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HAL Id: inria-00070792https://hal.inria.fr/inria-00070792
Submitted on 19 May 2006
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Microscopic Models for Chemical ThermodynamicsVadim A. Malyshev
To cite this version:Vadim A. Malyshev. Microscopic Models for Chemical Thermodynamics. [Research Report] RR-5200,INRIA. 2004, pp.22. inria-00070792
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ISS
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FR
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THÈME 1
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Microscopic Models for Chemical Thermodynamics
V. A. Malyshev
INRIA, France
N° 5200
May 2004
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Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)
Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30
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\]Ppj`_pms£7TtuPpjN\rT=v¡tj`igigT=\]oqj@vRsbvp`mec]jcgPpT^@tEc]j`vj`
X¢ 8 vª@TvpT=ig^`m b_pvRSUj`mT=t_pm^ni igT=^@tEc]j`vq\^nigT#^nopopigjabS^nc]j`vR\cgj£pvq^nige*igT=^`tc]j`vR\
PRTv\rj@ST¬igT=^@tEcg^`vNcg\^nigTXvª^n£R_pvRsp^`vRtT@¢O'PpT¡c]iu^nvR\r j`igS^c]j`v
j → j1\D^@ttj`SUoR^nvRTKs £ec]PRT¡TvpT=i]@eigT=sb\fcgi]£p_bcgj@v
(T, Kj) →(T1 = T + Kj − Kj1 , Kj1)
pT + Kj − Kj1 ≥ 0
¢O8 T + Kj − Kj1 < 0
cgPpTvªT^`\g\r_pSUT¬cgPR^cujj1(T ) = 0
¢O8 vjncgPpTij`iusp\=NT^`\g\r_RST¬T=vpTig`eDtj`vR\]Tigl^c]j`vT1 + Kj1 = T + Kj
¢ Tª^nm\rj.^`\g\r_RSTc]Pq^cUc]PpTl@Tmjtcfe
vj1
^cc]PpTSUj`SUTvNcjndc]PpTkf_pSUo £qTKtj@STK\_pvp j`igSmesp\rc]ig£R_bc]TKsj`vc]PpTcfj 0 spSUTvq\rj`vR^`m\]opPpT=i]Tjn iu^`sb_R\
T1(PpjTl`T=ij@vpTtj`_pms^`\g\]_pSUT*S_qtuP
mT=\g\=¢O'PpT\fcu^c]T \]oR^@tT'jn©c]PRTopi]jbtT=\g\
MN(t)\(c]PpTd\]TN_pT=vRtTj`©^`i]iu^eb\
Xi = ji, ~vi , i = 1, ..., N¢
O'PRTvnj
\cgPpTvN_RS£7TiUjni\r_RtuP%c]PR^nc
ji = jN = n1 + ... + nJ
¢ 5 jTl@TiKIsb_pTc]j.j`_pi^``igTT=ST=vNc^n£7j`_bcl`T=mjbtc]T=\Tt^`vcu^n`T*vR\rc]TK^`sc]PpT\fcu^cgT*jnX^¡SUj`mT=t_pmT^`\
Xi = ji, Ti¢
TvRjnc]TH
(u)N
cgPpT`T=vpTiu^cgj`i'jnc]PpTN0oR^`ircgtmT¬opigjbtTK\]\=bsbTRvRT=sj`v\]j`SUT^nopopigj`oRi]^cgTd _pvqtEc]j`v
\]oR^@tTUj`v(1, ..., J × R+)N
PptuP.Tmmvpjnci]c]TDsbjv.TaoRmtc]T=me@;£7T=t=^n_R\]Tc#\ N_pc]T\rcg^`vRsp^`igs®¢
7Rj`i¬Tap^nSUopmT`© j@i¬j`vpToR^ni]c]tmTTPR^l`TUtj@vNc]vN_Rj`_R\dc]SUTU|^nig`jlopi]jbtT=\g\M1(t)
c]P1c]PpT\rcg^nc]Td\]oR^@tT
(j, T )PptuP\3spTRvpTKs£e#cgPpTdiu^cgT=\
ujj1
@c]PpTT (t)
tj`SUoqj@vpTvNc3\I_RvpN_pT=meUsbTRvpT=s£eDvpc]^nmtj`vRspcgj@vR\^nvRs£eDcgPpT\rT @_RTvRtTXj`;cfeo7T¬c]iu^nvR\r j`igS^cgj@vR\¢
´³ b ± ° p` ´ 5 T=i]Tc]PRT|^nig`jlkf_pSUoopigjbtTK\]\\cgPpT j`mmjvpq¢«v%cgPpT*cgSUTvNc]T=i]l^nm
(t, t+dt)T=^@tuP¥9j`iusbTigT=sq¦oR^ni
(i, i′)jn©Sj@mTKt_pmT=\=`c]Poq^niu^nSUTcgTiu\
(j, T ), (j′, T ′)tj@i20
igT=\]o7j`vRsbvp@me@®PR^`\¬^ :tj`mm\rj`v :¬cgP.opi]j@£R^n£Rmcfe 1N
bjj′dt, bjj′ = bjj′ (T, T ′) = bj′j
¢UO'PpTv.^ccgPpT SUj`SUT=v@cjn®tj`mm\]j@vc]PpT oq^niu^nSUTcgTiu\0jn©c]PpTdoR^ni]c]tmT=\
i, i′^cIcgSUT
t+0£7T=tj`SUTdtj@i]igT=\]o7j`vRs 0
vp@me(j1, T1), (j
′
1, T′
1)¢IO'PpTsb\rc]ig£p_pc]j`vjn;c]PpTvpT=§oR^`ig^`SUTc]T=ig\\sbTqvpT=s£ec]PRT#tj`vqsbcgj@vR^nm
spTvR\]cgTK\P (b)(j1, T1, , j
′
1|(j, T ), (j′, T ′))cgPR^c¬^nigTsbTRvpT=sª j`i¬^nve
j, j′, T, T ′, j1, T1, j′
1
7cgPpTvT ′
1 = T + K + T ′ + K ′ − K1 − T1 − K ′
1
¢O;jXkf_R\rc] eDc]PR\ spTRvpc]j`vc \d^`\g\r_pSUTKsc]Pq^c
P (b)(j1, T1, j′
1|(j, T ), (j′, T ′)) = 0
SMSUT(VXW$Y$Z$Z
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F M ! '
T + K + T ′ + K ′ − K1 − T1 − K ′
1 < 0p^nvRs
P (b)(j1, T1, j′
1|(j, T ), (j′, T ′)) ≥ 0
T + K + T ′ + K ′ − K1 − T1 − K ′
1 ≥ 0¢Iªj`igTjl@TiKn j@id^nve
(j, T ), (j′, T ′)
∑
j1,j′1
∫
dT1P(b)(j1, T1, j
′
1|(j, T ), (j′, T ′)) = 1
T^`\g\r_pSUTdc]PpT¬\]^`SUT^n@i]T=TSUTvNc'^n£7j`_bcvpT l@Tmjbtc]T=\=@cgPR^c'\TK^`tuPDj`®c]PpT=S \sp\rc]ig£R_bc]TKsvRsbT=oqT=vRsbT=v@cgme^nvRs_pvp j`igSe*j`vc]PpT#tj`igi]TK\ro7j`vqsbvRTvRTig`eD\]opPpTigT`¢Tvpj`c]T
H(b)N
c]PRT#tj`igigT=\]oqj@vRsbvp@TvpT=ig^nc]j`iKj`vc]PpT#\g^nSUTX _pvRtc]j`vª\]oR^`tT`¢
p=1 b ± ° pn ´ O'PpTispTRvpc]j`v\U\]SUm^`i#cgjG\]mj£RvR^`i]eGigT=^`tc]j`voRi]jbtTK\]\=0£p_bccgPpToR^ni]c]tmT=\¬sbjvpjnctuPR^`vp`TcfeNo7T=\¬^`vRs|\]mjwT=vpTig`T=\=®\rjj`vRmeªigT=sb\rc]ig£p_pc]j`v¡j`3c]PRTiXvpTcgtT=vpTig`T=\3jbtt_Rig\=Nc]PR^nc\
T, T ′ → T1, T′
1
¢IO'PpT¬^n@i]T=TSUTvNctj`vRtTigvpvpl@TmjtcgTK\3PRj`msp\'^`\^n£7jl`T`¢ TIigcgT(c]PRT3tj@mm\]j`v¬opigj`£R^`£pmc]T=\ ^@\ 1
Nfjj′dt, fjj′ = fj′j
TI^`\g\r_pSUTI^`m\]j'c]PR^nc;v¬c]PpT3i]TK^`tEcgj@vj, j′ → j, j′
c]PpTUTvRTig`eªtj@vR\rT=i]l^nc]j`vT + T ′ = T1 + T ′
1
Ppj@msR\¢ T^`\g\r_RSTc]Pq^cfjj′
sbjvpjncspTo7TvRsj@v
T^nvqs
T ′¢ / \I j@i3cgPpTtj`vRspcgj@vR^nm7sb\rc]ig£p_bc]j`v
P (f)(T1|T, T ′)Tdmm7_R\]T cgPpTdj@vpT`
vNc]igjbsb_RtTKsDv ? z@ bPpT=i]TT1\cg^``Tv_pvp j`igSUmeDsb\rc]ig£p_pc]T=sj`vc]PRTXvNc]T=i]l^nm
[0, T + T ′]R^`vRsDj`
tj`_piu\]TT ′
1 = T + T ′ − T1¢ Tvpj`c]TXc]PpT#tj`igi]TK\ro7j`vqsbvR@TvpT=ig^nc]j`i
H(f)N
¢A_bcXvª9^@tEcK7^`vNe^`tcfeNo7T#SUjbsbTm0tj@_pms£qT_R\rTKsªIc¬\]^nc]\fRTK\cgPpT# j@mmjvRDtj`vqsbcgj@v1
TPq^`\'Tabo7j`vpT=vNc]^nm sb\fcgi]£p_bcgj@vvc]PpTm^nig`Tc]SUTmSUcd j`ivbRvpc]ToR^`ircgtmT\]eb\fcgTS¢
ª° bU= ° TSUjbsbT=m cd\]Sm^nigmecgjc]PpT¬9^@\fc £pvR^`i]eigT=^@tEc]j`vq\p^@\'ig^`vRsbj`S :tj@mm\]j`vR\:c]P.j@_bcg\]spTSUj@mTKt_pmT=\¬vG^`v1vbRvRcgTD£R^cgP PptuP.\¬@Tobc#^nctj`vq\fcu^nvNccgTSUo7Tiu^c]_Ri]T
β¢DO'PpT
T=vpTig`ejnT=^@tuP j`_bcu\rsbT¡SUj`mT=t_pmT¡\^`\g\r_RSTKsc]j£7T¡Tabo7j`vpT=vNc]^nmdc]P§oR^niu^nSUTcgTiβ¢&¡j`igT
Tap^`tEcgme@ j`i(TK^`tuPSUj@mTKt_pmTic]PpT=i]T'\;h0j@\g\rj@vopigjbtTK\]\c]PU\rj@ST'iu^cgT
h¢ TvRjnc]T
tik.k = 1, 2, ...
cg\kf_RSo SUj@ST=vNcg\= PpTv.c#_pvRsbT=i]@jT=\¬tj@mm\]j`vR\Xc]P.j`_bcu\rsbTSUj`mT=t_pmT=\=¢ / ccgPp\¬Sj@SUTvNcg\cgPpTDvpTcgtT=vpTig`e
Tj`cgPpTDSUj`mT=t_pmT
i\cgig^`vR\f j@i]SUTKs1^@\¬ j`mmj \=¢O'PRTDvpT vRTc]tT=vpTig`e
T ′^n c]Tic]iu^nvq\f j@i]S^cgj@vª\XtuPpjN\rT=v¡_pvp j`igSUmej`v¡c]PpTvNcgTigl^`m
[0, T + ξik]®PRTigT
ξik
^`i]T#¢ ¢ s®¢iu^nvqsbj`S l^nig^`£pmTK\DPR^lvpTao7j`vRTvNc]^nmXsb\rc]ig£p_bc]j`v§cgP sbT=vR\rcfe
β exp(−βx)¢ dT=vpjncgT|cgPpT
tj`igi]TK\ro7j`vqsbvRtj`vRsbc]j`vq^nm(Tabo7T=tcg^cgj@v.£NeP (β)(T1|T )
¢8 v.9^`tc=c]Pp\#opigjbtTK\]\^`Sj@_pvNcg\XcgjNvRsbT=oqT=vRsbT=v@c j@vpT0 oR^`ircgtmTXopigjtT=\g\rTK\bsbT=vpjncgT¬c]PpT#tj`igi]TK\ro7j`vqsbvR`T=vpTiu^cgj`i
H(β)N
¢
®µ ±1 ` djnc]Tc]Pq^c0£qj`c]P# j@i;_RvR^nige^nvRs\rmj £pvR^nige¬i]TK^`tEcgj@vR\®cgPpTvN_RS£7Tiu\nj
j`pcfeoqTjSUj`mT=t_pmT=\X^nigT#vpjnc¬tj@vR\rT=i]l@T=s£p_pccgPpT#cgjncg^`m;v_pS£qT=ij`3SUj`mT=t_pmTK\
N =∑
j nj
\¬tj@vR\]Tigl`T=s ¢«vGcgPpTDtj@v@cgig^`i]e@©9^`\rc#i]TK^`tc]j`vR\tj`vR\]Tigl`T
nj
¢*O'PpTopigjbtTK\]\MN(t)
\#sbTRvpT=sG£e¡cgPpTD\]_pSj`@TvpT=ig^nc]j@ig\
H(sf , sβ) = H(u)N + H
(b)N + sfH
(f)N + sβH
(β)Nj@v\]j`SUT^`opopigj`opig^nc]T( _RvRtEcgj@v\]oR^`tT3j`v
(1, ..., J×R+)NPpTigT
sf , sβ
^nigT\]j`SUTm^`i]@T3\]t=^nmvR9^@tEcgj`iu\PptuPUTl@TvNc]_R^`mmemmbcgTvRscgjvbqvpcfe@¢(O'Pp\0oRi]jbtTK\]\|£qT=mj@vp@\cgj#^¬tm^`\g\0jnqTmmR\rc]_RspTKstm^@\]\]t^nm©opigjbtTK\]\]T=\=¢
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!#"$ %&('()!( *$
O'PpT\fcu^c]T\roR^@tT j`©c]PpT _Rmm7opigjtT=\g\(\0c]PpTX\rT N_pTvRtTjn ^nigiu^e\Xi = ji, ~xi, ~vi , i = 1, ..., N
cgPpTv
nj
\cgPpTv_pS£7Ti(j`i\]_RtuPc]PR^nc
ji = jSUj`mT=t_pmT=\=
N = n1+ ...+nJ
¢O5 jTl`T=i=sb_RT'c]j¬j`_pi^``igTT=ST=vNc^n£7j`_pcl@TmjtcgTK\¢0TXt=^nv*cg^n@TvR\rc]TK^`scgPpT\fcu^cgTj`^Sj@mTKt_pmT¬^`\
Xi = ji, ~xi, Ti¢
7Rj`iITK^`tuPc]iu^kfTKtEc]j@i]eωj`
MN (t)TdspTRvpT'cgPpTdmjbt^`mR\roq^`tT speNvq^nSUt=\I^`\( j`mmj \=¢ 8 c3\ N_pc]T
\]SoRmT IcdspjNTK\vpj`cdtuPR^nvp@TXcfeNo7T=\=pT=vpTig`T=\=bl`T=mjbtc]T=\=b£p_bcj`vpmetjNj@igspvR^nc]TK\¢I8 I^ccgSUTtc]PpT
oq^ni]c]tmTXPq^`\'l`T=mjbtcfe~v(ω) = ~v(t, ω)
^nvRstjj`iusbvR^cgT~x(t, ω)
bcgPpTvª^c'cgSUTt + s
~x(t + s, ω) = ~x(t, ω) + ~v(ω)s¥rVK¦
_RvpmTK\]\c]PRTvpTacT=l`TvNc¥ kf_pSUoq¦tj@vRtT=i]vRvp¡c]Pp\oR^ni]c]tmT`(j` c]PpTc]iu^kfTKtEcgj`igeωjbtt_piu\#j`v%c]PpT
cgSUTvNc]Tigl^nm[t, t + s]
¢ T ^`\g\]_pSUTo7Tigjspt'£7j`_pvqsp^nigetj@vRsbc]j`vR\=nj@i=cgPR^cI\0cgPpT \]^`SUT`nT=m^@\fcgtigTqT=tEcgj@v* igj`S c]PpT£7j`_pvqsp^nige`¢
TsbT=vpjncgT¬c]PpT#igT=\]_pmc]vpUopigjtT=\g\XΛ,N (t)
¢8 cXsbTo7TvRsR\jnItj@_piu\rT^nm\]jj@vc]PpTvpc]^nm;tj`vRspL0cgj@vR\^nvqsj@v
sf , sβ
¢/ vpjncgPpTio7j@\g\r£pmcfej`_pmsD£7TXc]jtj@vR\rsbT=i\]jnc]igj`oRtsb¨©_R\]j@v
w(t)cgP ==Tigjspi] c ^nvRs*op_bc
~x(t + s, ω) = ~x(t, ω) + w(s)
vR\rc]TK^`s¬jn¥rVK¦¢M8 vR\rc]TK^`s¬j`bc]PpTl`Tmjbtcfe~v j`ij`vpT0oq^ni]c]tmT(Sj`c]j`vj@vpT3j`_pms¬PR^l@T(c]PpTtjl^nig^nvRtT
σ2 jnbcgPpTsp¨©_R\]j@v ¢(«vRT3\rPRj`_pmsj`vRmeXpa#\]j`SUTPpj1c]PRTsbT=oqT=vRsbT=vRtT T (σ2)¢0ªj@\rctj`vR\]spTiu^c]j`vq\
£7Tmj§^@sbSUccgPp\'@TvpT=ig^`m>==^nc]j`v ¢I5 jTl@Tic tj`_pms*£7TvpTKtT=\g\g^nigecgjcg^n@TvNc]j^`t=tj@_pvNc'sbT=To7TiigT=\]_pmcg\tj`vRtTigvpvRigT=^@tEc]j`v0 sb¨©_R\rj`vTN_R^nc]j`vR\=p\rT=T ? @
) +-!%'()# +-,8 vc]PpTvbqvpcgT*oR^`ircgtmT*mSUcUTsbj¡vpj`c`Tcv`T=vpTiu^nm¡^nig`jl¡oRi]jbtTK\]\=0£p_bcj`vpmeG\]j|t=^nmmTKsvRj`vpmvRT=^ni¡^`i]@jlopigjtT=\g\¢
´ ° 9p1 ° pn ´ dT=vpjncgTpt(j, T ), j = 1, ..., J
(cgPpTspTvR\]cgTK\jn cgPpTj@vpT0 oR^`ircgtmToRi]jbtTK\]\
M1^nc'c]ST
tJ∑
j=1
∫
pt(j, T )dT = 1
c]P1i]TK\ro7T=tccgj NT£7T=\]`_pTSUT=^`\]_pigTdT¢ 8 c]PpT=i]T^nigTj@vpme¡\]mj5_RvR^nigei]TK^`tEcgj@vR\=qcgPpTvGsb_pTc]j
cgPpTTvpT=i]@e tj`vq\rT=i]l^cgj@v ( j@iM1
cgPpTigTTab\rc¡¥ _pvqsbTiUSUms tj@vRsbc]j`vR\j`vujj1
¦cgPpTmSUcgvpspTvR\]cgTK\
π(j, T ) = limt→∞ pt(j, T )©sbT=oqT=vRsbvp*j@vªc]PpTvRcg^`mtj`vRsbc]j`vq\q£qTKt^n_q\rTc]Pp\dtuPR^`v
\\rc]igj`vp@meigT=sb_qt£pmT@¢TRvRT1vbRvpc]T1oR^ni]c]tmT|sbevR^`SUt=\
Xp0(t)
^`\Dc]PpTGtj`mmT=tEcgj@v¤jn#vRsbT=oqT=vRsbTvNcj`vpT0oR^`ircgtmTcgig^nkfT=tc]j`igT=\=qPpT=i]T
p0\Xc]PpTvRcg^`mIsb\rc]ig£p_bc]j`v| j`i¬T=^@tuP|oR^`ircgtmT@¢ 7pj@iXvbRvRcgToq^ni]c]tmTt=^`\]T
j@vpT'oR^ni]c]tmTc]iu^kfTKtEc]j@i]T=\^nigTsbTqvpT=sTap^@tEc]me#^`\0v^Rvpc]Tl`j@m_RST@nj`vRmecgPpTigT^nigT'vpj¬£qj@_pvRsp^`i]etj`vRspcgj@vR\p^`vRs*c]PpToR^`ircgtmT¬SUjl`TK\vc]PRTPpj`mT\roR^@tT@¢TRvRT¬c]PpT¬mSUc]vptj`vqtTvNcgig^nc]j`vR\
cj(t) = limΛ→∞ Λ−1 < nj(t >= pt(j)cbPpT=i]T
pt(j)\
cgPpTopigj`£R^`£pmcfeUc]PR^ncd^c'c]SUTt^SUj`mT=t_pmTPR^`\cfeo7T
j¢
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M ! '
ª° b ° 5 T=^ncDc]iu^nvq\f T=iD\^nm\rj^j`vpT0oR^`ircgtmTªoRi]jbtTK\]\*^nvRs v c]PpT1vpRvpc]T|oq^ni]c]tmTT=l`j@m_bcgj@v*\ sbTRvpT=s\]SUm^`i]meUc]jUcgPpTopigTlj`_R\j`vRT`¢O8 c mm ^`m\]jU£qT#¡^`i]@jl©¢
b ± ° Rn ´ O'PpTt^`\]TUcgP£pvR^nige¡igT=^@tEc]j`vq\\#Sj@i]Tvl`j@ml@T=s1£qTKt^n_q\rTj@vpTDt^`vpvpjncspTRvpTSUTK^nvRTmssbevR^nSUt\sbi]TKtEcgme* j`ivbRvpc]ToR^`ircgtmT#\]eb\fcgTS¢ T=l`T=ircgPpTmT=\g\T#sbTqvpT¬c]PpT\]j¡t^`mmT=sGvpj@vpmvpTK^ni|^nig`jlªopigjbtT=\g\=¢ 8 ctj`vR\]\rcg\#jn sbTcgTigSUvp\rc]tUTl`j@m_pc]j`v.jn'c]PRT*sbTvq\rc]T=\pt(j, T )
NsbTRvpTKsU£NeU\]j`SUT Aj`mc==SU^`vcfeNo7T T @_q^c]j`vN^`vRsUvpRvpc]Tdv_pS£qT=iIjn®vRsbT=oqT=vRsbTvNcvpPRj0SUj@`TvRTj`_q\¡^`i]@jl3kf_pSUoopigjtT=\g\rTK\® j@i;vNc]TigvR^`mspT`igTTK\ j`p i]T=T=sbj@SwjnbcgPpT'vRsplsb_R^nmoR^`ircgtmTK\¢O'PRT=\]T#cfjDTl@j`m_bc]j`vR\^nigTtj@vRtT=ircgT=sc]P1T=^`tuPªj`c]PpT=ivªcgPpT\]Tvq\rTT#Tabopm^`v|£qT=mj¢ 8 v¡c]PR\\]Tvq\rTSUT=^`vRTmssbevR^nSUt\vc]PpTvbRvRcgT#mSUcX£qTKtj@STK\'mjbt=^nm¢8 vªjncgPpTij`iusp\=b(T#^nigTj`£J0\]TiglvR#\rj@STmjbt=^nmRigT@j@v `TdvRTl`T=i\]TTd\]S_pmcg^`vpTj@_R\®kf_pSUoR\j`®cfj#oq^ni]c]tmT=\=`£p_bc'j`vRmekf_pSUoR\j`;j@vpT¬oR^ni]c]tmT 0Ic]PpToq^ni]c]tmTXc]PPptuPc:tj`mmsbTK\&:d\ ^p¢ \¢0vbRvpc]T=me*9^ni' igj`ScgPp\i]T=`j`v ¢
7Rj`i9^@\fc £pvR^`i]ei]TK^`tEcgj@vR\T#t=^nmm®c]PRT#vpRvpc]T#l@j`m_pSUT¬speNvq^nSUt=\'c]PRT¬c]STTl`j@m_pc]j`vjn0c]PpTspTvR\]cgTK\
pt(j, T )sbTRvpT=sU£eU\rj@SUT Aj@mc =S^nvcfeo7TdTN_R^nc]j`vª¥ T sbTRvpTcgPp\TN_R^nc]j`vU£7Tmjd¦E
cgj`@Tc]PRTiDc]P vbRvRcgT¡v_pS£7TijnvqsbTo7TvRspTvNc*cgSUT|vpPpj@SUj``T=vpTj@_R\¡^`i]@jlopigjtT=\g\rTK\ j`ivNc]T=i]vR^`m sbT@i]T=T=\jn; i]T=T=sbj@S jn0vRsblNsb_R^`m SUj`mT=t_pmTK\¢
/ \]\]_pSUT¬c]PR^nc T¬vpjpt(j, T )
¢IO'PpT=vop_bc
Pj(t; T1|T ) =∑
j′
∫
P(f)jj′ (T1|T, T ′)2fjj′pt(j
′, T ′)dT ′¥ z`¦
TUPR^l`T^\reb\rc]TS j`j`mSUj``j@i]jlTN_R^nc]j`vR\Xc]PGNvRjvpt(j, T )
spTRvpvpª¡^`i]@jlopigjbtTK\]\c]Pªsb\rc]ig£p_pc]j`vR\
pt(j, T )
∂pt(T1, j)
∂t=
∫
(Pj(t; T1|T )pt(j, T ) − Pj(t; T |T1)pt(j, T1))dT¥ +@¦
7Rj`i sb¨©TigTvNcSj@mTKt_pmT=\cgPpT=\]TopigjbtT=\g\]T=\'^nigTvRsbT=oqT=vRsbT=v@c£NesbTRvRcgj@v ¢/ cIcgPpT \g^nSUT'c]SUT'c]Pp\3\]eb\fcgTS&\0c]PRT TN_R^nc]j`vcgPi]TK\ro7T=tc(c]j
pt(j, T )¥ j`Aj`mc==S^nvcfeNo7TK¦
3T\]_p£R\rc]c]_bcgT¥z`¦dvNc]jG¥+N¦E¢ 8 cXmm0\]Tigl`T#_R\dc]jopigjl`T#Ta\rc]TvqtT^`vRsª_pvRN_pT=vpT=\g\dj`pt(j, T )
¢/ vRsc]PpT=v p_R\]vpUopi]j@£R^n£Rm\rc]tXvNcgTigopi]Tcg^nc]j`v Nc]jDspTc]T=i]SUvpTXc]PpTm^nig`Tc]SUT#£7TPR^lj`_piK¢
PpT=vª^nmm® j`_pi'cfeo7T=\'j`igT=^@tEc]j`vq\^`i]T¬opigT=\]TvNcc]PpTT N_R^cgj@v\
∂pt(j1, T1)
∂t=
=∑
j
∫
(P (t; j1, T1|j, T )pt(j, T ) − P (t; j, T |j1, T1)pt(j1, T1))dT
PRTigTP (t; j1, T1|j, T )
\XcgPpTD\r_RSjn j`_pi¬cgTigS\P (m), m = 1, 2, 3, 4
tj`igi]TK\ro7j`vqsbvRDc]j j@_piigT=^@tEcgj@vDcfeoqTK\RvNc]igjbsb_RtT=s^n£7jl`T`
P (1) = ujj1(T )δ(T + K − K1 − T1),
.0/ S1.32
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!#"$ %&('()!( *$
P (2) =∑
j′ ,j′1
∫
dT ′dT ′
12bjj′P(b)(j1, T1, j
′
1|(j, T ), (j′, T ′))
pt(j′, T ′)δ(T + K + T ′ + K ′ − K1 − T1 − K ′
1 − T ′
1),
P (3) = Pj(t; T1|T )δjj1 ,
P (4) = hδjj1P(β)(T1|T )
j@vR\rsbT=ivpj vbRvpc]T¬oR^`ircgtmT\reb\rc]TS vR3 PpT=i]TT=^@tuPoR^ni]c]tmTdPR^@\3vNc]T=i]vR^`mqoq^niu^nSUTcgTiu\
(j, v)¢ TvRjnc]T
McgPpT\]eb\fcgTS jn3^`mmopigj`£R^`£pmcfeSUT=^@\r_Ri]TK\ j`ic]PR\X\reb\rc]T=S c]P|c]PpT j`mmjvp
oRi]j@oqT=ircgTK\
tjj`iusbvR^cgT=\jn(cgPpT=\]Toq^ni]c]tmT=\X^nigTsb\fcgi]£p_bcgT=s|^@\ cgPpTPpj`SUj`@TvpT=j`_R\dh0j`\]\]j`v¡o7j`vNcqTmsjn0oR^ni]c]tmT=\'j@v
R3 c]Pª\rj@SUT¬spTvR\]cfe c
T=^@tuP|oR^`ircgtmTPR^`\¬^l`T=tc]j@ij`3oR^niu^nSUTc]Tiu\ j, Tsb\rc]ig£p_pc]T=s¥ vRsbT=oqT=vRsbTvNcgme¡j`5cg\¬tjNj@i20
sbvR^cgT#^nvRsjn;c]PpTj`c]PpT=ioR^ni]c]tmT=\u¦lN^U\rj@SUTXtj`SUSUj`vsb\rc]ig£p_bc]j`vp(j, T )
c]PpT#\g^nSUT j`i^nmm oR^`ircgtmTK\¢
j@vR\rsbT=iI^\rT @_RTvRtTj`7oRi]jbtTK\]\]T=\XΛ,N (t)
c]PN = N(Λ), Λ → ∞
\]jXc]PR^nc N(Λ)Λ → c > 0
¢O'PRTv^c'c]SUT
0cgPpTsb\rc]ig£p_bc]j`vj`
XΛ,N(0)tj@vl`Tig`TK\(cgjU\]j`SUT¬sb\fcgi]£p_bcgj@v£7Tmj`vp@vRcgjc]PpT
\]TcM¢
°´ ° * )X !( $ ) 1()( M
X) * ))X >X!( > ('()<! * ((
M ) (!($!) "!O ))X *! 1 ('()!( *$$# )< %
)<( ()X% !& !' ( X) $ %&('()!( * Xc(t)
2* X < XΛ,N (t)
%!)< 1()(+
M!& ! , 9 % %)
t
hIigjjnf¢'OjoRi]jl@TTa\rc]TvqtTjn0cgPpTc]PRTigSjbsbevR^`SUt¬mSUcdTopi]jbtTT=svcfjD\rc]T=oR\¢«vc]PpTqig\rcd\fcgToªTsbjvpj`ct^`i]T#^`£qj@_bcdtjj@igsbvR^nc]T=\=¢8 (cgPpTigT#^nigTj`vRme_pvR^nige*igT=^@tEcgj@vR\pc]PpT=i]T#^`i]TvpjoRi]j@£pmT=SU0'j@vpTU\rPpj@_pms¡vpj`cXo7Ti] j`igS c]PpTcgPpTigSUjspeNvq^nSUtmSUcK¢A_bcX j`i¬£pvR^nigeªigT=^`tc]j`vR\dj`vpT\]Ppj@_pms ¢
djnc]T|c]PR^ncT|t^`v¤i]T j`igS_pm^cgTª£pvR^nigetj`mm\]j@vR\v§^.Rvpc]T1l`j@m_pSUT|^@\ j@mmj \¢ / \*^`vTap^nSUopmTTcg^``T9^`\rc#£pvR^nige|tj@mm\rj`vR\=¢ / ve|oq^ni]c]tmT
ijncfeNo7T
j_RvRsbTig`jTK\%:tj@mm\]j`v|c]P
Cq«X%- Sj@mTKt_pmT¬jnOCR«X%-cfeoqTj′:
dt∑
j′
(fjj′ + fj′j)nj′(t)
N
/ c]T=ic]PpTvbRvRcgTl`j`m_pSUT¬mSUc T¬`Tc¥ z`¦ 0u¥ +@¦c]PR^nc \cgPpTopigj`£R^`£pmcfeDjn0tj`mm\]j@v\
dt∑
j′
(fjj′ + fj′j)cj′(t)
c
¥ N¦
O'PRT¬\g^nSUTt^`v£qTsbj@vpT¬ j`i'cgPpTTvpT=i]@eDsb\fcgi]£p_bcgj@v ¢
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VK M ! '
O;jDopigjl`TXc]PRTTab\fcgTvRtTjn(vbRvpc]ToR^ni]c]tmT#sbevR^nSUt\dj`vpT#\]Ppj`_Rmsopigjl`TXRiu\rc c]PpTTab\rc]T=vRtTj`\rj@m_pc]j`vR\jn cgPpT Aj@mc =S^nv*TN_R^cgj@v ^nvRsDc]PpT=vDc]PRTTa\rc]TvqtTdj` c]PRTX¡^`i]@jlopigjbtTK\]\I j`ic]PpTvRsblsp_R^nm®oR^`ircgtmT@¢IO'Pp\'\ N_pc]T\rcg^`vRsp^nius®¢° b !& ) ()<6 J1 )
cj′(t)!( ( 1!J)<) !& () G ! 1
) X & )R 'DG+ j! 1%"J 3($ %9 ) (+ 1& ! '()<!( *
«vc]PRTvpTacd\rc]TobTvpT=T=s*c]jopigjl`Tc]Pq^cc]PRTPpj`SUj`@TvpT=j`_R\h0j`\g\rj@vsb\fcgi]£p_bcgj@vvª\]oR^`tT\vl^`i]^nvNcK¢«vpToqjN\]\]£pmcfe|\¬c]jcu^n`TUqvpcgTDt_p£7T
Λ^`vRs.tj@vR\rsbT=ij`vpToR^`ircgtmT@ cgPpi]jv.^c
cgSUT0c]P_pvp j`igS sb\fcgi]£p_bcgj@vj@vNc]j
Λ¢ / \]\]_pSUTXc]PR^ncc]PpT¬oR^`ircgtmT¬SUjl`TK\vc]Pp\ t_p£qT¬c]P
\]j`SUT\]o7TT=sv(t)
0^nig£pc]iu^nige*iu^nvRsbj@S _RvRtEcgj@v|jn(cgSUT@¢¬O'PRTj`vRmeªtj@vRsbc]j`v¡\dc]PR^ncv(t)
spjNTK\vRjncsbT=oqT=vRsGj`vGc]PpTtjj`iusbvR^cgTj`cgPpTDoR^`ircgtmT@¢ / \g\r_pSUTD^`m\]joqT=i]jbsbtU£7j`_RvRsp^nige|tj`vRsbc]j`vq\cgPR^cX\XT=m^@\fcgti]T RT=tc]j`vªj`v¡c]PpT£qj@_pvRsp^`i]e@¢XO'PRTv¡cX\¬tmT=^`i c]Pq^cXc]PpT_pvp j`igS sb\rc]ig£p_bc]j`v|\vl^nig^`v@cK¢O^nvp
NvRsbTo7TvqsbTvNcoR^ni]c]tmT=\'^`vRsDo7Ti] j`igSUvpc]PpT=i]SUjbsbevR^`StmSUcT@Tc'c]PpT
^@\]\]Ti]c]j`v¢/ vpjncgPpTiXoqjN\]\]£pmcfe\cgjD_R\]T jNj@£J0 dj@£pi]_q\rPpvc]PRTj`igTS j`vªcgPpTvl^nig^`vRtTj`cgPpTh0j@\g\rj@v
o7j`vNc RT=ms®nspigT=tc]med j@ivbRvpc]Tl`j`m_pSUT`¢ICTT3`TvRTiu^nmnigT=\]_pmcg\ j`bc]Pp\;NvRs¬v ? V@ ¢;O'PpT^`\g\r_pSUobcgj@vcgPR^ccgPpTl`T=mjbtcgTK\^`i]T¬_pvR j@i]SUme*sb\fcgi]£p_bcgT=sj`vc]PRT¬\]opPpT=i]T¬\vpj`c'TK\]\]TvNcg^`m ¢
$w'0!)+"! +.;#+ # !%. "#"#
8 vj`_RicgPpT 3X£p£q\\fcu^c]Tmm'£7Tc]PpTª\]e\rc]T=S jnvRspTo7TvRsbT=vNcDoR^ni]c]tmT=\¥ sbT=^`m@^@\g¦¢ 5 T=i]TT@l@T^\]Tm9tj@vNcg^nvpTKs*opigT=\]TvNcu^c]j`v.¥ RavpUcgPpTvpjncu^cgj@vT¬_R\]TPpTigTK¦j`S^nv j`igS_pm^`\ j@ic]PpTtm^@\]\]t^nmsbT=^`m@^`\]T=\0^`vRsSUac]_pigT=\=nc]PUj`vpT'SUoqj@ircu^nvNcIsb¨©TigTvqtT c]PpT9^@\fcIsbT=`igTT=\j`R igTT=spj`S^`i]TX@^`_R\g\r^nv^nvRs\]mj§spT`igTTK\jn i]T=T=sbj@S ^nigTtj`vq\fcu^nvNcg\
Kj
psbT=oqT=vRsbvpj`vRmeDj@vj¢
/ TmmL0 vpjv*Tap^nSUopmTXj`vNc]T=i]vR^`m®TvpT=i]@eU _pvRtc]j`vR^`m \cgPpT N_R^`spig^nc]t 5^nSUmcgj`vp^nv
Ij =
dj−3∑
k=1
mj,kw2j,k
2
PRTigTmj,k, k = 1, ..., dj−3
^nigT0\]j`SUTItjNTKDtTvNcg\ ^`vRsc]PpT3l`T=tc]j@iyj = wj,k, k = 1, ..., dj − 3 ∈
Rdj−3 ¢*O'PpT=v dj − 3\c]PpTv_pS£qT=i#jn'vNcgTigvR^nmIsbT=`igTTK\¬jn3 igTTKsbj`Sjnc]PpTSUj@mTKt_pmTjncfeoqT
jdj
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O'P_R\TK^`tuPSj@mTKt_pmTjn;cfeoqTjPq^`\c]PRTTvpT=i]@e
Ej = Tj + Ij + Kj
TXtj`vR\]sbTi^¬Rvpc]TXv_pS£7Tinj
jn oR^`ircgtmTK\Ijn®cfeo7T=\j = 1, ..., J
v*^qvpcgTl@j`m_pSUTΛ¢O7Rj`i
cgPpTdsbTK^nm7@^`\3jn cgPpTjcfeo7TdoR^`ircgtmTK\(cgPpT@ig^`vRsUoR^`ircgcgj@v _pvRtc]j`vDj` c]PpT 3X£R£R\'sb\rc]ig£p_bc]j`v*\
Θ(j, β) =
∞∑
nj=0
1
nj !
(
nj∏
i=1
∫
Λ
∫
R3
∫
Ij
d~xj,id~vj,idyj.i
)
expβ(µjnj−
nj∑
i=1
(mjv
2j,i
2+Ij(yj,i))−Kj) =
.0/ S1.32
![Page 14: Microscopic Models for Chemical Thermodynamics · HAL Id: inria-00070792 Submitted on 19 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination](https://reader035.vdocument.in/reader035/viewer/2022081402/5f82df41a1f4ae3bf0511bab/html5/thumbnails/14.jpg)
!#"$ %&('()!( *$ V@V
=∞∑
nj=0
1
nj !Λnj β−
dj2
nj Bnj
j exp β(µj − Kj)nj = exp(Λβ−
dj2
j Bj exp βj µj)
PRTigT
Bj = (2π
mj
)3
2
dj−3∏
k=1
(2π
mj,k
)1
2 , µj = µj − Kj
3XT=vpTiu^nm®SUac]_pigTsb\rc]ig£p_bc]j`vj`Jcfeo7T=\'\ sbTRvpT=s£ecgPpT¬ j`mmjvpoR^ni]c]c]j`v _pvRtc]j`v
Θ =
J∏
j=1
Θ(j, β) = exp(Λ∑
j
λj expβµj), λj = β−dj2 Bj
TqvpT¬c]PpT@ig^`vRscgPpTigSUjbsbevR^nSUto7jncgTvNc]^nm
Ω = ΩΛ = −β−1 ln Θ = −β−1Λ∑
j
λj expβµj
O'PRT|mSUc]vp\]oR^`tT|sb\fcgi]£p_bcgj@v§j`Xcfeo7TjoR^`ircgtmTK\D\DcgPpT|h0j@\g\rj@v sb\fcgi]£p_bcgj@v cgP§iu^cgT
¥ tj@vRtT=v@cgig^nc]j`vq¦cj
R^`vRs
cj =< nj >Λ
Λ= β−1 ∂ ln Θ
∂µj
= λj exp βµj = exp(βµj − βµj,0 − βKj)
h3_bcc = c1 + ... + cJ
¢(O'PpTv
µj = β−1 ln(< nj >
Λλ−1
j ) = µj,0 + β−1 ln cj + Kj ,¥ y`¦
PRTigTµj,0 = −β−1 ln λj = −β−1(−
dj
2ln β + ln Bj)
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cj = 1¢
O'PpT¬vNc]TigvR^`m®TvpT=i]@evc]PpT=i]SUjbsbevR^`St\\sbTqvpT=s^`\c]PRTXSUTK^nvjncgPpT\r_RS jnTvpT=i]@TK\j`^`mmboq^ni]c]tmT=\=¢O'PpTtj`vRsbc]j`vq^nmbSUT=^`vT=vpTig`eD¥9`l`T=vcfeo7T
j¦oq^ni]c]tmT\ ¥ c]PpTm^%j`7T @_RoR^`ircgcgj@v
j`;T=vpTig`ep¦
< Ej >=dj
2β−1 + Kj
^`vRsU =
∑
j
< nj > (dj
2β−1 + Kj)
O'PRTXoRi]TK\]\]_pigTX\sbTRvpT=s^@\
P = −∂Ω
∂Λ= Λ−1β−1
∑
j
< nj >= β−1∑
j
cj =∑
j
pj
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Vz M ! '
PRTigTpj = β−1cj
^nigTc]PRToR^`ircg^`m;oRi]TK\]\]_pigT=\=¢ 8 vªcgPpTcgPpTigSUjbsbevR^nSUt#mSUccgPp\X\dT N_pl^`mT=v@ccgjc]PpT#sbTRvpc]j`v
βP = limV →∞
1
Λln Θ
O'PRTXTmm10vpjvTN_R^nc]j`v*j`0\rcg^nc]TX j`mmj \
PΛ = β−1∑
j
< nj >¥ n¦
7Rj`ij@vpTXcfeNo7TjcgPpTTvNc]igj`oe\ sbTRvpT=s^@\
Sj = −∂Ωj
∂(β−1)= Λλj exp(βµj)(
dj
2+ 1 − βµj) =< nj > (
dj
2+ 1 + βKj − βµj)
¥ Cb^`tu_pi0 OTc]igjspTd j@i]S_pm^N¦E¢O7pj@i'c]PpTSUac]_pigT¬c \c]PpT#\]_pS jnc]PpTK\rT
S =∑
j
Sj = Λ∑
j
λj exp(βµj)(dj
2+ 1 + βKj − βµj) =
∑
j
< nj > (dj
2+ 1 + βKj − βµj)
O;j`@TcgPpTi'c]Pc]PpTvNc]T=i]vR^`m®TvpT=i]@eUc]PpigTT¬jncgPpTiSUoqj@ircu^nvNc'c]PpT=i]SUjbsbevR^`StXoqj`c]T=v@cg^`m\d^nigT
T=v@cgPR^nmoe
H = U +PΛ =∑
j
(< nj > (dj
2β−1 +Kj)+β−1 < nj >) = β−1
∑
j
< nj > (dj
2+1+βKj),
¥ ¦3X£p£R\ i]T=TXT=vpTig`e
G = H − β−1S = β−1∑
j
< nj > (dj
2+ 1 + βKj − (
dj
2+ 1 + βKj − βµj)) =
∑
j
µj < nj >,
^`vRsD5 TmSUPpj`mc=¬ igTT¬T=vpTig`eF = U − β−1S
T¡t^`vsbTRvpTª^nm\]j|cgPpTªsbTvR\]c]T=\j`dc]PRTTac]Tvq\rl`T ¥ cgPR^c\^`\]eSopc]jncgt=^nmme mvpT=^`ivΛ¦
cgPpTigSUjbsbevR^nSUt#l^nig^`£pmT=\v|c]PpTc]PpT=i]SUjbsbevR^`St#mSc=¢ 7Rj`i¬Tap^`SoRmTTsbTRvRTcgPpTUmSUcgvp3X£p£R\ i]T=TXT=vpTig`eU j@i_pvpcdl`j@m_RST¬^`\
g = limΛ→∞
G
Λ=∑
µjcj
^`vRs*c]PpTT=v@cgi]j@oeDsbTvR\]cfe
s =∑
j
cj(dj
2+ 1 + βKj − β(µj,0 + β−1 ln cj + Kj)) =
∑
j
cj(− ln cj +dj
2+ 1 − βµj,0)
.0/ S1.32
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!#"$ %&('()!( *$ V&+
$w'0! )+"!
7Rj`i@l@Tv|iu^cgTUoR^niu^nSUTcgTiu\u, b, f, h
cgPpTvNcgTigvR^nmIsbT=`igTTK\j`3 igTTKsbj`S j`cgPpToR^`ircgtmTK\^nigTvJ0spTo7TvRsbT=vNc^nvqs spTvNc]t^`mmeGsp\rc]ig£R_bc]TKs ig^`vRsbj@Sl^nig^n£pmT=\=¢8 vjncgPpTij@igsR\;c]PpTsb\fcgi]£p_bcgj@v£7Tmj`vR@\Xc]j
M¢5 jTl`T=i=7cgPpTvpTc]tUTvpT=i]@TK\XSU^e|PR^l`TvpjncTao7j`vRTvNc]^nmIsb\rc]ig£p_pc]j`vR\=¢ T
mm® j`iutTc]PpTvpTcgt¬T=vpTig`T=\cgj£qTKtj`SUT¬Tabo7j`vpT=vNc]^nm®_R\]vRcgPpTmSUcsf → ∞
¢Tvpj`c]T3X£p£R\\rcg^cgTUjnIcgPpT
j0 cfeo7TsbT=^`m0@^`\¬^@\
Gj = Gj(β, µj)¢ TRvRT
M0 ⊂ Mc]PpT\]Tc
j`X^`mmSUT=^@\r_Ri]TK\×jGj(β, µj)
j@iU^`veβ, µ1, ..., µJ
^`vRsM0,β
0#cu\D\]_p£R\]TccgP%RaTKsβ¢ 8 v
oRPNeb\]t=^nm ^`vRs £pj`mj`@t=^nm®£7jj`b\'j`vvpj@vJ0T N_pm£pi]_pS c]PpT=i]SUjbsbevR^nSUt\=bc]PRTigT#^nigT¬\]j`SUT¬`T=vpTiu^nm\rcg^nc]T=ST=vNcg\=d\rT=T¡ j@iTab^`SUopmT ? V @ dPptuP Ppj@ms j`i*S^`vNe¤tj`vRti]Tc]T|Tap^nSUopmT=\= v§oR^ni]c]t_pm^nicgPpTemmdPRj`msv j`_pi*SjbsbT=m ¢ 7iu\fcgme@3cgPpTigTª\D^ \r_R£pS^nvp j`msv cgPpT¡\roq^`tTªj`XoRi]j@£R^n£pmcfeSUTK^`\]_pi]TK\Ij@vc]PpT\rcg^nc]T\]oR^@tT`sbTRvpT=s*£eD^Rvpc]TXv_pS£7Ti'j` SU^@tigj`oR^`ig^`SUTc]T=ig\=`^`vRsDSUj`igTjl@TiKcgPp\\r_p£RSU^`vp j`msD\vl^nig^`v@c'c]Pi]TK\ro7T=tccgjcgPpTX _pmm speNvq^nSUt=\¢ CTKtj@vRsbme`bT=^@tuP*o7j`vNcj`c]PR\S^`vp j@ms\0^opigjsp_RtEc(jn
kvRsbT=oqT=vRsbTvNc(SUT=^@\r_pigT=\=¢ 8 vj`_pi0t^@\rTT=^@tuP#oqj@vNc
ν ∈ M0\^opigjbsb_RtEc
ν = ν1 × ...× νJ
^`vRscgPpTXoqj@vNcg\j`M0
^nigTdvj`vpT0 cgj0 j`vpT c]Pc]PRTl@T=tEcgj`iM = (β, µ1, ..., µJ )j`oR^`ig^`STc]T=ig\=¢ djncgTdcgPR^c ^oqj@vNc'jn
M0\'^nm\]j_pvpN_pTme*sbTRvpTKs*£eUc]PRTl@T=tEcgj`i
(β, c1, ..., cJ )¢
O'PRTcgPpigs`T=vpTiu^nm®\rcg^nc]TSUT=v@cdtj`vRtTigvR\'sb¨7T=i]T=v@ccgSUT#\gt^`mTK\Nc]Pq^c T¬sp\gt_R\g\]T=s^n£7jl`T`¢ °´ ° " * ) ( $ ()
Cc(t) = limsf→∞
Xc(t), Oc,β(t) = limsh→∞
Cc(t)
IB!)X'" t ( D !()R (1
M0 ) (!($!) !O $ X+ X9<&$
Cc(t)(9!)X' !& $
u, b, h%"9 %!)X (>
M0,β
! ) !$!() !O B $ X Oc,β(t)
hIigjjnf¢%O'PRTTa\rc]TvqtTjnd£7jncgP%mSUcg\U'^`\v9^`tcopigjl`TKs.v ? z&@¢O'PpTj@vpmesb¨©TigTvqtT\cgPR^cc]PpT\ro7TTKsp\¥9ig^nc]TK\g¦
fjj′S^e£qTsb¨©TigTvNc j@isb¨©TigTvNcdcfeo7T=\=¢ / \v ? z@0j`vRTt^`v¡tj`vq\rsbTi
N0oR^`ircgtmT#speNvq^nSUt=\^`vRsopigjl`T¬cgPR^cc]PpTopigjtT=\g\PR^@\d^D_RvpN_pT_pvp j`igS sb\rc]ig£p_bc]j`vRcgPR^cX\
NoR^ni]c]tmT=\^nigTc]PpigjvD_pvR j@i]SUmej@v
[0, E]PpTigT
E =∑N
i=1 Ti
\tj@vR\rT=i]l@T=s®¢M8 c j@mmj \(cgPR^cv
N → ∞mSUc
Ti
mmPR^l`TXTabo7j`vpT=v@cg^`m sb\fcgi]£p_bcgj@vβ exp(−βx)
j`i \]j`SUTβ > 0
¢O'P_R\=0v cgPpT*oRi]jbtTK\]\
Cc(t)cgPpT*vpTcgt*TvpT=i]@TK\#PR^l@TD¡^abTmmsb\fcgi]£p_bcgj@v^cU^nve¡cgSUT
SUj@ST=vNc=¢B7pj`i#c]PpTopi]jbtT=\g\Oc,β(t)
SUj`igTjl`T=i= ^nc^nve|c]STtcgPpTcgTSUoqT=ig^nc]_pigTD\TN_R^`mIcgj
β
cgPR^c\dc]PpT=i]T\dPpT=^ncdTaptuPR^`vp`Tc]PªcgPpTT=vlNigj`vpSUTvNc¥9j`o7Tv|\]eb\fcgTSD¦E¢ O'Pp\X^nm\rjU j@mmj \' igj`S? z&@ ¢O'PpTi]TK\r_RmcgvpopigjtT=\g\
Cc(t)j`v
M0t=^nv#£7TsbTRvpTKs®_R\]vpdc]PpTTl@j`m_bc]j`vj`
cj(t)^`vRs¬ j@i]S_pm^
¥y`¦^nm\]j£e*spTc]T=i]SUvp\fcgt¬T=l`j`m_bcgj@vj`c]PRTl`T=tc]j@i
M(t) = (β(t), µ1(t), ..., µJ (t))
O'PRTigT\^_pvpN_pTRaTKs¡^c]c]iu^`tc]vp*o7j`vNcM(∞)
j`ic]Pp\¬\reb\rc]TS¢O'Pp\ j@mmj \d i]j@S tj`vl`T=i20@TvRtT#jnj`vpT0oR^`ircgtmT#|^nig`jl*oRi]jbtTK\]\
Mt
c]j*c]PpTU\fcu^cgj@vR^nigesp\rc]ig£R_bc]j`v ©Icu\Xigi]TKsb_Rt£pmcfe\^`\g\]_pSUT=s®¢
NTc _R\S^``T¬Riu\fc \]j`SUTigTS^nigb\^`£qj@_bcdtj@vR\]Tigl`T=sN_R^nvNc]c]T=\=¢ djnc]T¬cgPR^cN =
∑
j < nj >¥ j`iRvpc]TΛ¦^nvqs ∑
j cj
¥ j`iUvpqvpcgTl@j`m_pSUTK¦^nigTtj@vR\]Tigl`T=s ¢1O'PpT=v igj`S c]PpTTN_R^nc]j`vj`
SMSUT(VXW$Y$Z$Z
![Page 17: Microscopic Models for Chemical Thermodynamics · HAL Id: inria-00070792 Submitted on 19 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination](https://reader035.vdocument.in/reader035/viewer/2022081402/5f82df41a1f4ae3bf0511bab/html5/thumbnails/17.jpg)
V= M ! '
\rcg^nc]T¥ n¦Ic j`mmj \(c]Pq^cP\'tj@vR\]Tigl`T=s¥ j`ipabT=s
β¦ENc]PpT¬\g^nSUT j@iIc]PRT@ig^`vRsUoqj`c]T=v@cg^`m ¢(O'P_R\
vj`_pidSjbsbT=mN, P, Λ
^nigT#tj@vR\]Tigl`T=s ¢ jncgT`bcgPR^c_pvRsbT=ic]PpTK\rT#tj`vRspcgj@vR\'T=^@tuP\]_p£pS^`vp j@msj`M0,β
sbTqvpT=s£ec]PRTTN_R^cgj@v∑
j
λj exp βµj =∑
j
cj = c
\^nm\]jvl^nig^`v@cK¢
$w'0!)+"! +!# " 0"#.;#+
7R_pi]c]PpT=ij`v TDtj@vR\]sbT=i#j`vRme1c]PpTopigjtT=\g\Oc,β(t)
¢¡O'PpT=v%^nmmIcgPpTigSUjspeNvq^nSUtUo7jncgTvNc]^nm\^`i]T _RvRtEcgj@vR\¡¥ j@ipabT=s
K1, ..., KJ
¦j`vM0β
j`iDTap^nSUopmTcgPpT¡T=v@cgPR^nmoeH'j`iUcgPpT 3X£p£R\U igTT
T=vpTig`eG¢
ª° K9 ³ j@vR\rsbT=icfjDsp¨©TigTvNcdopigjbtTK\]\]T=\µ
(1)j (t)
^nvRsµ
(2)j (t), j = 1, ..., J
qj`vM0,β
R j`iTap^nSUopmTXc]Pªsb¨©TigTvNc i]TK^`tc]j`v*iu^cgT=\=¢ / \g\r_pSUT^nm\]jcgPR^c j@i \rj@SUT
T > 0
µ(1)j (0) = µ
(2)j (0), µ
(1)j (T ) = µ
(2)j (T ), j = 1, ..., J
cgPR^cX\dc]PRT=\]Tcfo¡oRi]jbtTK\]\]T=\dPR^l`T#c]PpTU\]^`SUT#vRcg^`m0^`vRsRvq^nmoqj@vNcg\=¢¬O'PpTvªcgPpT%5 T=\g\m^w\g^e\cgPR^ccgPpT#sb¨7T=i]T=vRtTK\'£qTcfT=Tvvpcg^`m^nvqs*qvR^nm T=vNc]PR^`mopT=\^`i]TXcgPpT#\]^`SUT j@i £7jnc]PopigjbtTK\]\]T=\=¢I8 v9^@tEcKnc]PR\3m^Ppj`msp\I^`_bc]j@SU^nc]t^`mme#vj`_pi(SUjbsbTm`£7T=t=^n_R\]T£7jnc]PoRi]jbtTK\]\]T=\(^`i]TsbTK\]ti]£7T=s£e#cfjoq^c]Pq\'j`v
M0,β
c]PcgPpT#\]^`SUTXvpc]^nm^`vRsRvR^`m®oqj@vNcg\=R^`vRsc]PpTT=v@cgPR^nmoeD\^ _pvRtEcgj@vj`vc]PpTvl^nig^`v@c'S^nvp j`ms
M0,β
¢O'PpT\]SUopmT=\rc tm^`\g\rqt=^c]j`v*jn;igT=^@tEcgj@vR\3\'v*cgTigSU\'j` c]PRTTvNc]PR^`moe
H¢I8
∆H = H(∞)−H(0) < 0
cgPpTv|c]PpTi]TK^`tc]j`vª\t=^nmmTKsªTabjncgPpTigSUt`pcgPpTPRT=^cQ\ `jTK\c]j*c]PpTTvli]j@vpSUTvNc=q
∆H > 0cgPpTi]TK^`tEcgj@vª\TvqsbjncgPpTigSUt^nvRscgPpTPpT=^nc\ cu^n`T=vª igj`S c]PpTTvligj`vRST=vNc=¢O'Pq^c\
∆H = Q¢
9; ´ µ ´ T^`\g\r_pSUT¬ _pi]c]PRTi j@vc]PR^ncc]PpT=i]T^`i]T¬vpjD\]mj £pvR^`i]eDigT=^@tEcgj@vR\SUj@i]T=jl`TiTtj`vR\]sbTiXSUj@\rc]mecgPpTUt^`\]T
J = 2¢¬O'PR^cX\=®tj@vR\rsbT=idcgPpT\]eb\fcgTS c]PªcfjDcfeo7T=\
^`vRs*cfjUigTl`T=ig\]£pmTXigT=^`tc]j`vR\1 2
¢0O'P_R\'T¬PR^l`T#zoR^`ig^`STc]T=ig\µ1, µ2
^nvqsDpabTKsβ¢
NTc_R\i]T=SUvRs.PRjcgPpT*T @_Rm£Ri]_pStj`vRspcgj@vµ1 = µ2
^nopo7T=^`ig\vtuPpT=St^`mIcgPpTigSUjspe0vq^nSUt=\¢ 7pj@icgPpTTac]T=vR\]l@Tl^nig^n£pmT
X =< n1 >cgPpTªtj`igigT=\]oqj@vRsbvp1tj`vkf_pN^cgTl^nig^n£pmT
A¥ c]PpT=i]SUjbsbevR^`St j`iutT¦3\#¥ ^`\g\r_pSUvpN =< n1 > + < n2 >
pabT=sq¦'t^nmmT=sG¥9tuPpT=St^`m¦'^Kvpcfe
A = −∂G
∂X|β,P,N = −µ1 + µ2 = −∆G0 − β−1 ln
c1
c2, ∆G0 = µ1,0 − µ2,0 − (K1 + K2)
∆G0\t^nmmT=sªcgPpTU igTTTvpT=i]@eªjnIcgPpTi]TK^`tc]j`v ¢ djncgTcgPR^cvR\rc]T=^@s¡j`l`TKtEcgj`iu\
(µ1, ..., µJ) j`i
cgPpTo7j`v@cu\'jnM0,β
j@vpTt^nv_R\]Toqj@vNcu\(c1, ..., cJ)
¢IO'PpT=vAt=^nv^nm\rjU£7T#sbTqvpT=s^`\
A = −∂g
∂c1|β, P, c
.0/ S1.32
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!#"$ %&('()!( *$ Vy
O'PRTXT N_R^cgj@vj`0\rcg^nc]T¥ igTm^cgj@v£7TcfTT=vX^nvRs
A¦\
c1 =c
1 + exp(−βA − ∆G0)
- @_Rm£Ri]_pS&o7j`vNcg\(^nigT'sbTRvRT=s^`\0o7j`v@cu\0PpTigTA = 0
nc]Pp\0`l`TK\µ1 = µ2
¢ 7Ri]j@S ¥ y@¦c0 j`mmj \cgPR^ccgPpTDT @_Rm£Ri]_pStj`vqsbcgj@v
µ1 = µ2vtuPpTSUt^`m(c]PpT=i]SUjbsbevR^`St\#_pvp@_RTme1sbTRvpT=\#c]PpT
N_pj`c]TvNc c1,e
c2,e
j`c]PRTTN_pm£pig_pS sbTvq\rc]T=\cj,e
¢IO'PpTT @_Rm£Ri]_pS tj`vR\rcg^`v@c\ sbTRvpT=s^`\
κ =c1,e
c2,e
= exp(−β∆G0)¥@¦
¡j`igTjl`T=i=N j@i ^U`l`Tvcc]PRT#TN_pm£pig_RS tj`vRsbc]j`v_pvRN_pT=mesbTRvRT=\ ^ª¥ pabT=sR¦'o7j`vNcdj`v
M0,β
cgPR^c \c]PRTvl^nig^`vNc63X£p£R\SUTK^`\]_pi]T@¢
´ ´ ´ 9 ±|´ °° ± ´ ° µβ
O'Pp\(m^\g^eb\cgPR^c#3X£p£q\0 i]T=TTvpT=i]@eGPR^`\
cg\3SUvpS_RS ^ncIc]PpT pabT=sUo7j`v@c^nvRsG(t)
\3Sj@vpjncgj`vpt'vUc]ST@¢I8 c3\ITlspTvNc3vUc]PpT ltvpcfej`cgPpTT N_pm£pi]_pS oqj@vNcK¢X«vpTt=^nv¡\g^eSUj`igT`q(cgPpToRi]jbtTK\]\
cj(t)tj`igi]TK\ro7j`vqsp\'c]j\rj@ST¡^nig`jl
oRi]jbtTK\]\=¢NTcd^nve*|^nig`jlUopigjtT=\g\cgPcfj\fcu^cgT=\
1, 2£7T`l`T=v\]_RtuPc]PR^nc j`i \]j`SUTtj@vR\fcu^nvNc
C
p1(t) = Cc1(t), p2(t) = Cc2(t), π1 = Cc1,e, π2 = Cc2,e
¥rV=@¦
PRTigTpj(t)
^nigTXcu\opigj`£R^`£pmc]T=\^c'cgSUTtR^`vRs
πj
^nigTXcg\d\fcu^c]j`vq^nigeopi]j@£R^n£RmcgTK\¢YT=SUvRs*c]Pq^c' j`i ^Rvpc]T¬i]igT=sb_qt£pmT¬¡^`i]@jltuPR^`vcgP*cgPpT¬ig^nc]TK\
wjj′cgPpT¬TvNc]igj`oeUjn;c]PpT
o7j@\]c]l`T'SUT=^`\]_pigTp = (p1, ..., pJ)
igTm^cgl@T3cgj¬c]PpT\rcg^cgj@vR^nigeSUT=^@\r_pigTπ = (π1, ..., πJ )
\IsbTRvpTKs^@\
SM =∑
pj lnpj
πj
= C∑
cj lncj
cj,e
¥rV`VK¦
dj§Tmm®opigjl`Tc]PR^ncc]PpT93X£p£R\' igTT¬T=vpi]@eg^nvqs¡^`i]@jlUTvNc]igj`oe
SM
^nigTT @_q^nm®_poc]j^S_pmcgoRmt^nc]l`T^nvRs^@spsbc]l`Ttj@vR\fcu^nvNcg\=¢
°´ ° #!()X') t! ! (+ # *) ' )$ '
g(t)
g(t) = µc +1
βCSM (t)
!MX µ = µ1 = µ2
R( >&!I ! g(t)
+ %9 %)< )X( 6<
pj(t) !& '( ) J)RX & ! () !1
hIigjjnf¢ 7Rj`i'c]PRT93X£p£q\' i]T=T¬TvpT=i]@eDsbTvR\]cfe*TX@Tc_R\]vR¥y`¦
g = limΛ
G
Λ=∑
j
cjµj = β−1∑
j
cj ln cj +∑
j
cj(µj,0 + Kj) =¥fVz`¦
SMSUT(VXW$Y$Z$Z
![Page 19: Microscopic Models for Chemical Thermodynamics · HAL Id: inria-00070792 Submitted on 19 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination](https://reader035.vdocument.in/reader035/viewer/2022081402/5f82df41a1f4ae3bf0511bab/html5/thumbnails/19.jpg)
V&F M ! '
= β−1∑
j
cj ln cj +∑
j
cj(µ − β−1 ln cj,e) = µc + β−1∑
j
cj lncj
cj,e
/ ccgPpT#\g^nSUTXc]STSM =
∑
pj lnpj
πj
= C∑
cj lncj
cj,e/ \SM
\¬NvRjv|cgjsbTKtigT=^@\rTUsb_pigvR¡^`i]@jlTl`j@m_pc]j`v \]TT ? y&@c]PpT\]T=tj`vRs1^`\g\rT=ircgj@v|jnc]PpTcgPpTj@i]T=S j`mmj \^@\'T=mm¢
NTc_R\\]Ppj&vpjc]PR^ncXcgPpTigTU\¬_RvpN_pTtuPpj@tTj`sbevR^`SUt=\®cgPR^c\Xc]PpTUiu^cgT=\vjj′
PptuP@l@T'TN_pm£pig_pS tj@vRsbc]j`v
µ1 = µ2¢ -3^`tuP¡^`i]@jl#tuPR^nvUcgPUcfj\rcg^nc]T \(igTl`T=ig\]£pmT@£qTKt^`_R\rT
igTl@Tiu\r£pmcfeDtj@vRsbc]j`vπ1v12 = π2v21
j`mmj \SSUTKsb^nc]T=me* i]j@S dj@mSUj@`j`igjlTN_R^cgj@v
dπ1
dt= π2v21 − π1v12
O'PRTvπ1
π2=
c1,e
c2,e
¥rV+@¦
8 v9^@tEc' i]j@Sπ1
π2=
v21
v12^`vRs.¥fV&+@¦cd j`mmj \cgPR^cvjj′
^nigTX_RvpN_pT=mesbTqvpT=s_pocgjD\rj@STtj`vR\rcg^`v@cCRPptuPªsbTc]T=i]SUvpT=\
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dj T1`l`T1^`v Tap^`SoRmT1jn\]_RtuP opigjbtTK\]\*v j`_pit^@\rT@¢ / \g\]_pSUTK1 < K2
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ujj′j`v
T
ujj′ (T )T @_q^nm\\rj@STtj`vR\rcg^`v@cu\
wjj′
Tj + Kj − Kj′ ≥ 0^nvRs
ujj′(T ) = 0j`c]PpT=i]\]T`¢%O'PpT=v%cgPpTopigjtT=\g\
Oc,β(t)t=^nv%£7T`l`T=v
Tabopmtc]T=me@¢ dT=vpjncgTgβ(r) = P (|ξ| > r)
j@icgPpTN^n_R\g\]^`vi=¢ l©¢ξcgPSUTK^nv
0^nvRs vl`T=ig\]T
cgTSUo7Tiu^c]_Ri]Tβ¢
8 cX\T=^@\rec]j*\]TT#c]PR^ncdc]PRTopigjbtT=\g\Oc,β(t)
t^nvª£7TigT=sp_RtTKsc]jcgPpT|^nig`jl*tuPR^`v|j`v1, 2c]Piu^cgT=\
v21 = w21, v12 = gβ(K2 − K1)w12YT=m^nc]j`vcgP1«vR\g^n`T=i3cgPpTj@i]e*vj`_RiTap^nSUopmTX\'cgPpT¬ j`mmjvpq¢0O'PRT q_ba\sbTqvpT=s^`\
J1 = X1
j@i'vc]PRT¬c]PpT=i]SUjbsbevR^nSUtXmSUc
J1 =dc1
dt/ vRs i]j@Sc]PRTTN_R^cgj@vR\dc1
dt= c2u21 − c1u12, c2 = c − c1
T¬PR^l`T
J1 =1 − exp(−βA)
u−121 + u−1
12 exp(−βA)
.0/ S1.32
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^nvG^nig£pcgig^`i]esb\fcgi]£p_bcgj@vp0(j, T )
j`cgPpTl`TKtEcgj`i
(j, T )\`l`T=v ¢(O'PpTvª^nc^`ve
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V`¢K(0) < K(∞)
c]PR\SUT=^`vR\IcgPR^cc]PRTvpTcgtTvpT=i]@e`Nop_pSUo7T=s_poDcgjcgPpTX\]eb\fcgTS c]Pc]PpTPpT=^nc=p\c]iu^nvq\f j@i]SUT=s*c]jcgPpT#tuPpTSUt^`m®TvpT=i]@e
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8 vNc]T=i]TK\fcgvRU\]c]_R^nc]j`vR\^`opoqTK^ni' j@iJ > 2
p£p_bc T¬mm vpj`ctj`vR\]sbTi'cgPpT=\]Tt^`\]T=\'PRTigT`¢
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.0/ S1.32
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Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, 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 - 38330 Montbonnot-St-Martin (France)
Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399