J=A:JBIH<
GHU :JKLBKEHZevg_cr__ jZa\blb_l_hjbb^eyg_Z^^blb\guoh[t_dlh\g_fukebfh[_ahkgh\hiheZ ]Zxsbo b^_c b...
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J=A:JBIHBKEHZevg_cr__ jZa\blb_l_hjbb^eyg_Z^^blb\guoh[t_dlh\g_fukebfh[_ahkgh\hiheZ ]Zxsbo b^_c b f_lh^h\ – 24, 130, 131]. JZkkfhljbfkemqZcwdklj_fmfZwgljhibbR_gghgZ–Ze__ jZkkfhljbf wdklj_fmf bgnhjfZpbb jZaebqby Dmev[ZdZ– E_c[e_jZ −
m p I ( p : u ) = ∑ log 2 i pi ui i
(1.9.14)
^ey ^\mo \ur_ jZkkfhlj_gguo kemqZ_\ b ihemqbf khhl\_lkl\_ggh jZk ij_^_e_gby pi = ui ,
(1.9.15) m
pi = ui 2 τTi + −1 ( τ ) , + ( τ ) = ∑ 2 τTi Xi
(1.9.16)
i
bk\hckl\Znmgdpbb Γ ( τ ) ∂ log 2 Γ ( τ ) ∂ 1 = E (T ) , − log 2 Γ ( τ ) = I ( p : u ) . ∂τ ∂ (1 τ ) τ
(1.9.17)
>ey ijhba\h^ghc kemqZcghc bgnhjfZpbb jZaebqby bf__f khhlgh r_gb_ ∂I ( pi : ui ) ∂τ
= Ti − E (T ) .
(1.9.18)
JZkij_^_e_gb_ ^Z_l wdklj_fZevgu_ agZq_gby bgnhjfZpbb jZaebqbybjZkoh`^_gby m p I ( p : u ) = ∑ log 2 i pi = τE (T ) − log 2 Γ ( τ ) , ui i
(1.9.19)
m u I (u : p ) = ∑ log 2 i ui = −τE 0 (T ) + log 2 Γ ( τ ) , pi i
(1.9.20)
J ( p : u ) = I ( p : u ) + I (u : p ) = τ E (T ) − E 0 (T ) .
(1.9.21)
Imklv jZkij_^_e_gb_ p ijhba\hevgh_ Z jZkij_^_e_gb_ u ijbgZ^
56
e_`bliZjZf_ljbq_kdhfmk_f_ckl\mjZkij_^_e_gbc ui = 2 τ0Ti + −1 ( τ0 ) , + ( τ 0 ) = ∑ 2 τ0Ti . m
(1.9.22)
i
Lh]^Z bkihevam_f agZq_gby wgljhibb R_gghgZ–@ JZkkfhljbf kemqZc baf_g_gby h^ghf_jgh]h iZjZf_ljZ Ih hij_^_ e_gbx iZjZf_lj g_ bkiulu\Z_l nemdlmZpbc b ke_^h\Zl_evgh g_ bf__l klZlbklbq_kdbooZjZdl_jbklbdH^gZdhfh`ghihklZ\blvaZ^Zqmh[hp_gd_ g_nemdlmbjmxs_]h iZjZf_ljZ ih j_amevlZlZf gZ[ex^_gbc ih^\_j`_g guo klZlbklbq_kdbf hldehg_gbyf Ijb wlhf bkihevamxlky b^_b l_hjbb iZjZf_ljbq_kdh]hhp_gb\Zgby ImklvkemqZcgZy\_ebqbgZ θ*i dhlhjZyg_aZ\bkblhl θ \ujZ`Z_l kyq_j_aba\_klgu_^bkdj_lgu_\_ebqbguoZjZdl_jbamxsb_kemqZcguc h[t_dl LZdZy \_ebqbgZ gZau\Z_lky hp_gdhc bklbggh]h agZq_gby g_ nemdlmbjmxs_]hiZjZf_ljZ θ Ijb\u[hj_hp_gdb_kl_kl\_gghij_^ih
( )
eh`blv qlh __ kj_^g__ agZq_gb_ E θ* [ueh [ebadh d iZjZf_ljm θ . ey ^bki_jkbb hp_gdb kijZ\_^eb\h nmg^Zf_glZevgh_ bgnhjfZpbhggh_ g_jZ\_gkl\h JZh– DjZf_jZ>@
59
∂b ( θ ) 1 + ∂θ , = 2 m ∂ ln p (θ ) i ∑ pi ( θ ) ∂θ i 2
( )
( )min
D θ* ≥ D θ*
(1.10.9)
^Zxs____gb`gxx]jZgvG_dhlhju_h[h[s_gby^ey ijb\h^ylky \>-@Ijb b ( θ ) = 0 g_jZ\_gkl\h h[jZsZ_lky\
( )
( )min = = θθ−1 .
D θ* ≥ D θ*
(1.10.10)
@ H ( p ) = −k ∑ ( ln pi ) pi = β ( E − F ) m
(1.11.1)
i
^hklb]Z_lky kh]eZkgh ijb jZ\gh\_kghf dZghgbq_kdhf jZkij_^_ e_gbb=b[[kZ
60
{
}
pi = exp − k −1β ( H i − F ) .
(1.11.2)
A^_kv β = − k τ ln 2 = 1 T _klv h[jZlgZy Z[khexlgZy l_fi_jZlmjZ H i – ^bkdj_lgu_agZq_gbywg_j]bbqZklbpu E = E ( H ) –kj_^gyywg_j]byqZk lbpuK\h[h^gZywg_j]by m
{
F = − kT ln ∑ exp −k −1β H i i
}
(1.11.3)
hij_^_ey_lkybamkeh\byghjfbjh\dbjZkij_^_e_gby >bnn_j_gpbZevgu_ khhlghr_gby jZ\gh\_kghc klZlbklbq_kdhc l_j fh^bgZfbdb aZfdgmluo kbkl_f bf_xl kh]eZkgh b ke_ ^mxsbc\b^ TdH ( p ) = dE , dF = − H ( p ) dT .
(1.11.4)
>eyhldjuluokbkl_fgZoh^ysboky\hdjm`_gbbkl_fi_jZ lmjhc T0 nbabq_kdZy jZaf_jgZy bgnhjfZpby jZaebqby bf__l kh]eZkgh agZq_gb_>@ m p 1 I ( p : p0 ) = k ∑ ln i pi = −δH + δE = p T0 i 0i 1 = − H ( p ) − H ( p0 ) + ( E − E0 ) ≥ 0 T0
(1.11.5)
kjZ\_gkl\hflh]^Zblhevdhlh]^Zdh]^Z pi = p0i A^_kv^eyg_jZ\gh\_k ghcbjZ\gh\_kghcwgljhibcbkhhl\_lkl\_gghkj_^gbowg_j]bcbf__f H ( p ) = −k ∑ ( ln pi ) pi , H ( p0 ) = − k ∑ ( ln p0i ) p0i , m
m
i
i
m
m
i
i
E = ∑ H i pi , E0 = ∑ H i p0i .
(1.11.6) (1.11.7)
Khklhygb_ ihegh]h jZ\gh\_kby k hdjm`_gb_f hibku\Z_lky dZghgb q_kdbfjZkij_^_e_gb_f
{
}
p0i = exp − k −1β 0 ( H i − F0 )
(1.11.8)
kh[jZlghcl_fi_jZlmjhc β 0 = 1 T0 bk\h[h^ghcwg_j]b_c F0 . Bkihevah\Zgb_^bZ]jZffuwgljhiby-wg_j]byijb^Z_lijhklhcbgZ ]ey^guckfukebgnhjfZpbbjZaebqbyGZjbkijb\_^_gh]_hf_ljbq_ 61
kdh_ij_^klZ\e_gb_l_jfh^bgZfbq_kdbo\_ebqbg [12]. H(p) b
q
d –δH I(p:p0) e
δE
a
Rmax
c E
Jbk
Wgljhiby-wg_j]by ^ey nbabq_kdbo kbkl_f
Djb\Zy ab bah[jZ`Z_lbaf_g_gb_nmgdpbb H ( E ) hlijhba\hevgh]h khklhygbykhhl\_lkl\mxs_]hlhqd_ a = ( H , E ) diheghfmjZ\gh\_kghfm
\lhqd_ b = ( S 0 , E0 ) Balhqdb b ijh\h^blkydZkZl_evgZyijyfZy bc Lh ]^Z kh]eZkgh hlj_ahd ad iZjZee_evguc hkb H ( p ) _klv bg
nhjfZpby jZaebqby Dmev[ZdZ–E_c[e_jZ I Hlj_ahd ac , iZjZee_ev guc hkb E , bah[jZ`Z_l fZdkbfZevgmx jZ[hlm Rmax \uihegy_fmx kbk l_fhc gZ^ hdjm`_gb_f FbgbfZevgZy jZ[hlZ Rmin = − Rmax k\yaZgZ k ba f_g_gb_f iheghc wgljhibb aZfdgmlhc kbkl_fu kbkl_fZhdjm`_gb_ hl k\h_]h gZb[hevr_]h agZq_gby _keb kbkl_fZ g_jZ\gh\_kgZ \ujZ`_gb_f ∆H i = − Rmin T0 . @ H ( p ) ≥ H ( p0 ) .
(1.11.11)
LZdbf h[jZahf g_ ijb[_]Zy d jZkkfhlj_gbx \j_f_gghc w\hexpbb wgljhibbfh`ghhij_^_eblvbaf_g_gby__ijbi_j_oh^Zof_`^mjZaebq gufbkhklhygbyfbkbkl_fuIjbq_fwgljhibyijbh[j_lZ_lk\hckl\ZagZ dhhij_^_e_ggh]h nmgdpbhgZeZ lhevdh ijb \uiheg_gbb ^hihegbl_evgh]h mkeh\byihklhygkl\Zkj_^g_cwg_j]bb IjbgpbifZdkbfmfZwgljhibbihdZau\Z_lqlhijbkihglZgghfi_ j_oh^_ hl ijhba\hevgh]h g_jZ\gh\_kgh]h khklhygby d jZ\gh\_kghfm kl_ i_gv g_hij_^_e_gghklb jZamihjy^hq_gghklb kbkl_fu m\_ebqb\Z_lky b ijbjZ\gh\_kbb^hklb]Z_lfZdkbfZevgh]hagZq_gby >eyg_h[jZlbfuobgnhjfZpbhgguonbabq_kdboijhp_kkh\djZ\_g kl\m ^h[Z\ey_lky agZd g_jZ\_gkl\Z b \ ^bnn_j_gpbZevghf \b^_ bf__f>@ dI ( p : p0 ) ≥ − dH ( p ) +
1 dE . T0
(1.11.12)
JZkkfZljb\Z_fuckemqZckhhl\_lkl\m_llZdhcnbabq_kdhckblmZpbb \l_jfh^bgZfbd_g_aZfdgmluokbkl_fdh]^Z\kbkl_f_g_ijhbkoh^blba f_g_gby wg_j]bb ( dE = 0 ) Ba \ul_dZ_l \ujZ`_gb_ g_]wgljh ibcgh]hijbgpbiZ;jbeexwgZ>@
dI ( p : p0 ) + dH ( p ) ≥ 0 ,
(1.11.13)
h[h[sZxs_]h ijbgpbi DZjgh–DeZmabmkZ h \hajZklZgbb wgljhibb ^ey aZfdgmluokbkl_f dH ( p ) ≥ 0 .
(1.11.14)
Ijb hlkmlkl\bb baf_g_gby jZ[hlu ( δR = 0 ) \uihegy_lky h[sbc ijbgpbimf_gvr_gbybgnhjfZpbbjZaebqby>@ dI ≤ 0 .
(1.11.15)
LZdbf h[jZahf \aZbfgh_ baf_g_gb_ wgljhibb b nbabq_kdhc bg nhjfZpbb jZaebqby khijh\h`^Z_lky ihl_j_c bgnhjfZpbb mihjy^h q_gghklb khklhygbc kbkl_fu Ba ke_^m_l qlh ijbg_h[jZlb
fuo y\e_gbyo m\_ebq_gb_ wgljhibb dH ( p ) > − dI ( p : p0 ) [hevr_ q_f mf_gvr_gb_ jZaebqZxs_c bgnhjfZpbb Ke_^h\Zl_evgh ijhbkoh^bl kZfhjZkiZ^ g_jZ\gh\_kghc kbkl_fu b kihglZggu_ i_j_oh^u ijb\_^ml
63
\ dhgp_ dhgph\ d iheghfm jZ\gh\_kghfm khklhygbxg_aZfdgmlhc kbkl_ fukfZdkbfZevghcg_hij_^_e_gghklvxjZamihjy^hq_gghklvx \khklhy gbyo Kh]eZkgh;jbeexwgm>@baf_g_gb_jZ[hlu^eyjZkkfZljb\Z_fh ]hkemqZygZ\_ebqbgm kT0 khhl\_lkl\m_lh^ghcgZlmjZevghc_^bgbp_ba f_j_gbydhebq_kl\ZbgnhjfZpbbgbl \l_ieh\hckj_^_ bZ]jZffZwgljhiby-wg_j]byiha\hey_lgZclb]_hf_ljbq_kdh_ij_^ klZ\e_gb_ dhwnnbpb_glm ihe_agh]h ^_ckl\by DI> ij_h[jZah\Zgby wg_j]bb ^ey h[jZlbfuo ijhp_kkh\ Ba\_klgh qlh DI> ij_h[jZah\Zgby wg_j]bb\ujZ`Z_lkyhlghr_gb_fkh\_jr_gghcfZdkbfZevghcjZ[hluhl djulhckbkl_fudaZljZq_gghcwg_j]bb\hdjm`_gbb η=
Rmax Rmax + δE
.
(1.11.18)
>Zggh_jZ\_gkl\hij_^klZ\ey_lkh[hchlghr_gb_hlj_adh\ ac ec dhlhjh_ fh`gh aZibkZlv \ \b^_ η = ad eb Lh]^Z DI> ij_h[jZah\Zgby wg_j]bb \ujZ`Z_lkydZdhlghr_gb_bgnhjfZpbbjZaebqbydbaf_g_gbxwgljhibb ijbi_j_oh^_hlijhba\hevgh]hkhklhygbydjZ\gh\_kghfm>@ η=
I I = ≤ 1. −δH H 0 − S
(1.11.19)
K ^jm]hc klhjhgu bgl_jij_lZpby g_jZ\_gkl\Z gZ fbdjh kdhibq_kdhf mjh\g_ hagZqZ_l qlh kl_i_gv mihjy^hq_gghklb khklhygbc hldjulhckbkl_fuijbkihglZgghfi_j_oh^_kbkl_fuf_gvr_q_fbaf_ g_gb_kl_i_gbjZamihjy^hq_gghklb BgnhjfZpbyjZaebqbyDmev[ZdZy\ey_lkyagZdhhij_^_e_gghcnmgd pb_c Eyimgh\Z Ihwlhfm qlh[u khklhygb_ ihegh]h jZ\gh\_kby [ueh mk lhcqb\uf g_h[oh^bfh \uiheg_gb_ ke_^mxs_]h g_jZ\_gkl\Z ^ey ijhba \h^ghc
64
dI ( p : p0 ) dt
=−
(
d H ( p ) − H ( p0 ) dt
) ≤ 0.
(1.11.20)
Ba ke_^m_l aZdhg \j_f_gghc w\hexpbb wgljhibb ( H –l_hj_fZ;hevpfZgZ>@ dH ( p ) ≥0 dt
(1.11.21)
ijb ijb[eb`_gbb d khklhygbx ihegh]h jZ\gh\_kby Ijhbkoh^bl kZfh jZkiZ^fZdjhkdhibq_kdhckbkl_fuijbkihglZgguoi_j_oh^Zo Imklv kbkl_fZ gZoh^blky \ ehdZevghf qZklbqghf jZ\gh\_kbb k l_fi_jZlmjhc T Ih^klZ\eyy \ hkgh\gh_ mjZ\g_gb_ ihemqbf 1 1 dI ( p : p0 ) ≥ − − dE , , T T 0
(1.11.22)
]^_ agZdg_jZ\_gkl\Z khhl\_lkl\m_l g_h[jZlbfuf bgnhjfZpbhgguf ijh p_kkZfijbdhglZdl_ehdZevghjZ\gh\_kghckbkl_fukhdjm`_gb_f
65
=eZ\Z2 D @ KemqZcgZy \_ebqbgZ _klvhi_jZlhjT \a\_r_ggh_kj_^g__dhlhjh]hjZ\gh E (T ) =
SpT ρ . Spρ
(2.1.1)
D\Zglh\ucZgZeh]\_jhylghklghcghjfbjh\db_klv Spρ = 1 ,
66
ρ ≥1
(2.1.2)
bke_^h\Zl_evghkj_^g__agZq_gb_hi_jZlhjZaZibr_lkylZd E (T ) = SpT ρ .
(2.1.3)
>eyfZljbqgh]hij_^klZ\e_gbyba b \ul_dZ_lnhjfmeZ E (T ) = ∑ Tii ρii + ∑ ∑ Tij ρ ji ,
(2.1.4)
i j ≠i
i
ba dhlhjhc ke_^m_l qlh \_jhylghklgZy ljZdlh\dZ kijZ\_^eb\Z dh]^Z ρij = 0 ijb i ≠ j @ > 1) klZlbklbq_kdb kemqZcguo h[t_dlh\ dhlhju_
bf_xl fgh`_kl\h d\Zglh\uo khklhygbc G = {G1 ,! , Gm } ]^_ m – qbkeh khklhygbc H[t_dlZfb fh]ml y\eylvky gZijbf_j ihke_^h\Zl_evghklb kbf\heh\ ba ZenZ\blZ b fZl_jbZevgu_ qZklbpu k khhl\_lkl\mxsbfb kh klhygbyfb \ \b^_ kbf\heh\ b djZlghklb kh[kl\_ggh]h agZq_gby wg_j]bb qZklbpubli Ijb\_^_fhkgh\gu_hij_^_e_gby Hij_^_e_gb_ Ghkbl_e_f bgnhjfZpbb y\eyxlky ^\Z fgh`_kl\Z Z fgh`_kl\h \k_o d\Zglh\uo khklhygbc G = {G1 ,! , Gm } kh\hdmighklb kemqZcguo h[t_dlh\ N = {N1 ,! , N m } [ fgh`_kl\h kemqZcguo \_ebqbg T = {T1 ,! , Tm } oZjZdl_jbamxsbokh\hdmighklvh[t_dlh\
Hij_^_e_gb_@ JZkkfZljb\Zy \gZqZe_ jZkij_^_e_gby N1i b N 2i ih khklhygbyf Gi , dZdg_aZ\bkbfu_\uqbkebfbgnhjfZpbxjZaebqby\\b^_jZaghklbd\Zg lh\uowgljhibc
(
( I12 )N = − H N − H N 1
1
2
)
= log 2
∆+N
2
∆+N
,
(2.7.1)
1
]^_ m
m
i
i
m
m
i
i
∆Γ N = ∏ ∆Γ N , ∑ N1i = N1 , 1 1i
∆Γ N = ∏ ∆Γ N , ∑ N 2i = N 2 . 2 2i
(2.7.2)
(2.7.3)
83
Wlh h[hkgh\Zggh khhl\_lkl\m_l hij_^_e_gbx bgnhjfZpbb \ \b^_ g_]wg ljhibbih;jbeexwgmH^gZdhmq_ljZaebqbyf_`^m^\mfy]jmiiZfbqZk lbp oZjZdl_jbamxs_]hky qbkeZfb K i = N1i − N 2i b K = N1 − N 2 ^Z_l ^h
(
ihegbl_evgh_keZ]Z_fh_ log 2 ∆+N
∆+ N
2
2 −K
) ey klZlbklbd N_jfb–>bjZdZ b ;ha_–Wcgrl_cgZ bf__fkhhl\_lkl\_ggh\ujZ`_gby m D1i τΩ = ∑ log 2 D2i 2 τTi +α i m D1i τΩ = − ∑ log 2 D2i 2 τTi +α i
Gi ,
(2.7.37)
Gi .
(2.7.38)
Wdklj_fmf bgnhjfZpbc jZaebqby jZaebqguo klZlbklbd khhl\_lkl \m_lbofbgbfmfmqlhe_]dh^hdZau\Z_lky ey gZoh`^_gby bgnhjfZpbb jZaebqby \hkihevam_fky hkgh\guf khhlghr_gb_f aZibkZgguf\ke_^mxs_f\b^_ m r G ! Gi ! ∏∏ i i k G2ik ! i k G ! . = log 2 m r 1ik + log 2 m r Gi ! Gi ! ∏∏ ∏∏ i k (G2ik − K ik ) ! i k G2ik ! m
( I12 )N
1
r
∏∏
(2.8.7)
A^_kv qbkeZ K ik = G1ik − G2ik oZjZdl_jbamxl mq_l jZaebqby f_`^m ^\mfy kh\hdmighklyfb khklhygbc ^ey dhlhjuo bf__f ke_^mxsb_ khhl ghr_gby r
r
k
k
N1i = ∑ kG1ik , N 2i = ∑ kG2ik ,
(2.8.8)
r
r
m
m
k
k
i
i
Gi = ∑ G1ik = ∑ G2ik , N1 = ∑ N1i , N 2 = ∑ N 2i .
(2.8.9)
Bkihevam_fnhjfmemKlbjebg]Zbijb[eb`_ggh_khhlghr_gb_ G2ik !
(G2ik − C1ik )
!
≈ G2Kikik
(2.8.10)
ijb K ik bjZdZb;ha_–Wcgrl_cgZ Di = Di =
1 2
−τTi −α
+1
1 2 −τTi −α − 1
,
(2.8.25)
.
(2.8.26)
>eyijhba\hevgh]hagZq_gby r bf__fjZkij_^_e_gb_^eyiZjZklZlb klbdb>@ Di =
1 2
−τTi −α
r +1 , − 1 2( r +1)(−τTi −α ) − 1 −
(2.8.27)
]^_bkihevah\ZeZkvnhjfmeZ^eydhg_qghckmffu r
∑ xk =
k =0
x r +1 − 1 . x −1
(2.8.28)
>Ze__ jZkkfhljbf gh\uc kemqZc iZjZklZlbklbdb ^ey dhlhjh]h \dZ`^hfkhklhygbbfh`_lgZoh^blvkyg_f_g__ s bg_[he__ r h[t r
r
k =0
k =s
_dlh\Lh]^ZkmffZ ∑ aZf_gy_lkygZkmffm ∑ bijh\h^y\uqbk
92
e_gbyihemqbfjZkij_^_e_gb_ Di =
Ni Gi
=
r k τT +α ∑ k2 ( i )
k =s r
∑2
k (τTi +α )
=
k =s
=
1 2
−τTi −α
r − s +1
−
−1 2
( r +1)( −τTi −α )
−2
(2.8.29)
s ( −τTi −α )
b\ujZ`_gb_ m r k τT +α τΩ = ∑ log 2 ∑ 2 ( i ) Gi = i k =s 1 r T s + τ +α τT +α m 2( )( i ) − 2 ( i ) = ∑ log 2 Gi . (τTi +α ) − 1 i 2
(2.8.30)
Ijb s = 0 ba b \ul_dZxl nhjfmeu ljZ^bpbhgghc iZjZklZlbklbdb b ?keb s = r − 1 lhbf__fjZkij_^_e_gb_ Di =
Ni Gi
=
1 2
−τTi −α
2
+
− 1 2(r +1)(−τTi −α ) − 2(r −1)(−τTi −α )
,
(2.8.31)
lZd`_aZ\bkys__hlh^gh]hiZjZf_ljZ r dZdbijb s = 0 . bjZdZ k = 0, 1 ) G
m
N =∑ i
2
i −τTi −α
+1
;
(2.8.32)
[ klZlbklbdZ;ha_–Wcgrl_cgZ k = 0,! , ∞ ) m
N =∑ i
Gi
2 −τTi −α − 1
;
(2.8.33)
\ iZjZklZlbklbdZ k = 0,! , r ) m 1 r +1 N = ∑ −τT −α − Gi ; i − 1 2(r +1)( −τTi −α ) − 1 i 2
(2.8.34)
] iZjZklZlbklbdZ k = s,! , r ) 93
m 1 r − s +1 − N = ∑ −τT −α Gi ; 1 + −τ −α −τ −α r T s T ( ) i −1 2 ( i ) − 2 ( i ) i 2
(2.8.35)
^ iZjZklZlbklbdZ k = r − 1, r ) m 1 2 N = ∑ −τT −α − Gi . i − 1 2( r +1)(−τTi −α ) − 2( r −1)(−τTi −α ) i 2
(2.8.36)
GZdhg_p gZoh^bf wdklj_fmf bgnhjfZpbb jZaebqby ijb aZ ^Zgghklbke_^mxsboagZq_gbc m
m r
i
i
m
m r
i
i
E N (T ) = ∑ Ti N1i = ∑ ∑ kTi G1ik , 1
(2.8.37)
k
N1 = ∑ N1i = ∑ ∑ kG1ik .
(2.8.38)
k
Bkihevamy \ZjbZpbhgguc f_lh^ hdhgqZl_evgh ihemqbf qbkeZ khklhygbc k τT +α (2.8.39) G1ik = 2 ( i ) G2ik , r k τT +α Gi = ∑ 2 ( i ) G2ik ,
(2.8.40)
r k τT +α N1i = ∑ k 2 ( i ) G2ik
(2.8.41)
k
agZq_gbyqbk_eh[t_dlh\ k
bbkdhfh_jZkij_^_e_gb_ D1i =
N1i Gi
=
r k τT +α ∑ k 2 ( i ) G2ik k r
∑2 k
k ( τTi +α )
,
(2.8.42)
G2ik
dhlhjh_ ^ey jZaebqguo agZq_gbc r [m^_l khhl\_lkl\h\Zlv lhc beb bghc jZkkfZljb\Z_fhcklZlbklbd_
Ijbeh`_gbyd\Zglh\uof_j Imklv kemqZcgufb h[t_dlZfb y\eyxlky qZklbpu jZkkfZljb\Z_ fu_\klZlbklbq_kdhcnbabd_>@Lh]^Z\_ebqbgu Di = N i Gi _klvlZd gZau\Z_fu_ kj_^gb_ qbkeZ aZiheg_gby \ i -khklhygbb. D\Zglh\u_ kh 94
−r G G klhygby Gi = ( 2π= ) dri dpi jZ\gyxlky[_ajZaf_jgufwe_f_glZfnZah\h]h
ijhkljZgkl\Z ]^_ = – ihklhyggZy IeZgdZ r – qbkeh kl_i_g_c k\h[h^u G G qZklbpu ri b pi _klv agZq_gby dhhj^bgZl b bfimevkh\ Ijb i_j_oh^_ d g_ij_ju\guf ZgZeh]Zf bf__f khhl\_lkl\mxsb_ jZaf_jgu_ nbabq_kdb_ d\Zglh\u_ wgljhibb bgnhjfZpbb jZaebqby jZkoh`^_gby b f_ju g_lhq ghklb\gZlZo Z klZlbklbdZFZdk\_eeZ–;hevpfZgZ H N ( D ) = − k ∫ ( ln D ) DdX ,
(2.9.1)
D I N ( D : D0 ) = k ∫ ln DdX , D 0
(2.9.2)
D J N ( D : D0 ) = k ∫ ln D − D0 ) dX , D ( 0
(2.9.3)
H N ( D : D0 ) = −k ∫ (ln D0 ) DdX ,
(2.9.4)
[ klZlbklbdZN_jfb–>bjZdZ H N ( D ) = −k ∫ D ln D + (1 − D ) ln (1 − D ) dX ,
(2.9.5)
(1 − D ) dX , D I N ( D : D0 ) = k ∫ D ln + (1 − D ) ln D0 (1 − D0 )
(2.9.6)
D (1 − D ) J N ( D : D0 ) = k ∫ ( D − D0 ) ln dX , D0 (1 − D0 )
(2.9.7)
H N ( D ) = −k ∫ D ln D + (1 − D ) ln (1 − D ) dX ,
(2.9.8)
\ klZlbklbdZ;ha_–Wcgrl_cgZ H N ( D ) = −k ∫ D ln D − (1 + D ) ln (1 + D ) dX ,
(2.9.9)
(1 + D ) dX , D I N ( D : D0 ) = k ∫ D ln − (1 + D ) ln D0 (1 + D0 )
(2.9.10)
D (1 + D ) J N ( D : D0 ) = k ∫ ( D − D0 ) ln dX , D0 (1 + D0 )
(2.9.11)
95
H N ( D ) = −k ∫ D ln D + (1 − D ) ln (1 − D ) dX ,
(2.9.12)
] d\ZabdeZkkbq_kdZyklZlbklbdZ H N ( D ) = k ∫ ln DdX , I N ( D : D0 ) = k ∫ ln
(2.9.13)
D dX , D0
(2.9.14)
J N ( D : D0 ) = 0 , H N ( D0 ) = k ∫ ln D0 dX .
(2.9.15)
−r G G A^_kv dX = ( 2π= ) drdp b k –nmg^Zf_glZevgZyihklhyggZy;hevp
fZgZ>eyjZkij_^_e_gbcbf__f
∫ DdX = N ,
∫ D0 dX = N 0 .
(2.9.16)
>ey \k_o klZlbklbd bf_xl f_klh ^bnn_j_gpbZevgu_ khhlghr_gby klZlbklbq_kdhcnbabdbjZ\gh\_kguokbkl_f dH N ( D ) = β ( dE N − µdN )
(2.9.17)
bg_jZ\gh\_kguohldjuluokbkl_f\hdjm`_gbb dI N ( D : D0 ) = − dH N ( D ) + β 0 dE − β 0µ 0 dN ,
(2.9.18)
]^_ β 0 b µ 0 _klvh[jZlgZyl_fi_jZlmjZbobfbq_kdbcihl_gpbZehdjm`_ gby
JZ\gh\_kgu_ jZkij_^_e_gby fZdkbfbabjmxsb_ wgljhibx bf_xl khhl\_lkl\_ggh^eyjZaebqguoklZlbklbdke_^mxsbc\b^
{
}
D = exp −k −1β ( H − µ ) , D= D=
1
exp
{ k −1β ( H − µ )} + 1
exp
{k
D=
1
}
β ( H − µ) −1
−1
1 , k β (H − µ) −1
(2.9.19) ,
(2.9.20)
,
(2.9.21)
(2.9.22)
]^_ β = 1 T – h[jZlgZy l_fi_jZlmjZ H = H ( X ) _klv wg_j]by qZklbpu E N = E N ( H ) –ihegZywg_j]bykbkl_fuqZklbp µ –obfbq_kdbcihl_gpbZe
96
JZkkfhljbfg_dhlhju_ijhkl_crb_nbabq_kdb_ijbf_ju B^_Zevguc ]Za Ij_g_[j_]Z_f \aZbfh^_ckl\b_f qZklbp \ ]Za_ G qlh ijb\h^bl d hij_^_e_gbx nmgdpbb =ZfbevlhgZ H ( p ) = p 2 2m ^ey qZklbpu k fZkkhc m Bkihevamy mkeh\b_ ^ey jZkij_^_e_gby (2.9.1 ihemqbfqbkehqZklbp
N = ( 2π= )
−r
G p 2 2m − µ G G 32 −r µ ∫ exp − kT drdp = ( 2π= ) V exp kT ( 2πmT ) (2.9.23)
bhdhgqZl_evghjZkij_^_e_gb_FZdk\_eeZ
(
D px , p y , pz
)
G p2 exp − = , 32 V ( 2πmkT ) 2mkT N
(2.9.24)
kghjfbjh\dhc
∫ D ( p x , p y , p z ) dp x dp y dp z = V . N
(2.9.25)
Nhlhgguc]ZaWe_dljhfZ]gblgh_baemq_gb_ij_^klZ\ey_lkh[hc b^_Zevguc nhlhgguc ]Za \ dhlhjhf nhlhgu g_ \aZbfh^_ckl\mxl f_`^m kh[hc b obfbq_kdbc ihl_gpbZe m^h\e_l\hjy_l mkeh\bx µ = 0 Wg_j]by nhlhgZ _klv H = cp = =ω ]^_ ω – kh[kl\_ggZy qZklhlZ baemq_gby \ ^Zg ghfh[t_f_ V Lh]^Zkh]eZkgh b bf__fqbkehnhlhgh\ N=
V ω2 d ω ∫ π 2c 3 e =ω kT − 1
(2.9.26)
bihegmxwg_j]bxnhlhggh]h]ZaZ E N = ∫ ρ ( ω) d ω ,
(2.9.27)
]^_ ρ ( ω) hij_^_ey_lky ih nhjfme_ IeZgdZ ^ey ki_dljZevgh]h jZkij_^_ e_gbywg_j]bb ρ (ω) =
V =ω2
π 2 c 3 e =ω kT − 1
(
)
.
(2.9.28)
Mihjy^hq_gghklvnhlhggh]h]ZaZIjhba\_^_fhp_gdmmihjy^h q_gghklb khklhygby we_dljhfZ]gblgh]h baemq_gby k jZkij_^_e_gb_f D
97
hlghkbl_evgh D0 >@HldjulZy kbkl_fZ b hdjm`_gb_ ij_^klZ\eyxl kh [hc bkoh^gh_ b \g_rg__ ihey baemq_gbc k qZklhlhc ω b l_fi_jZlmjZfb T b T0 khhl\_lkl\_ggh Lh]^Z i_j_oh^ f_`^m khklhygbyfb hibku\Z_lky jZkij_^_e_gbyfbIeZgdZ −1
−1 =ω =ω 1 D = exp − − 1 , , D0 = exp kT kT 0
(2.9.29)
V ω2 V ω2 , ∫ D π 2c 3 d ω = N ∫ D0 π 2c 3 d ω = N 0 .
(2.9.30)
Kh]eZkgh aZibr_fbgnhjfZpbxjZaebqby^ey;ha_-]ZaZ 1 + D ) V ω2 ( D I N ( D : D0 ) = k ∫ D ln − (1 + D ) ln dω = D0 (1 + D0 ) π2c 3 = − (H − H0 ) +
1 ( E − E0 ) . T0
(2.9.31)
AgZq_gby wgljhibb b wg_j]bb bkoh^gh]h q_jgh]h baemq_gby \ qZk lbqghfjZ\gh\_kbbgZoh^ylkyihnhjfmeZf H = H ( D ) = − k ∫ D ln D − (1 + D ) ln (1 + D ) E = ∫ ( hω) D
4 V ω2 d ω = ET −1 , (2.9.32) 2 3 3 π c
V ω2 d ω = σT 4V , π 2c 3
(2.9.33)
]^_ σ –ihklhyggZyKl_nZgZ @ G G G U r , k dkdrG N =∫ . G 3 =ω k ( 2π )
( ) ()
(2.9.38)
IhegZy wg_j]by b wgljhiby l\_j^h]h l_eZ ^Zxlky ke_^mxsbfb \u jZ`_gbyfb 99
G G G G dkdr , EN = ∫ U r , k 3 2 π ( )
( )
G U rG, k H N = k ∫ ln G =ω k
(2.9.39)
G G ( ) dkdr . 2 π ( ) ( ) 3
(2.9.40)
2.10IZjZf_ljbah\Zggu_d\Zglh\u_f_ju JZkkfhljbfiZjZf_ljbah\Zggu_f_ju\bamqZ_fuoklZlbklbdZo>ey q_]h\\_^_fiZjZf_lj ε jZg__ij_^klZ\e_gguc\jZ[hlZo>@bi_j_ ibr_fjZ\_gkl\Z – \\b^_ D1i 1 ± εD1i
=
D0i
D2i
1 ± εD0i 1 ± εD2i
,
(2.10.1)
]^_jZkij_^_e_gb_ D0i =
1 2 −τTi −α B ε
(2.10.2)
b iZjZf_lj ε = 0 ^ey klZlbklbdb FZdk\_eeZ–;hevpfZgZ AgZd iexk fbgmk ijb ε = 1 khhl\_lkl\m_l klZlbklbd_ N_jfb–>bjZdZ ;ha_– Wcgrl_cgZ @ – >ey klZlbklbd N_jfb–>bjZdZ b ;ha_–Wcgrl_cgZ kh]eZkgh (2.10.7)– bf__fiZjZf_ljbah\Zggu_d\Zglh\u_bgnhjfZpbhggu_ f_ju m 1 H ε ( p ) = −∑ pi log 2 pi B (1 ± εpi ) log 2 (1 ± εpi ) υi , ε i
106
(2.11.15)
m 1 ± εpi p 1 I ε ( p : u ) = ∑ pi log 2 i B (1 ± εpi ) log 2 υ , 1 ± εui i ui ε i
(2.11.16)
m p (1 ± εpi ) J ε ( p : u ) = ∑ ( pi − ui ) log 2 i υi , 1 u u ± ε ( ) i i i
(2.11.17)
m 1 H ε ( p : u ) = − ∑ pi log 2 ui B (1 ± εpi ) log 2 (1 ± εui ) υi . (2.11.18) ε i
@jZkkfZljb\Zxlkywgljhibbk υi = 1 b ε > 0 \ke_^mx s_f\b^_ m 1 H ε ( p ) = −∑ pi log 2 pi − (1 + εpi ) log 2 (1 + εpi ) + pi , (2.11.19) ε i m 1 H ε ( p ) = −∑ pi log 2 pi − (1 + εpi ) log 2 (1 + εpi ) − ε i 1 − (1 + ε ) log 2 (1 + ε ) , ε
(2.11.20)
m 1 1 H ε ( p ) = −∑ pi log 2 pi − 2 (1 + εpi ) log 2 (1 + εpi ) − , (2.11.21) ε ε i m 1 H ε ( p ) = −∑ pi log 2 pi − 2 (1 + εpi ) log 2 (1 + εpi ) − ε i 1 − 2 (1 + ε ) log 2 (1 + ε ) , ε
(2.11.22)
dhlhju_ lZd`_ y\eyxlky g_dhlhjufb dhf[bgZpbyfb ihklhygguo \_ ebqbg iZjZf_ljbah\Zgghc f_ju b f_ju ijb\_^_gghc \jZ[hl_>@ 1m 1 H ε ( p ) = − ∑ (1 + εpi ) log 2 (1 + εpi ) + 1 + log 2 (1 + ε ) . (2.11.23) ε i ε
107
=eZ\Z KL:LBKLBQ?KD:YFH>?EVJ?GVB Ijb\h^ylky hkgh\gu_ f_lh^u b b^_b l_hjbb bgnhjfZpbb ihkljh _gghc gZ klZlbklbq_kdhc fh^_eb J_gvb ey g_kemqZcghc \_ebqbgu bf_xs_c ihklhyggh_ agZq_gb_ C kijZ\_^eb\hjZ\_gkl\h
Aϕ ( C ) = C .
(3.1.3)
Ihkdhevdm ijb C = 1 ba \ul_dZ_l Aϕ (1) = 1 lh bf__f k\hckl\h ghjfbjh\Zgghklb\a\_r_ggh]hkj_^g_]hgZ_^bgbpm Wd\b\Ze_glgu_kj_^gb_ G_h[oh^bfuf b ^hklZlhqguf mkeh\b _fjZ\_gkl\Z Aϕ (T ) = Aχ (T )
(3.1.4)
^ey\k_oT b p y\ey_lkymkeh\b_ χ = αϕ + β ,
(3.1.5)
]^_ α b β –ihklhyggu_b α ≠ 0 Kj_^gb_knmgdpbyfb ϕ b χ gZau\Zxl kywd\b\Ze_glgufbkj_^gbfb H^ghjh^ghklv Imklv ϕ (T ) g_ij_ju\gZ \ hldjulhf bgl_j\Ze_
(0, ∞ ) bimklv Aϕ ( aT ) = aAϕ (T )
(3.1.6)
^ey \k_o iheh`bl_evguo T , p b a Hq_\b^gh qlh kijZ\_^eb\h dh]^Z ϕ = T q beb ϕ = log 2 T ey d\Zglh\hc \_ebqbgu T y\eyxs_cky wjfblh\uf hi_jZlhjhf bf__f\ujZ`_gb_ Aϕ (T ) = ϕ−1 Spϕ (T ) ⋅ρ , Spρ = 1 ,
(3.1.20)
111
]^_ ρ –hi_jZlhjkf_rZggh]hkhklhygbykemqZcgh]hh[t_dlZ @ ϕ (T ) :
Aϕ (T ) : m
E (T ) = ∑ Ti pi ,
T
(3.1.21)
i
1ε
m N ε (T ) = ∑ Ti ε pi i
Tε
(3.1.22)
m
∑ (log 2 T ) pi
N (T ) = 2 i
log 2 T 2 εT
sin T
,
(3.1.23)
1 m H ε (T ) = log 2 ∑ 2 εTi pi , ε i
(3.1.24)
m S (T ) = arcsin ∑ (sin Ti ) pi , i
(3.1.25)
g_dhlhju_badhlhjuojZkkfZljb\Zxlky\klZlbklbq_kdhcl_hjbbbgnhj fZpbb
Ihemghjfu JZkkfhljbffgh`_kl\h\k_okhklhygbckemqZcgh]hh[t_dlZhibku
\Z_fuojZkij_^_e_gb_f p = { p1 ,! , pm } bfgh`_kl\hkemqZcguo\_ebqbg T = {T1 ,! , Tm } @
112
m q ∑ Ti pi N q (T ) = i m ∑p i i
1q
.
(3.2.2)
?keb\_kZ pi m^h\e_l\hjyxlmkeh\bx\_jhylghklghcghjfbjh\db m
∑ pi = 1 ,
(3.2.3)
i
lh\ujZ`_gby b ijbfmlke_^mxsbc\b^ m
E (T ) = ∑ Ti pi ,
(3.2.4)
i
1q
m N q (T ) = ∑ Ti q pi i
.
(3.2.5)
Ijb pi = 1 ba bf__f kj_^gb_ Zjbnf_lbq_kdh_ b ]_hf_ljbq_ kdh_ E (T ) = N1 (T ) =
1 m ∑T , m i i
−1
1 m M (T ) = N −1 (T ) = ∑ Ti −1 . m i
(3.2.6)
0 ) nmgdpby y\
ey_lky h^ghjh^guf nmgdpbhgZehf gme_\hc kl_i_gb hlghkbl_evgh p lh _klv\uihegy_lkyk\hckl\hh^ghjh^ghklb Iheh`bl_evghklvbghjfbjh\Zgghklv>ey\k_ch[eZklbbaf_ g_gby qbkeZ q ∈ R nmgdpby y\ey_lky iheh`bl_evghc b g_m[u\Zxs_c ?keb q < 0
(N
q
(T ) = ∞ ) .
(q > 0)
b g_dhlhju_ Ti jZ\gu gmex lh N q (T ) = 0
>eyg_kemqZcghcihklhygghc\_ebqbgu C bf__fjZ\_gkl\h
113
m q ∑ C pi N q (C ) = i m ∑p i i
1q
=C,
(3.2.7)
badhlhjh]hke_^m_lk\hckl\hghjfbjh\ZgghklbnmgdpbbgZ_^bgbpm N q (1) = 1 .
(3.2.8)
Ihemghjfu N q (T ) . >ey agZq_gbc 1 ≤ q < ∞ qbkeh N q (T ) _klv ihemghjfZ\ehdZevgh\uimdehfijhkljZgkl\_ Lq ^eydhlhjh]hkijZ\_^ eb\ukhhlghr_gby 1) N q ( aT ) = a N q (T ) ,
(3.2.9)
2) N q (T1 + T2 ) ≤ N q (T1 ) + N q (T2 ) g_jZ\_gkl\hFbgdh\kdh]h
(3.2.10)
]^_ a ≠ 0 _klv ijhba\hevgh_ qbkeh Ijb 0 < q < 1 g_jZ\_gkl\h g_ \uihegy_lky G_\ujh`^_gghklv >ey ihemghjfu ^himklbfh N q (T ) = 0 ijb T ≠ 0 Wlbfk\hckl\hfihemghjfZhlebqZ_lkyhlghjfuijb q = 2 12
m N 2 (T ) = ∑ Ti 2 pi i
,
(3.2.11)
^eydhlhjhc N 2 (T ) = 0 ijb T = 0 . G_jZ\_gkl\h =_ev^_jZ Ijb 0 < q ≤ ∞ b 0 < q ≤ ∞ \uihegy_lky g_jZ\_gkl\h=_ev^_jZ N r (T1T2 ) ≤ N q (T1 ) N q′ (T2 ) ,
(3.2.12)
]^_^eyijhba\hevguokemqZcguo\_ebqbg T1 b T2 bf__f 1q
m N q (T1 ) = ∑ T1qi pi i
r m N r (T1T2 ) = ∑ (T1iT2i ) pi i
114
1 q′
m , N q′ (T2 ) = ∑ T2qi′ pi i
,
1r
.
(3.2.13)
Dhg_qgu_ qbkeZ q b q′ _klv lZd gZau\Z_fu_ khijy`_ggu_ ihdZaZ l_ebdhlhju_m^h\e_l\hjyxljZ\_gkl\m 1 1 1 + = , 0 0 ]^_ 0 < r < q lh_klv N q (T ) –\hajZklZxsZynmgdpby ijbm\_ebq_gbbagZq_gby q . Ijb r = 1 ba ihemqbf\Z`gh_g_jZ\_gkl\h E (T ) ≤ N q (T ) .
(3.2.18)
IhemghjfZ[hevr_kj_^g_]hagZq_gbykemqZcghc\_ebqbgu 0 ^ey dhlhjuo N q (T ) dhg_qgh nmgdpby ln N q (T ) y\ey_lky kljh]h \uimdehc hl 1 q >ey ex[uo 0 < a < 1 bf__l f_klhg_jZ\_gkl\h log N q (T ) ≤ a log 2 N r (T ) + (1 − a ) log 2 N ν (T ) ,
(3.2.20)
]^_ r > 0 , ν > 0 ZqbkeZ r , q b ν m^h\e_l\hjyxljZ\_gkl\m
115
1 a 1− a . = + q r ν
(3.2.21)
7. >bnn_j_gpbjm_fhklv nmgdpbc N q (T ) b log 2 N q (T ) Nmgdpbb N q (T ) b log 2 N q (T ) [_kdhg_qgh^bnn_j_gpbjm_fu\dZ`^hclhqd_bg q
m
l_j\ZeZ \ dhlhjhf hgb dhg_qgu Nmgdpby N q (T ) = ∑ Ti q pi bf__l i ij_^_e m
lim ∑ Ti q pi = 1 ,
(3.2.22)
q →0 i
dh]^Z q klj_fblkydbbf__l\wlhclhqd_i_j\mxijhba\h^gmx
(
)
m m d m q lim ∑ 2 q log 2 Ti pi = ln 2∑ (log 2 Ti ) pi . ∑ Ti pi = q→ 0+ 0 i dq i i
(3.2.23)
Ijbwlhf N q (T ) b ln N q (T ) bf_xlij_^_eu m
∑ (log 2 Ti ) pi
lim N q (T ) = N (T ) = 2 i
,
(3.2.24)
lim log 2 N q (T ) = ∑ ( log 2 Ti ) pi .
(3.2.25)
q →0
m
q →0
i
?keb T q b lnT bgl_]jbjm_fulh N (T ) ≤ N q (T ) ,
(3.2.26)
ijbq_f jZ\_gkl\h ^hklb]Z_lky ebrv \ lhf kemqZ_ dh]^Z T ihklhyggZ ihqlb\kx^m