ) ( ) .ˆ1 ! 1 131 = ∞→− l vl UTr V Lim l b λ
- 87. 9.6.39.6.3))9.6.3a9.6.3a(( ))11(( ))22(( S ( )EB. A( )DF. S NW A NW αψiψ W NWNPNWN N P S N ,...,1ˆ,...,1,...,1ˆ,...,1 ′′′∑=′′ ′ ( )[ ] ,,...,1ˆ,...,11,..,1ˆ,...,1 NWNPNWN N P P A N ′′′−∑=′′ ′ ′
- 88. p′!N1,...,rrN ′′ p′ [ ]p′ ( ) ( ) ( ) ( )1,2,32,3,13,1,23,2,1 ′′′′′′′′′=′′′′ ororp ( ) ( ) ( ) ( )2,1,33,2,11,3,23,2,1 ′′′′′′′′′=′′′′ ororp p′ p′
- 89. 22((1111 ))33(( ))44(( ))55(( ( ){ },111ˆ1 22 11 / 1 / 1 λπ rrAS eWW −′− =′=′ ,2,1ˆ1,22,1ˆ2,12,1ˆ2,1 22 / 2 WWW AS ′′±′′=′′ 3,2,1ˆ3,1,23,2,1ˆ3,2,13,2,1ˆ3,2,1 33 / 3 WWW AS ′′′±′′′=′′′ 3,2,1ˆ2,3,13,2,1ˆ1,3,2 33 WW ′′′±′′′+ .3,2,1ˆ1,2,33,2,1ˆ2,1,3 33 WW ′′′±′′′+
- 90. ))9.6.129.6.12(( 9.6.139.6.13(( 9.6.149.6.14(( >′>=<′< 1ˆ11ˆ1 11 UW >′′<+>′><′>=<′′< 2,1ˆ2,12ˆ21ˆ12,1ˆ2,1 2112 UUUW >′><′><′>=<′′′< 3ˆ32ˆ21ˆ13,2,1ˆ3,2,1 1113 UUUW >′′><′<+ 3,2ˆ3,21ˆ1 21 UU >′′><′<+ 3,1ˆ3,12ˆ2 21 UU >′′><′<+ 2,1ˆ2,13ˆ3 21 UU >′′′<+ 3,2,1ˆ3,2,1 3U
- 91. 9.6.179.6.17(( 9.6.189.6.18(( 9.6.199.6.19(( ,1ˆ11ˆ1 11 WU ′=′ ,2ˆ21ˆ12,1ˆ2,12,1ˆ2,1 1122 WWWU ′′−′′=′′ 3,2,1ˆ3,2,13,2,1ˆ3,2,1 33 WU ′′′=′′′ 3,2ˆ3,21ˆ1 21 WW ′′′− 3,1ˆ3,12ˆ2 21 WW ′′′− 2,1ˆ2,13ˆ3 21 WW ′′′− ,3ˆ32ˆ21ˆ12 121 WWW ′′′+
- 92. −′− =′=′ 2/ 2 121 1 ˆ11 1 ˆ1 λπ rr eUA S U ( )[ ] 3,2,1 3 ˆ3,2,113,2,1/ 3 ˆ3,2,1 ,2 1 ˆ11 1 ˆ22,1 2 ˆ1,22,1 2 ˆ2,12,1 / 2 ˆ2,1 UPP P ASU UUUU AS U ′′′′′ ∑ ′ ±=′′′ ′′±′′±′′=′′ ,3 1 ˆ22 1 ˆ11 1 ˆ3 3 1 ˆ12 1 ˆ31 1 ˆ2 3,2 2 ˆ1,23,2 2 ˆ2,11 1 ˆ3 3,2 2 ˆ3,13,2 2 ˆ1,31 1 ˆ2 UUU UUU termsofsetssimilartwo UUU UUU ′′′+ ′′′+ + ′′±′′′+ ′′±′′′+ (6) (7) (8) .....
- 93. ))99(( ))1010(( s lUA lU( )lnUn ≤ l( )1,2,...,lαm α ,2,1,0, 1 ==∑ = α α αα ml l m ( )( ){ } ( )( ){ } ( ){ } , 321 groupsm ghi groupsm efcd groupsm ba abc ,,....,
- 94. 11 22 ))1111(( ( )[ ] { }[ { } { } ],ˆ ˆˆ ˆˆ1 3 22 11 ×′′′× ′′′′× ′′±∑′ ′ ′ efUIHG efUFEcdUDC bUBaUA P P
- 95. 1111 77 [ ],...,, CBA ′′′p′[ ]l′′′ ,..,2,1 [ ]p′ lUl A S l ,..,1ˆ,..,1 ′′ 2=l( ){ }12 12 =m0=αm ( )( ){ }21 21 =m l 0=αm 77.. ( )1,2,..,l
- 96. 6688 44 2211 11 UU ′′ 11 rr =′ 22 rr =′ 2r 1r 3=l( ){ }123 ( ){ } ( ){ }231 ( ){ } ( ){ }132( ){ } ( ){ }123
- 97. ))1212(( lUB NW lU NNN TH Ω+= ˆ NN T Ωˆ NW ( ) ,ˆexp3 −= N HN N W βλβ
- 98. ))1313(( ))1414(( ( ) ( ), 1 , 1 2 2 2 exp3ˆexp30 ∏ = ⋅ ∏ = ∇=−= N i iw N i imN TN N W β β λβλβ ( ) .2 2 2 exp3, ∇≡⋅ im iw β λβ ( )1;βW 11 1W ′ 11 1U ′ ( ) ( )β0 NWN
- 99. ( )βNW NΩ ( ) ( ) ( )ββ 0 NN WW → 0→ΩN 0→β ∞=Ω N NΩ ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ′′Ω−′′−′Ω−′−∫ ′′∫ ′+ ′Ω−′−∫+= ′ ββββββ ββ β ββββ βββ 0 ˆ 0 ˆ 0 00 0 ˆ 0 0 0 N W NN W NN Wdd N W NN Wd N W N W
- 100. 1515 ))1616(( ∞=Ω ∧ N ≡)(1 βW 1 1′ ββ = 0=β ≡ ),1;(βw ≡)(2 βW 1 1′ 2 2′ + 1 1′ 2 2′ + 1 1′ 2 2′ β′ β ′′ β′ )1;(βw= )2;(βw )2;()1;()ˆ)(2;()1;({ 12 0 βββββββ β ′′Ω−′−′−′+ ∫ wwwwd )}2;()1;()ˆ)(2;( )1;()ˆ)(2;()1;({ 12 12 0 0 ββββ ββββββββ β β ′′′′Ω−′′−′ ′′−′Ω−′−′−′′′+ ∫ ∫ ′ www wwwdd ....+
- 101. ))1717(( ))1818(( =)(3 βW 1 1′ 2 2′ 3 3′ + 1 1′ 2 2′ 3 3′ + 1 1′ 2 2′ 3 3′ + + 1 1′ 2 2′ 3 3′ 1 1′ 2 2′ 3 3′ ....++ + )1;(βw= )2;(βw )3;(βw )3;()}]2;()1;()ˆ)(2;()1;({[ 12 0 ββββββββ β wwwwwd ′′Ω−′−′−′+ ∫ Two similar terms Terms of higher order
- 103. ))2020(( ))2121(( ))2222(( =)(βlU =)(2 βU 1 1′ 2 2′ β′ + 1′ 1 2 2′ β ′′ ....+ ....++=)(3 βU + + ....+ lβ
- 104. 17172121 ))2323(( ))2424(( ))2525(( l l ∇ ∇−−= −= 2 2 2 2 1 2 2 6 )0( 222 2 exp 2 exp)ˆexp( )()()( m h m h H WWU ββ βλ βββ )2,1;(βB ) 2 ˆexp( 12 ˆ6)( 212 ˆ)2,1;( HWB βλββ −Ω−=Ω−≡ )(2)2 2 2 1 ( 2 2)(2)2,1;( β β β β U m hU B ∇+∇− ∂ ∂ =
- 105. ))2626(( 00 )(2 βW 12 ∧ Ω− β =Ω−≡ )(ˆ)2,1;( 212 ββ WB + 1 1′ 2 2′ + 1 1′ 2 2′ 1 1′ 2 2′ ....+ = 1 1′ 2 2′
- 106. 2020 ))2828(( 00 ))2929(( 2525 β′ β′ β ′′ B β ∫ ′′−′−′= β βββββββ 0 )}2,1;()2;()1;({)(2 BwwdU )(2)2 2 2 1 ( 2 2)(2)2,1;( β β β β U m hU B ∇+∇− ∂ ∂ = 1 1′ 2 2′ β′
- 107. 2222 ))3030(( 1 1′ 2 2′ 3 3′ β′ β ′′ 1 1′ 2 2′ 3 3′ β ′′ β′ , , ....etc and 2′ 1 1′ 2 3 3′ , ....etc
- 108. . (31) . 2 )}1;()3,2;()2,1;( 0 0 )3;()2;()1;({ )}3;()2,1;()3,2;( 0 0 )3;()2;()1;()(3 Binordershigherofterms Borderoftermsotherfour wBB wwWdd wBB wwwddU + + ′′×′′′′−′× ′ ′′−′−′−′′′+ ′′×′′′′−′× ∫ ∫ ′ ′−′−′′−′′′= ∫ ∫ ββββ β β ββββββββ ββββ β β βββββββββ 3 ( )U β
- 109. :یییی ییییییی یییی یی یییی:یییی ییییییی یییی یی یییی 13.7.913.7.9.(.( ))11(( >′<>′><′>=<′′′< NWNWWN N WN 1 ˆ...2 1 ˆ21 1 ˆ1,...,2,1ˆ,...,2,1 1 NW W ∧ ) N
- 110. 17.6.917.6.919.6.919.6.9 ))22(( ))44(( 2 /2 1111 )({1ˆ11ˆ1 λπ rreWU −′=>′>=<′< − 0...ˆˆˆ 432 ==== UUU )2,1;(βB lb otherwise lforUTr Vll b 0 11)ˆ( )1(! 1 131 = == − = λ
- 111. 7.7.97.7.98.7.98.7.9 ))55(( ))66(( ))77(( AS lU /∧ 1>l 2 1 11,2 1,2 2 1 1 2 ,SIA U U U′ ′ ′ ′= ± ) ) ) / 3 1 1 1 11,2 ,3 1,2,3 2 1 3 2 1 1 3S A U U U U U′ ′ ′ ′ ′ ′ ′= ) ) ) ) ) 1 1 13 1 1 2 2 3 ,U U U′ ′ ′+ ) ) ) l ( )[ ] / 1 11,..., 1,..., 1 1 2 ... S A p l p l U l A U B U ∧ ∧ ∧′ ′ ′ ′ ′ ′= ± ∑
- 112. ))88(( ))99(( 6.6.96.6.9 ))1010(( ( ) ( ) rdlUlUUll l V l 3 111 1 1...2111!1 ∫ −±−= ∧∧∧ − ( ) ( ) l l UTrl ±−= ∧ − 1 1 1!1 ×−= ∧ )!1()( / lUTr AS l ,ˆ 2/ , 3 11 2 mp pp epWppUp β δλ − ′ ∧ =′=′ pp ′,δ ( ) ( ) 1 / 1 3 1 1 ( ) l S A l l l b Tr U l Vλ − ∧ − ± =
- 113. ))1111(( 111199 ))1212(( 28.6.928.6.9(( ∫∑ ∞ −− ∧ →= 0 22/ 3 32/3 1 22 4 dppe h V eUTr mpl p lmpll l ββ π λλ ( ) 1/ 5/ 2 1 / lS A lb l − = ±
- 114. :ییی ییی یییییییی:ییی ییی یییییییی 30.5.930.5.9 ))1313(( 9.5.79.5.79.5.89.5.81313 ))1414(( 0r )(2 βW ( ) ( ){ }6 * 2 ˆ1 , 2 1, 2 1 ,2 1,2 E a a a W e αβ λ ψ ψ − ′ ′ ′ ′= ∑ ( ) ( ) ( )2 22 . / 1 . / 2 /3 / 4 3/ 2 , 1 1 2p r P R m R Ri p m p e e e e v v λβ λ ′− − −− − ∑ h h %
- 115. ))1515(( ))1616(( ))1717(( ( ) ( ){ } mk ml klmklm errdk / , 0 * 22 3 hβ λ − ∞ ∑∫ Ψ′Ψ )( → rklmψ r ( ) ( ) ( ) , 2 ,, 2/1 r r Yr kl mlklm χ π ϕθ =Ψ ( ) ( ) ( ),,, 3* kkrdrr mmllklmmlk −′=ΨΨ ′′′′′ ∫ δδδ
- 116. ))1818(( ))1919(( 1515 ))2020(( ( ) ( ),cos 4 12 , 1 , θ π ϕθ l l m ml P l Y + =′′∑−− ( ) ( )rrrr ′′= /.cosθ ( ) ( ) ( ) ( ){ } .cos12 2 0 /* 2 3 22 ∑ ∫ ′+ ′ ∞ − l mk klkll errdkPl rr hβ χχθ π λ
- 117. 14142020 2121 9.7.239.7.23 ))2222(( ( ){ }22 21212 32/1 2 2/exp 2 2,1ˆ2,1 λπ π λ rrrr rr W −−′+′− ′ =′′ ( ) ( ) ( ) ( ){ } .cos12 0 /* 22 ∑ ∫ ′+× ∞ − l mk klkll errdkpl hβ χχθ ( ) ( ){ } ( ) ( ){ }( )0** rrrr klklklkl χχχχ ′−′ { })()( * rr klkl χχ ′ 2U
- 118. 2222 :: 2 0 λ r )()0( sorl = 0, rrr >′ ( ) ( ) ( ) ( ){ }krrkkrkrkrrk sinsinsinsin 00 ′−−−′ ( ) ( ){ }02coscos 2 1 krkrrkkrrk −+′−+′= ( ) ( ){ }krrk sinsin0 ′− ( ) ( ){ }.coscos 2 1 krrkkrrk −′−+′= { })()( rr klkl χχ ′
- 119. 2121l0=l ( ){ } { } 2 2 1/2 3 2 2 / 2 1 2 1 22 0 2 1,2 1,2 exp / 2 ... k m U r r r r dk e rr βλ π λ π ∞ − ′ ′ ′ ′= − + − − ′ ∫ h ) ( ){ }22 2121 2 2/exp 2 λπ π λ rrrr rr −−′+′− ′ = ( ) } ( ){ }{ ( ){ } ( ){ } 2 2 2 0 2 22 2 exp / 2 exp 2 2/ 2 , exp / 2 exp / 2 or r r r r r r r r r r r otherwise π λ π λ π λ π λ ′ ′ ′− + − − + − 〉 × ′ ′− + − − −
- 120. ))2525(( 9.7.79.7.7 ))2626(( 2424 ))2727(( .2,12,1 2 1 2 3 1 3 1 3 232 rdrdrdU V b s s ∫∫ ∧ = λ s b2 [ ] 1 3 2 3 221132 2,1ˆ1,22,1ˆ2,12ˆ11ˆ2 2 1 rdrdUUUU V bs ∫∫ ++= λ R r { } 22 200 2 2 2 0 2 ( )2 2 (0) 2 2 2 2 2 2 0 0 1 1 1 4 1 4 2 2 rr rr r r b b e e r dr e r dr r r r ππ π λ λ λ π π πλ λ ∞ − − − − − = − + − ÷ ÷ ÷ = − ÷ ∫ ∫
- 121. ))2828(( 9.7.299.7.299.7.319.7.31 ))2929(( { }{ }211 ' .... UUU∑ s iUs b2 ( ) ( ){ } ,ˆˆˆ 21 121 nn UUUTr UhlenbeckBeth− { } { } { } { } { }' ' ' 1 1 1 1 2 1 1 2 2... ... ... ...s lU U U U U U U U U U= + + +∑ ∑ ∑
- 122. 88 ))3030(( ( ) [ ]{ }[ ]∑ ∑〈 ′′+′′− ii nn nn l UjiUIJiiUJIUTrl 2 21 , 1221 ˆ,ˆ,,ˆ,ˆ!2 1n2n02 ≥n 01 ≥n221 −=+ lnn )( ∧ s lUTr ),.......,1( l′′)!2( −l )( ∧ s lUTr ij ),.......,1( l 2 )1( −ll
- 123. ))3131(( ))3232(( ))3333(( 10109.109.10(((( 3333 [ ]{ } ..ˆ2,1ˆ,2,1ˆ,ˆ! 2 1 21 1221 2,1 ′′+′′ ′ ∑ nn nn UUIJUJIUrTl 2 2 1 1, 1 / 23 1 1 1 ˆ k m k kk U k e β λ δ − ′′ = h ( ) { } ( ), 1 2 6 2 1 2 2 1 2 02 2 2 ˆ, , 0 2 i i E E k k k k ro k k U k k e e r k k β βλ δ π ′− − + + ′ ′ = − + ′−
- 124. ))3434(( ))3535(( 313132323333 ))3636(( 33333535 2121 2 1 2 1 kkkkkk ′−′=′−= ( ) ( ). 22 2 2 2 1 2 2 2 2 1 2 kk m Ekk m E ′+′=′+= hh .ˆˆ 2,121,22,1221 kkUkkkkUkk +
- 125. ))3737(( 1 2 1 1 1,2 1,2 6 2 , 02 2 2 lim 0 E E k k k k k k ro e e r k k β β λ δ π ′− − ′ ′+ +′ → − + ÷− ( ) ( ) ( )1 2, 1 2 1,2 1,2 2 2 2 28 1 2 1 2 20 03 2 2 lim 0 4 k E k k k k k k k k kr e r k k β λ δ π → − ′ ′+ +′ ′ ′+ − + = + ′− ( ) 8 2 2 2 20 1 2 03 exp ( ) 2 2 r k k o r m λ β π = − − + + h s lb
- 126. ))3838(( 3838 ))3939(( ( ) 3 3 3 1, 2 1 1 1 2 1 1 23 1 1 ˆlim ,..., , ,..., ... ! s llv r r r U r r r d rd r d r l Vλ −→∞ ∫ ( ) 3 3 2 1 1 2 1 23 1 1 ˆ0, ,..., 0, ,..., ... ! s ll r r U r r d r d r l λ − = ∫ ( ) ( ) 3 3 1 1 133 1 1 ,..., ,..., ... , ! 2 s l l l ll k k u k k d k d k l λ π− ∫ s s l lU U )
- 127. ( ) ( ) 6 2 3 1 32 2 2 , 2 1 16 0 5 2 1exp 2 1exp 32 21 h pdpd m p n m p n r nn +− ′ +−−= ∫∫ ∑ ββ π λ ( ) ( ) 12/exp 1 12/exp 11 2 2 12 1 16 −− −− mpzmpz ββh ( ) ( ) 2 2/33 3 2 0 21 2 3 8 12/exp 41 = −∫ ∞ − zg mpz dpp λ π β π h ( ) ( ) ( ) ( ) +− − ′ − ∫∫∑ + − 2 2 2 1 22 1 2 1 , 3 3 0 8 313 2 exp 2 exp 2 ! 2 1 8! 1 21 21 kk mm k n r l l nn nn l hh β βλ π λ πλ 2 3 1 3 2 2 2 2 2 exp kdkd m k n −× h β 40
- 128. ( ){ }2 2/3 02 zg r λ − ( ) ( ){ } , 2 2 02 2/3 0 2/5 3 +−=≡∑ λλ λ r Ozg r zg kT p zb l ls l 41 42