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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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

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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Xi = ji, Ti¢

TvRjnc]TH

(u)N

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\]oR^@tTUj`v(1, ..., J × R+)N

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c]P1c]PpT\rcg^nc]Td\]oR^@tT

(j, T )PptuP\3spTRvpTKs£e#cgPpTdiu^cgT=\

ujj1

@c]PpTT (t)

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(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

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t+0£7T=tj`SUTdtj@i]igT=\]o7j`vRs 0

vp@me(j1, T1), (j

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1, T′

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spTvR\]cgTK\P (b)(j1, T1, , j

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j, j′, T, T ′, j1, T1, j′

1

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1 = T + K + T ′ + K ′ − K1 − T1 − K ′

1

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1 < 0p^nvRs

P (b)(j1, T1, j′

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∑

j1,j′1

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H(b)N

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T, T ′ → T1, T′

1

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1

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[0, T + T ′]R^`vRsDj`

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H(f)N

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h¢ TvRjnc]T

tik.k = 1, 2, ...

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[0, T + ξik]®PRTigT

ξik

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β exp(−βx)¢ dT=vpjncgT|cgPpT

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H(β)N

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N =∑

j nj

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nj

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ji = jSUj`mT=t_pmT=\=

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oq^ni]c]tmTXPq^`\'l`T=mjbtcfe~v(ω) = ~v(t, ω)

^nvRstjj`iusbvR^cgT~x(t, ω)

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´ ° 9p1 ° pn ´ dT=vpjncgTpt(j, T ), j = 1, ..., J

(cgPpTspTvR\]cgTK\jn cgPpTj@vpT0 oR^`ircgtmToRi]jbtTK\]\

M1^nc'c]ST

tJ∑

j=1

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cj(t) = limΛ→∞ Λ−1 < nj(t >= pt(j)cbPpT=i]T

pt(j)\

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j¢

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∂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 )

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P (1) = ujj1(T )δ(T + K − K1 − T1),

.0/ S1.32

!#"$ %&('()!( *$

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j′ ,j′1

∫

dT ′dT ′

12bjj′P(b)(j1, T1, j

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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 )

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Ij =

dj−3∑

k=1

mj,kw2j,k

2

PRTigTmj,k, k = 1, ..., dj−3

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Ej = Tj + Ij + Kj

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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

!#"$ %&('()!( *$ 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

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k=1

(2π

mj,k

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2 , µj = µj − Kj

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Θ =

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j=1

Θ(j, β) = exp(Λ∑

j

λj expβµj), λj = β−dj2 Bj

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Ω = ΩΛ = −β−1 ln Θ = −β−1Λ∑

j

λj expβµj

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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`¦

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dj

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cj = 1¢

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j

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∑

j

< nj >= β−1∑

j

cj =∑

j

pj

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Vz M ! '

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βP = limV →∞

1

Λln Θ

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PΛ = β−1∑

j

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dj

2+ 1 − βµj) =< nj > (

dj

2+ 1 + βKj − βµj)

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S =∑

j

Sj = Λ∑

j

λj exp(βµj)(dj

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∑

j

< nj > (dj

2+ 1 + βKj − βµj)

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T=v@cgPR^nmoe

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j

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∑

j

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j

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G

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µjcj

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s =∑

j

cj(dj

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∑

j

cj(− ln cj +dj

2+ 1 − βµj,0)

.0/ S1.32

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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

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

!#"$ %&('()!( *$ 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

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=\

\]j`SUTtj@SSUj@v c]SUT\gt^`mT.¥9\]o7TT=sjn£qj`c]Pi]TK^`tEcgj@vR\u¦#^nvqs\igi]T=mT=l^`vNcc]jGcgPpTigSUjspeNvq^nSUt=\¢O'PRTj`igTS\opigjl`TKs®¢

dj T1`l`T1^`v Tap^`SoRmT1jn\]_RtuP opigjbtTK\]\*v j`_pit^@\rT@¢ / \g\]_pSUTK1 < K2

¢ / \g\r_RSTvRj&c]PRTD\rSUopmTK\fc#o7j@\g\]£pmTsbTo7TvqsbTvRtTjn

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

!#"$ %&('()!( *$ V

° ± ° µ 9=9; ´ / \]\]_pSUTcgPR^c^cc]SUTt = 0

^nvG^nig£pcgig^`i]esb\fcgi]£p_bcgj@vp0(j, T )

j`cgPpTl`TKtEcgj`i

(j, T )\`l`T=v ¢(O'PpTvª^nc^`ve

t > 0c]PpT#sbT=vR\]cgTK\

pt(j, T ) j`i ^`vNe*oR^ni]c]tmT¬mm®£qT

β exp(−βT )pt(j)¥fV=N¦

j@i \rj@SUTpt(j)

¢O'PR\ t=^nv£qT#\]Ppjv^`\' j@mmj \=¢ / \'c]PRT#vNcgTigvR^nm sbT=`igT=\g\jn i]T=T=sbj@S jn0vpRvpc]Tv_pS£qT=ijn0oR^ni]c]tmT=\'^`i]T¬­¢ ­¢ s ¢(ig^`vRsbj`S l^`i]^n£RmTK\Nc]PRTvc]PpT=i]T¬Ta\rcd^p¢ \¢c]PRTmSUcu\

T (t) = limΛ→∞

1

Λ

∑

i:xi∈Λ

Ti(t), K(t) = limΛ→∞

1

Λ

∑

i:xi∈Λ

Kji(t)

Tab\rc ^cd^nveUcgSUTt¢O8 voR^ni]c]t_Rm^`i=b^p¢ \=¢0 j`i ^nveUpabTKsl^`m_pTK\'jn

Kji(t)

c]PpTmSUcg\

T (t) = limΛ→∞

1

Λ

∑

i:xi∈Λ

Ti(t, ~K(t))

Tab\rc ^nvRs^nigTXTN_R^nm­¢O5 T=i]T ~K(t) = Kji(t), i = 1, 2, ...

¢ªj@i]T=jl`T=i3 j@i ^nveD`l`T=v ~K(t)

c]PpT=i]T\^U\rT N_pTvRtT¬jn7kf_pSUoSUj@ST=vNcg\

t1 < t2 < ... < tn < ...

j`q9^@\fcI£RvR^`i]etj`mm\]j@vR\(^nvqsPRT=^c(cgig^`vR\r TiKnPptuPUspjvpjnc3tuPR^`vp`T'oR^`ig^`SUTc]T=ig\ji

¥ ^nvRsc]P_R\Kji

¦j`0c]PpTSj@mTKt_pmT=\=¢#8

sf

^nvqssβ

¢ c]T=vRscgjDvpRvpcfeT#PR^l@T#^p¢ \¢cgPpTigT#mm£qTD:EvbRvpc]T :¬v_pS£7Tij`9^@\fctj`mm\]j@vR\^nvRsGPpTK^ccgig^`vR\f T=ig\¬£7TcfTT=v ^nveªcfj_RvR^nige|igT=^@tEc]j`vq\¢ 8 c j@mmj \Xc]PR^nc^nvecgSUT

tTPR^l@TX^opigjbsb_Rtc STK^`\]_pigT¥rVN¦¢

Tmm0\rc]_qsbec]PpTU\rT N_pTvRtTK(t)

¢ / \X^nvec]SUTSUj@ST=vNct ≥ 0

TPq^l`TT (t) = β−1 op_bc^`m\]j

T (0) = β−1 j`i tj@vNc]vN_Rcfe@¢ j cgPpTigTXcfjUo7j@\g\r£pmc]T=\

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

zb¢K(0) > K(∞)

'c]PR\SUTK^nvR\UcgPR^ccgPpT|tuPpT=SUt=^nmTvRTig`e\Uc]iu^nvq\f j@i]SUT=s%c]j.cgPpT¡vpTcgtTvpT=i]@e`bPptuP`jT=\'j@_bc ^`\c]PRTPpT=^nc=¢

8 vNc]T=i]TK\fcgvRU\]c]_R^nc]j`vR\^`opoqTK^ni' j@iJ > 2

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.0/ S1.32

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