Microscopic Models for Chemical Thermodynamics
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Microscopic Models for Chemical Thermodynamics V. A. Malyshev INRIA, France. N° 5200 May 2004. ISSN 0249-6399. ISRN INRIA/RR--5200--FR+ENG. THÈME 1. apport de recherche.
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(181) l@eUT¬T=c]tEspPRcgj`T\gitl@_RT=\gtE\]cgT=j`si ^n£7jl`T`¢ ¢ °´ ° " * ) ( $ () 7. u, b, f, h. M. j. ×j Gj (β, µj ). β, µ1 , ..., µJ. Gj = Gj (β, µj ) M0,β. β. ν ∈ M0 M = (β, µ1 , ..., µJ ) (β, c1 , ..., cJ ). k. ν = ν1 × ... × νJ. sf → ∞ M0 ⊂ M. M0 M0. Cc (t) = lim Xc (t), Oc,β (t) = lim Cc (t) sf →∞. IB!)X'" t (
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(188) T mNj_R ^`mI\'cg jigj`S ¥y`¦
(189) O'^nmPp\]Tji]£TKe*\r_RspmcgTc]vpT=i]SUopigvpjt\fT=cg\gt¬\ T=l`j`m_bj`cgv j@vj` t=c]^nPRv#T£7l`TT=tsbc]Tj@Ri vpTKs®_R\]vpdc]PpTTl@j`m_bc]j`vj` ^`vRs¬ j@i]S_pm^ @O'\TPRvR^`Tt\gig\]T#T _pjnSU\T=j`^s®vp¢ _pTvp0oR N^`_pircgT tRmT#aTK}|s¡^n^igc]`c]jiul*^`toRc]i]vpjb*tTKo7\]\j`vNc c]j*c]PpTU \fj`cu^icgc]Ppj@vR\¬^nig\reeb\rspc]T\rSc]ig¢£RO'_bPpc]j`\ v ©j@mIj cu\X\d igi]i]j@TKS sb_Rttj`£pvl`m T=cf2i e 0 ¥ j`iN RTcvp_Rc]T\S¦^``^nT¬vqRs iu\fc \]j`SUT¥ igj`TiUSvp^nigqbvp\
(190) ^`cgT£qj@l@_bj`cdm_ptSUj@vRTK\]¦ Tig^nl`igT=T s tNj@_RvR^n\]TvNigc]l`c]T=s T=\=¢1¢ O' dPpjnT=c]vT¬cg PRigj`^S c c]PpTT
(191) N_R^nc]j`vj` c. fjj 0. N N. [0, E]. N→∞. 0,β. c,β. N i=1. i. β exp(−βx). Ti. Cc (t) Oc,β (t). Cc (t). β>0. t. β. cj (t). M0. M(t) = (β(t), µ1 (t), ..., µJ (t)) M(∞). Mt. N=. Λ. SMSUT(VXW$Y$Z$Z. P. j cj. P. j. < nj >.
(192) =V M ! ' \rvcg^nj`c]T_pid¥sbn¦STIqjbvpc
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(194) T^^nNc ig_RT#^tcgj@\'j@vRv t\]j@TvRigl`\]T=Tsigl`¢ T= sjn¥ cg j`T`ibcgpPRab^T=cs _pvR¦EsbNT=c]iPpc]T¬Pp\gTK^n\rSUT#Ttj` vRj@spiIc]cgPRj@TvR@\'igT=^`^@vRtusUPoq\]j`_pc]£pT=Sv@cg^`^`vpm ¢( j@O'msP_Rj`\ P. β. N, P, Λ. M0,β. X. λj exp β µ ˆj =. X. cj = c. \^nm\]jvl^nig^`v@cK¢ $w'0!)+" ! +!# " 0"#.;#+ T=7R_Rvp_pvRTi]tEigc]cg`Ppe T=j@i vR\¡j`¢ v ¥ j@
(195) iTDptabj@T=vRs \]sbT=i#j`vRme1c]Pp¦Tj`opv igjtT=\g\
(196) j`iDTap^n¢¡SUO'opmPpTT=v%cgPp^nT¡mIT=cgv@PpcgPRTig^nSUmoje speNvq'^nSUj`iUtUcgPpo7Tjncg3XTvNc]£p£R^n\Um\ ^`igTi]TT qj`v R j`i Tªap°^nKSU op mTX9³ c]Pªsb j@¨©vRT\rigTsbvNT=c ii]cfTK^`jDtc]spj`¨©v*Tiuig^TcgvNT=cd\=¢opig/ jb\gt\rTK_p\]SU\]T=T\ ^nm\]jcgPR^^ncvR s j@i \rj@SUT j. j. Oc,β (t). M0β. K1 , ..., KJ. H. G. . . (1). (2). µj (t). µj (t), j = 1, ..., J T >0. M0,β. cgcgPRPR^^cXccg\dPpT#c]PRsbT=\]¨7T T=i]cfT=vRo¡tTKoR\'i]£qjbtTTKcf\]\]T=T=T\dvPR^vpl`cgT#^`c]m PpTU^nvq\]s*^`SUqT#vR^nmvRT=cgvN^`c]PRm0^`^`mvRopsT=\Rvq^`^ni]TXmoqcgPpj@T#vN\]cg^`\=SU¢¬O'TPp j@Tivª£7cgjnPpc]T%P5 opT=ig\gjb\tTKm^\]w\]T=\=\g¢I^e8 v \ 9oq^@^tEc]cKPqn\'c]PRj`v \3m^Ppj`mspc]\IP^`_bcgPpc]j@T#SU\]^`^nSUc]tTX^`mvpe#c]v^nm j`^`_pvRi(sSURjbvRsb^`Tm®m`oq£7j@T=vNt=cg^n\=_RR\]^`TvR£7sjnc]c]PPpToRT=i]jbv@tcgPRTK\]^n\]mT=o\(eD^`i]\T^sbTK _p\]tvRi]tEcg£7j@T=sv£j`e#vcfc]
(197) PpjT vl^nO'igPp^`Tv@c'\]SSU^nopvpmT=\r j`c mts m^`\g\rqt=¢ ^c]j`v*jn;igT=^@tEcgj@vR\3\'v*cgTigSU\'j` c]PRTTvNc]PR^`moe I¢ 8 c¢ gPpcgPpTTi]v|TK^`c]tEPpcgTj@i]vªTK^`t\c]Tj`vqvªsbjn\cgPpt=T^nigmSUTKsªtT^nabvRjnscgPpcgTPpigTSUPpt`T=p^nccgPpT\ cuPR^nT=`^T=c vª igj`\ S `jc]TKPp\Tc]Tj*vlc]PpigTj`vRTvSlT=vNi]j@c=vp¢SUO'TPqvN^c=cq\ 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\ SU^`vRj@s*i]T=cfjl`jUTiigT
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(200) ¢ jDcfeo7T=\ vq¥ c]^nPpSUT=N i]TSUt=c \jb¢ _Rsb\ 7pevRj@i]T=i^`SUScgPpvRtTs. Tj`PRaiujc]tT=TvR¦3\]cg\#Ppl@T*T¥ ^`Tl\g@\r^n_p_RigSU^nm£p£Rvpmi]T _pStj`vRspcgj@v cgPpTªtj`igigT=p^n\]opaboqT=o7j@sqvRT=¦'^`sbigt\vp^n1mvT=tsGj`tuPpv¥9kftuT=_pPpSNT=^Stcg^`TmIt^`lcgmPp^n¦'igT^ig^nKSU£pvpjmT spcfe e 0 (1). (2). (1). (2). µj (0) = µj (0), µj (T ) = µj (T ), j = 1, ..., J. M0,β. M0,β. H. H(0) < 0 ∆H > 0 ∆H = Q. ∆H = H(∞) −. Q. . J =2. 12. µ 1 , µ2. β. µ1 = µ 2. X =< n1 > N =< n1 > + < n2 >. A. c1 ∂G |β,P,N = −µ1 + µ2 = −∆G0 − β −1 ln , ∆G0 = µ1,0 − µ2,0 − (K1 + K2 ) ∂X c2. cgPpTo7j`\v@tcu^n\'mjnT= sªcgPpTU j@igTvpTTTtvp^nT=vi]@_Reª\]TjnoqIj@cgPpvNTcu\ i]TK^`tc]j`v ¢ d¢IjnO'cgTPpcgT=PRv ^ct=vR^n\rvc]T=^n^@ms¡\rjUj`£7T#l`TKsbtETcgqj`vpiu\ T=s^`\ A=−. ∆G0. (µ1 , ..., µJ ). M0,β. (c1 , ..., cJ ). A=−. j`i. A. ∂g |β, P, c ∂c1. .0/ 1S .32.
(201) Vy.
(202) !#"$%&('()!(*$. O'PRTXT N_R^cgj@vj`0\rcg^nc]T¥ igTm^cgj@v£7Tcf
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(205) igNT't_pj`sbvqmTsbR£pvRigcgT=_pj@sSv ^`\0sbTo7vqj`\rv@c]cu\0T=\ vPpTtuig¢IPpT TO'SUPpTt^`T m(@n_Rc]c]PpPpmT=£R\0i]SUi]`_pjbl`SsbTKe\ vRtj`^`vRS\rcgt^`\#v@c_p¢ vp7R\ @i]sbj@_RTS TRme1vp¥ T=y@sbs¦ T^`Rc0\vp j`T=\#mjc] PpT\ ¥ @¦ cg}¡PRj`^igc Tj\
(206) l`c]T=PRi=TN j@vi l^n^Uig`^`l`vNT6c v 3X£pc]£RPR\T#SU
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(208) v¡ ^n vR\g± ^s e´ SU j`igT`\3q° Sµ (j@cgvpPpjnT cgO'oRj`vpi]Ppjbt'\(tTKmvU\]^\ c]S\g^T@ebI¢ \ t8cgc3j`PRig^\Ii]#c TKT\r3Xlo7j`sp£pvqT£qspvN\0c3\' i]c]vUT=jTc]\rTPpj@vpT ST=li]T@te}¡vp^ncfige`PRj^`j`l \ oRi]jbtN TKT\]cd\=¢^nve*}|^nig`jlUopigjtT=\g\
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(210) j\fcu^cgT=\ £7T`l`T=v\]_RtuPc]PR^nc j`i \]j`SUTtj@vR\fcu^nvNc ¥rV={@¦ PRTYigT T=SUvRs*c]^nPqig^TXc'cu \j`i opig^ j`R£Rvp^`£pc]T¬mc]i]T=igT=\sb^_qc'tcg£pSUmT¬T }¡R^`^`i]@vRjs ltuPR^`^nigvTXcg\dcgP*\fcu^cgPpc]T¬j`vqig^n^nc]igTKe\ opi]j@£R^ncg£RPpT¬mcgTvNTKc]\ig¢ j`oeUjn;c]PpT o7^@\j@\]c]l`T'SUT=^`\]_pigT igTm^cgl@T3cgj¬c]PpT\rcg^cgj@vR^nigeSUT=^@\r_pigT \IsbTRvpTKs ¥rV`VK¦ ^S d_pmjcg§oRm
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(212) # *) ' ) $ ' c1 =. A=0. µ 1 = µ2. µ1 = µ 2. c1,e c2,e. cj,e. κ=. c1,e = exp(−β∆G0 ) c2,e. M0,β. c. . . β. G. G(t). cj (t). 1, 2. C. p1 (t) = Cc1 (t), p2 (t) = Cc2 (t), π1 = Cc1,e , π2 = Cc2,e. pj (t). t. πj. wjj 0 π = (π1 , ..., πJ ). p = (p1 , ..., pJ ). SM =. X. pj ln. X cj pj =C cj ln πj cj,e g. t. g(t). g(t) = µc +. !MX . SM. 1 SM (t) βC. R( >&!I ! + %9%)< )X (µ = µ
(213) = µ6
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(222) igN\]_R£p^cgmT@j@v £qTKt^`_R\rT = β −1. /. cj ln cj +. cj (µ − β −1 ln cj,e ) = µc + β −1. j. j. j. SM =. SM. cj ln. X. pj ln. X pj cj =C cj ln πj cj,e. vjj 0. µ1 = µ 2 π1 v12 = π2 v21. dπ1 = π2 v21 − π1 v12 dt. O'PRTv ¥rV+@¦ 8 v9^@tEc' i]j@S ^`\]j`vRSUs.T¥f&V t+@j@¦ScdSU j@j`v mjc] SU\
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(225) T N_R^cgj@vR\
(226) T¬PR^l`T π1 c1,e = π2 c2,e v21 π1 = π2 v12. vjj 0. C. K1 < K 2. T ujj 0 (T ). ujj 0. Tj + K j − K j 0 ≥ 0. ujj 0 (T ) = 0 gβ (r) = P (|ξ| > r). wjj 0. Oc,β (t). 0. ξ. β. Oc,β (t). {1, 2}. v21 = w21 , v12 = gβ (K2 − K1 )w12. . J1 = X˙ 1. J1 =. dc1 dt. dc1 = c2 u21 − c1 u12 , c2 = c − c1 dt J1 =. 1 − exp(−βA) + u−1 12 exp(−βA). u−1 21. .0/ 1S .32.
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