Nonequilibrium Thermodynamics of open driven systems
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1 1 Boynam Otal Imagng Center (BIOPIC) 2 Beng Internatonal Center for Matematal Resear (BICMR) Peng Unversty, Cna Nonequlbrum ermoynams of oen rven systems Hao Ge
2 A sngle boemal reaton yle B + AP 1 C + ADP (1) 1 B + P 2 C (2) 2 Equlbrum onton: eq [ AP] eq [ ADP] [ P ] eq
3 QSS v.s. NESS Very slowly angng [AP], [ADP] an [P] Quas-steay-state (QSS) Close system e emal nets of te nternal system (B+C) are te same. Any new termoynams? Nonequlbrum steay state (NESS)
4 Heat aton After an nternal lowse yle, te tratonal eat aton urng AP yroly ( ) ( + + ) B AP AP ADP C P. ADP C P B
5 ermoynams of a lose sotermal system Close system F t S t lose lose e f lose lose e f lose lose ; lose.. Gbbs states free energy never nrease n a lose, sotermal system; wle Prgogne states tat te entroy routon s non-negatve n an oen system. ey are equvalent.
6 As a subsystem of te lose larger system S t B, C Close system e lose lose + S AP, ADP, P Heat aton oul not be nferre ust from te nternal emal nets, regarle of te onrete meansm tat ow te nternal system s oule wt te surrounngs. t.
7 Wt regeneratng system Oen rven system: regeneratng system eeng te onentratons of AP, ADP an P γ 1 2[ AP] [ ADP] [ P ] 1 2 1
8 ere s an external ste for te regeneratng system onvertng ADP+P ba to AP after ea omleton of a nternal reaton yle. e mnmum wor (non-pv) t as to o s ust te free energy fferene between ADP+P an AP,.e. W µ µ µ mn AP ADP e total eat aton of su a reaton yle s Extra eat aton + ext W Drven energy nut of te nternal system e extra eat aton mn B ext W mn P ( AP ADP logγ. Dretly nterrete from te emal nets of te nternal system P )
9 Master equaton moel esrbng te nternal system Conser a motor roten wt N fferent onformatons R 1,R 2,,R N. s te frst-orer or seuo-frst-orer rate onstants for te reaton R R. No matter startng from any ntal strbuton, t wll fnally aroa ts statonary strbuton satsfyng N 1 ( ) ( t) t ( Self-aembly or self-organzaton ) eq eq Detale balane (equlbrum state)
10 Coule wt energy soure Aume only one of te transton s nvolve n te energy soure,.e. AP an ADP. ~ ~ 12 12[ AP], 21 21[ ADP] ermoynam onstrans µ µ 1 µ + µ B log µ µ 2 D ; µ B µ D ~ log ~ B log eq D eq,
11 Heat aton oen ( t) B > + B ( )( ( t) ( t) ) ( ( t) ( t) )( µ µ ) D In te NESS ne B > ( ) log. Solely eenent on te equaton of nternal ynams. 1 st Law: Energy nut (mnmum wor)eat ate
12 Energy transuton effeny A meanal system oule fully reversbly to a emal reatons, wt a onstant fore retng te meanal movement rven by te emal graent. Energy Inut (Cemal) ne + or Energy Inut (Meanal) ne + Energy Outut (Meanal) Energy Outut (Cemal) Energy outut Energy outut η ne Energy nut + Energy outut 1 e ne ne
13 Mesoso termoynam fores n NESS e eat aton for ea onformaton transton s ( Q log ) B + s s e steay-state entroy fferene S B log + s ermoynam fore for su a transton A Q + S B log Q + (Clausus equalty) etale balane! S s
14 Deomoston of mesoso termoynam fores A log B A + δa δa B log B log Entroy routon Houseeeng eat Free energy aton e oen Q > ( t) J A ( ) t J A > f ( ) t J δa > oen e Q + f
15 Wat s f? F F t B Relatve entroy f log Generalze Free energy
16 wo orgns of rreversblty f, Q, e f + Q. e araterzes total tme rreversblty n a Marov roe. Wen system reaes statonary, f. Wen system s lose (.e., no atve energy rve, etale balane) Q. Boltzmann: f e > but Q ; Prgogne (Bruel sool, NESS): Q e > but f. f n rven systems s self-organzaton.
17 QSS v.s. NESS S t lose e lose lose. Close system e lose e oen S t oen e oen oen lose oen
18 wo ns of Seon Law S t e S f ( Q ) t S t ( e ) S t ( ) Q ( f ) In etale-balane ase, tey are equvalent. e f, Q In non-etale balane ase, te new one s stronger tan te tratonal one.
19 Summary Regeneratng system aroa woul stngus quateay-state an nonequlbrum-steay-state, an suly an equlbrum termoynam founaton for te exreon of eat aton n nonequlbrum steay state of subsystems, wtout te nee to now envronment ; s new ersetve yels an extene Seon Law, w emerges only from rven ynams wt external regeneratng system. Hene te NESS ersetve suggests new ngreents for te nonequlbrum termoynams. A omreensve framewor for bot equlbrum an nonequlbrum statstal means s roose (sotermal).
20 Anowlegement Prof. Hong Qan Unversty of Wasngton Deartment of Ale Matemats
21 ans for your attenton!
22 Evoluton of entroy System Meum S tot Ssystem + Smeum S system S S e wo fferent ersetves er S Stot J X S, S e an S tot, rater tan S, S e an S tot are te state funtons of te nternal system. Detale balane J X Generalze flux Generalze fore I. Prgogne: Introuton to termoynams of rreversble roees. 3 r e. (1967).L. Hll: Free energy transuton n bology. (1977)
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