Stochastic theory of nonequilibrium statistical mechanics

Transcription

1 Ge, H.: Sochasic heory of Nonequilibrium Saisical Physics (review). Advances in Mahemaics(China) 43, 6-74 (04) Sochasic heory of nonequilibrium saisical mechanics Hao Ge Being Inernaional Cener for Mahemaical Research Biodynamic Oical Imaging Cener Peing Universiy, China h://bicmr.u.edu.cn/~gehao/

2 BICMR: Being Inernaional Cener for Mahemaical Research

3 BIOPIC: Biodynamic Oical Imaging Cener

4 Summary of Ge grou Sochasic heory of nonequilibrium saisical mechanics SP06,08 PRE09,0,3,4 CP; SA5 Sochasic modeling of biohysical sysems PCB08,3; PA Phys. Re. Science3;Cell4;MSB5 Sochasic Biohysics (Biomah) Nonequilibrium landscae heory and rae formulas PRL09,5 RSI; Chaos Saisical machine learning of singlecell daa

5 Which ind of hysical/chemical rocees can be described by sochasic rocees? Mesoscoic scale (ime and sace) Single-molecule and single-cell (subcellular) dynamics rajecory ersecive

6 Sochasic single-molecule and single-cell ineics Single-molecule enzyme ineics Single-cell ineics Eldar, A. and Elowiz, M. Naure (00) Lu, e al. Science (998) Choi, e al. Science (008)

7 he fundamenal equaion in nonequilibrium hermodynamics Second law of hermodynamics Clausius inequaliy er ds Q ds rewrie 0 Enroy roducion Q Rudolf Clausius (8-888) More general Carl Ecar (90-973) P.W. Bridgman (88-96) Nobel Prize in 946 ds er d d i e S S d i S X 0 Ilya Prigogine (97-003) Nobel Prize in 977

8 Mahemaical heory of nonequilibrium seady sae ime-indeenden(saionary) Marov roce Min Qian (97-) Reciien of Hua Loo-Keng Mahemaics Prize ( 华罗庚数学奖 ) in 03

9 he simles hree-sae examle a seady sae: single-molecule enzyme ineics

10 Reversible Michaelis-Menen enzyme ineics wo reversible Michaelis-Menen reacions Ge, H.:. Phys. Chem. B (008) Ge, H., Qian, M. and Qian, H.: Phys. Re. (0) Kineic scheme of a simle reversible enzyme. From he ersecive of a single enzyme molecule, he reacion is unimolecular and cyclic.

11 Simulaed urnover races of a single molecule S P () Min, e al. Nano Le, (005) Ge, H.:. Phys. Chem. B (008) ( ) : S P ( ) : P S he number of occurrences of forward and bacward cycles u o ime

12 Seady-sae cycle fluxes and nonequilibrium seady sae ( ) lim [ S] VS V KmS [ S] K ms [ P] KmP [ P] K P mp. Ge, H.: PCB (008) Ge, H., Qian, M. and Qian,H.: Phys. Re. (0) Michaelis-Menen ineics ( ) lim [ S] VS KmS [ S] [ P] K K ms mp ; ( ) lim [ P] VP KmP [ S] [ P] K K 0 3[ S] Chemical oenial difference: B ln ln( ) 0 B [ P] 3 ms mp. 0 0

13 Waiing cycle imes S E ES EP E ES EP E P he ineic scheme for comuing he waiing cycle imes, which also serves for molecular moors. ; ;.

14 Generalized Haldane equaliy P E ( ) PE ( ). Suerosed disribuions! Average ime course of forward and bacward ses Carer, N..; Cro, R. A. Naure (005) Waiing cycle ime is indeenden of wheher he enzyme comlees a forward cycle or a bacward cycle, alhough he robabiliy weigh of hese wo cycles migh be raher differen. Ge, H.:. Phys. Chem. B (008);. Phys. A (0) ia, e al. submied (05): general Marovian juming roce Ge, e al. submied (05): diffusion on a circle

15 Nonequilibrium seady sae Ge, H., e al. Phys. Re. (0) 3 3 B P S log ) ( ) ( P P E E Ge, H.. Phys. Chem. B (008);. Phys. A (0) Generalized Haldane Equaliy Flucuaion heorem of flucuaing chemical wor n E E B e n W P n W P ) ) ( ( ) ) ( ( ) ( W B e Second law in erms of equaliy Free energy conservaion 0 log er 3 3 B Enroy roducion: Free energy diiaion Free energy inu S P 0. m Equilibriu radiional Second law W

16 ransien flucuaion heorem for cyclic chemical wor ' l l ' l ' l ' l,v l P v l,v l P v l,v l P v l,v l P v v v, ' e g g g g g,log log,log, log, Ge, H.: PA (0) v W v W ' log, log e e e W W W W

17 Maser equaion model describing he sysem Consider a moor roein wih N differen conformaions R,R,,R N. is he firs-order or seudo-firs-order rae consans for he reacion R i R j. No maer saring from any iniial disribuion, i will finally aroach is saionary disribuion saisfying N j 0 j ji d i ( ) d i Self-aembly or self-organizaion j j ji i eq j ji eq i Deailed balance (equilibrium sae)

18 NESS hermodynamic force and enroy roducion rae NESS hermodynamic force NESS flux i j A ji B log i j ji NESS enroy roducion rae er ne i j A 0

19 Energy ransducion efficiency a NESS A mechanical sysem couled fully reversibly o a chemical reacions, wih a consan force resising he mechanical movemen driven by he chemical gradien. Energy Energy Inu (Chemical) Inu (Mechanical) ne h d or ne h d Energy Ouu (Mechanical) Energy Ouu (Chemical) Energy ouu Energy inu ne h d Energy ouu Energy ouu h e all diiaed! ne d ne Ge, H.: PRE (03)

20 From flucuaion heorems o he decomosiion of er a ransien sae

21 ime-deenden case di d j j ji i Quasi-saionary disribuion N j 0 j ji i If { ()} saisfys he deailed balance condiion Bolzmann s law Free energy Ei ( ) / eq i ( ) i ( ) E j ( ) / e j E j ( )/ F( ) B log e j e

22 Mahemaical equivalence of arzynsi and Haano-Sasa equaliies arzynsi equaliy: local equilibrium e Same heorem for ime-deenden Marov roce W / F / (0) eq (0) Ge, H. and Qian, M., MP (007); Ge, H. and iang, D.Q., SP (008); e W. eq F ( 0) (0) Haano-Sasa equaliy: wihou local equilibrium e Q ex / (0) (0) S 0 0 Q ex /. Are hese inequaliies already nown in he Second Law of claic hermodynamics? Do hey only hold for he whole ransiion roce beween wo seady saes? he radiional Clausius inequaliy can be in a differenial form.

23 Decomosiion of mesoscoic hermodynamic forces i j A B log i j ji A A A B log i j ji Enroy roducion Houseeeing hea e A 0 i j Q h ( ) ( ) A ( ) 0 i j f d ( ) ( ) A ( ) i j Free energy diiaion 0 Q f Ge, H., PRE (009); e h d Ge, H. and Qian, H., PRE (00) (03)

24 wo origins of nonequilibrium e 0 for any ime In he absence of exernal energy inu and a seady sae. Q h 0 for any ime In he absence of exernal energy inu f d 0 for any ime A seady sae Ge, H., PRE (009); Ge, H. and Qian, H., PRE (00) (03)

25 Decomosiion of enroy roducion rae f d 0, e Q f d h Q 0, h 0. Ge, H., PRE (009); Ge, H. and Qian, H., PRE (00) (03) ds d e Q o ds d ds d Q ex Q o e 0 Q o Qh f 0 he new Clausius inequaliy is sronger han he radiional one. d

26 Second-order sochasic sysem: ime-reversibiliy and anomalous behavior m d X d F. X,X

27 wo differen definiions of enroy roducion raes EPR B B D x lim 0 F log F P P s : 0 r s x D m s v log s : 0 s dxdv P F F x, v Sinney, R.E., and Ford, I.., PRL (0); Lee, H.K., Kwon, C., and Par, H., PRL (03) EPR B D x x v x D m v log dxdv Kim, K.H. and Qian, H., PRL (004) EPR 0, EPR 0 corresond o ime-reversibiliy and Maxwell-Bolzmann disribuion for hermodynamic equilibrium resecively hermodynamic equilibrium Ge, H.: PRE (04)

28 When he exernal force is only deenden on osiion F x,v xv Gx D B x x x Ge, H.: PRE (04) EPR EPR hermodynamic equilibrium 0 G x x ; x U x hermal equilibrium Mechanical equilibrium Flow of ineic energy along he saial coordinae d d E ineic ineic x W x, Qx, x x (measurable) Hea flux q x ineic x E ineic xv x EPR 0 x 0 q hermodynamic equivalence beween mesoscoic and macroscoic scales

29 he enroy roducion rae in he small-noise limi Decomosiion of enroy roducion rae saial EPR EPR EPR saial EPR over x x ˆ over x ˆ dx Celani, e al.: PRL (0); Ge, H.: PRE (04) Enroy roducion rae of he overdamed-limi over n 6 B x x x x ˆ x dx Anomalous conribuion of EPR Hence he overdamed aroximaion only ees he dynamics raher han he second law of hermodynamics.

30 Local recirocal relaion beween linear coefficiens L xx L qx over x q x x L xq x ˆ L x xx L qx X X L x x qq L xq L n qq X q X 8 6 Recirocal relaion beween Sore effec (hermal diffusion) and Dufour effec B q B x x 3 ˆ x Ge, H.: SA (05) Come from he second momen of velociy along each dimension. Always hold, even for far-from-equilibrium sysem.

31 Summary Sochasic heory of nonequilibrium saisical mechanics is rigorously sudied and furher develoed, including he cyclic dynamics wih an unexeced equaliy, flucuaion heorems and decomosiion of enroy roducion rae. A sronger Clausius inequaliy emerges only in he nonequilibrium case; Recenly we are ineresed in he second-order sochasic sysem wih emeraure gradien.

32 Acnowledgemen Prof. Min Qian Peing Universiy Prof. Hong Qian Universiy of Washingon Fundings: NSFC, MOE of PRC

33 hans for your aenion!