The Jarzynski Equation and the Fluctuation Theorem

Transcription

1 The Jarzynski Equation and the Fluctuation Theorem Kirill Glavatskiy Trial lecture for PhD degree 24 September, NTNU, Trondheim

2 The Jarzynski equation and the fluctuation theorem Fundamental concepts Statiscical physics Recents developments Practical applications Fluctuations 2

3 Scope «there are a few relations that describe the statistical dynamics of driven systems which are valid even if the system is driven far from equilibrium...» Gavin E. Crooks, Physical Review E, 61(3), p.2361, 2000 the Jarzynski equality the Fluctuation theorem 3

4 Outline General introduction «Characters in play» Crash course in statistical mechanics Thermodynamics and its range of validity The Jarzynski equality and the Fluctuation theorem The contents of the theorems Applications and experimental verification Discussions and critique 4

5 Part A.I «Characters in play» Scope of the theorems Main authors A.I 5

6 Jarzinsky equality Equilibrium (reversible process): Work = Δ Energy Non-Equilibrium (irreversible process): Work = Δ Energy + Lost Work depends on the process path Work Δ Energy Christopher Jarzynski University of Maryland, Assoc. Prof. Chemistry and Biochemistry «Nonequilibrium Equality for Free Energy Differences» Physical Review Letters, 78(14), p.2690, 1997 A.I f(work) = f(δ Energy) 6

7 Fluctuation theorem The Second Law of Thermodynamics: Macroscopic processes irreversible Motion of molecules: Newton's equations reversible in time ΔS 0 F=ma Denis J. Evans The Australian National University, Prof., Research School of Chemistry «Probability of Second Law Violations in Shearing Steady States» Evans et al, Physical Review Letters, 71(15), p.2401, 1993 A.I Probability ( Δ S ) Probability ( -Δ S ) 7 grows exponentially with time

8 Fluctuation theorem Garry P. Morriss Debra J. Searles The University of New South Wales, Australia Griffith University, Australia E. G. D. Cohen Giovanni Gallavotti The Rockefeller University, USA Universita di Roma La Sapienza, Italy Gavin E. Crooks A.I 8 and others... Lawrence Berkeley National Lab., USA

9 Part A.II Crash course in statistical mechanics Distribution function Lyapunov exponent A.II 9

10 Distribution function Thermodynamic variables are averages of microscopic properties 3 2 k T = E kin {x 1, v 1 ; x 2, v 2 ; x N, v N } E i N particles: microscopic configuration k E k K energy intervals for N particles : distribution function Extensive properties: N E tot = A.II k 0 k 1 0 Ei i=1 K = k E k E k k=1 Ensemble averages: K E = 1 k E k E k K k =1

11 Lyapunov exponent t 0 t 0 Molecular motion reveals the similar behavior: dynamical systems Divirgense of particle's trajectory : t t e 0 Lyapunov exponent A.II p r o b a b i l i s t i c 11 d e s c r i p t i o n

12 Statistical mechanics Link between microscopic and macroscopic properties: Distribution function i, E kin Lyapunov exponent 0 Detailed balance P A B =P B A A.II 12

13 Part A.III Thermodynamics and its range of validity Equilibrium systems Fluctuations Non-equilibrium prcesses A.III 13

14 Equilibrium system There are configurations with the same distribution function M configurations with the same distribution function: M-1 M There are configurations with different distribution functions In equilibrium, the same distribution function belongs to the most of configurations Equilibtium state is described by this distribution function: the most probable distribution Meaningfull only for systems With large number of molecules, N With no external perturbations A.III i = 14 H 1 exp i Z kt Gibbs canonical distribution

15 Fluctuations Large number of molecules: Small number of molecules: All the distributions are incarnated equally often: A.III 15 there is no most probable distribution No way to introduce the state functions

16 Non-equilibrium processes Microscopic configuration evolves in time: non-equilibrium fluctuations Relaxation Steady states Time-dependent conservative Non-conservative Transition between steady states A.III 16 Aging state

17 Thermodynamics number of particles Global equilibrium Equilibrium thermodynamics Local equilibrium Fluctuations T,p Non-equilibrium thermodynamics T r, p r Thermodynamics? Fluctuations A.III 17 Newton's dynamics process rate

18 Part B.I Contents of the theorems Transient Fluctuation theorems Jarzynski equality Crooks Fluctuation theorem B.I 18

19 Transient Fluctuation theorems D. J. Evans, E. G. D. Cohen, G. P. Morris, D. J. Evans, D. Searles, G. Gallavotti and E. G. D. Cohen, G. Gallavotti and E. G. D. Cohen, Phys Rev Lett, 71(15), p.2401, 1993 Phys Rev E, 50(2), p.1645, 1994 Phys Rev Lett, 74(14), p.2694, 1995 J. of Stat Phys, 80, p.931, 1995 Dynamical systems Second law vs microscopic reversibility There are two kinds of microscopic trajectories: ordinary trajectories ΔS 0 0 anti-trajectories ΔS 0 P anti P ordinary ~e t dissipation Anti-trajectories are less mechanically stable, then their corresponding trajectories B.I 19 P = P e t

20 Jarzynski equality C. Jarzynski, Phys Rev Lett, 78(14), p.2690, 1997 C. Jarzynski, Phys Rev E, 56(5), p.5018, 1997 P W rev = F W irr = f x dx F f W1 Process average: 2 t W2 W K t WK 1 t same schedule: 1 t = 2 t = = K t diffrent work: W 1 W 2 W K 20 V K 1 W j K 1 e =e x B.I W W kt F kt W F

21 Crooks Fluctuation theorem G. E. Crooks, J. of Stat Phys, 90, p.1481, 1998 G. E. Crooks, Phys Rev E, 60(3), p.2721, 1999 G. E. Crooks, Phys Rev E, 61(3), p.2361, 2000 Path ensemble: Initial thermal equilibrium (canonial distribution) The process, perturbing from equilibrium Direction of the process: Forward (F) Reverse (R) 2 2 R t Q kt Any function defined for the process: Path function: R A t and A t Fluctuation theorem A F B.I t e W F H kt = A F t R P F =e P R Jarzynski equality e =e 21 F 1 [ F t ]= [ R t ] exp Detailed balance: 1 F t W W kt FH kt

22 Contents of the theorems Transient Fluctuation theorems P = P e t Jarzynski equality e =e W kt FH kt Family of theorems Reduce to the common expressions in linear regime Crooks Fluctuation theorem B.I 22 A t e W F H kt = A F t R

23 Part B.II Applications and experimental verification B.II 23

24 Applications Physical processes Colloids Biological machines Turbulent flow Chemical reactions B.II Energy conversion in ATP 24

25 MD Simulations Relaxation ln [ P t = P t = ] = B.II 25

26 Experiments pulling biomolecules: a bead in an optical trap W = f x dx B.II 26

27 Experiments J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002 Prerequisites: small number of molecules both, Eq and Neq regimes Expectations: WJE1 WJE2 WJE1 ΔF Conditions: 40 folding-unfolding cycles 7 datasets with different molecules Reversible work: slow rate B.II 27 folding and unfolding curves coincide

28 Part B.III Discussions and critique B.III 28

29 Relevance Aiming for a new understanding of Nature Does the macroscopic description contain all the necessary information? Mechanism of Life Arrow of Time A family of the relations must be treated together It is the consistency of different approaches, which matters a lot A complex verification is needed Is it a coincidence for special processes or a general result? Do experiments correspond to the required conditions? B.III 29 The physical meaning of the used quantities

30 Debates E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004 Received: 23 June 2004 Accepted: 29 June 2004 Published: 13 July 2004 C. Jarzynski, Received: Accepted: Published: J. of Stat Mech: T&E, P09005, August August September 2004 «The communities accepting the Jarzynski equality consists overwhelmingly of chemists and biophysicists, while the physicists have divided opinions... E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004 B.III 30»

31 Cohen arguments E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004 C. Jarzynski, J. of Stat Mech: T&E, P09005, 2004 The Jarzynski equality (JE) is not an equality in any mathematical sense, but can be a useful approximate equality in certain important fields, e.g. study of single molecules in solution Correct accounting for the heat exchange The system is subjected not only to the mechanical work, but also to the simultaneous energy exhange with the surroundings Temperature of the initial equilibrium state Usage of the temperature of the surroundings for every irreversible path e =e W kt FH kt makes no physical sense A rigorous derivation is possible only for «linear regime», which is already known B.III 31 Essentially reversible isothermal experiments were performed True irreversible processes, have so far not been considered experimentally

32 Vilar arguments J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, , 2008 L. Peliti, J. Horowitz and C. Jarzynski, J. M. G. Vilar and J. M. Rubi, J. of Stat Mech: T&E, P05002, 2008 Phys. Rev. Lett. 101, , 2008 (Comment) Phys. Rev. Lett. 101, , 2008 (Reply) JE is not general: there are systems, where it does not hold Harmonic oscilator W e W Jarzynski: e =1 The Jarzynski definition of the work is not general: Parameter א is not necessarily the (generalized) coordinate F =e W = dt H t Hamiltonian is defined up to an arbitrary time-dependent function JE holds, but not between the work and free energy Z t exp [ f x dx ] exp [ dh x, t ] = exp [ F Z t ] Z 0 B.III The experiments confirm JE beacuse of specialy chosen conditions Yet, the agreement is good, maily close to relatively slow perturbations 32

33 My arguments Why does an irreversible process average depends on an equilibrium state ( work vs free energy )? «The microscopic history of the system and environment will differ from one realization to the next, simply because the initial microstate differs from one realization to the next process average 1 A = N B.III N Ai i=1...» canonical average 1 A = N N Ai Ai i=1 diversity is not only due to initial configuration does ρ = 1 for an irreversible process? 33

34 Kulinskii arguments recent communications The definition of the work is misleading microscopic energy: H micro {x1, v 1 ; x 2, v 2 ; x N, v N } microscopic work: W micro= dh = H NB! equality for any process: molecules do not know about heat! macroscopic configuration: H macro {T, } W macro= force dx macroscopic work: Because of averaging over microscopic degrees of freedom we loose information: Q W micro W macro Jarzynski: e W macro =e F JE does not hold for a simple process W =0 F ~ln 34 no work on the system V2 V1 and = dt H W t Irreversible adiabatic expansion of ideal gas into vacuum increase of entropy and free energy e W e F

35 The End The story just begins, doesn't it? 35

36 Bibliography Books & Reviews: Kvasnikov. Thermodynamics and Statistical Physics. Editorial, 2002 Landau, Lifshitz. Statistical Physics. Pergamon Press, 1980 Rumer, Ryvkin. Thermodynamics, Statistical Physics and Kinetics. Nauka, 1972 Bustamante, Liphardt, Ritort. The Nonequilibrium Thermodynamics of Small Systems, Physics Today, p43, 2005 Evans, Searles. The Fluctuation Theorem, Advances in Physics, 51(7), p1529, 2002 Jarzynski. Nonequilibrium Fluctuations of a Single Biomolecule. Lecture Notes in Physics, 711, p201, Ritort. Nonequilibrium fluctuations in small systems: From physics to biology. arxiv, cond-mat: Articles: V. L. Kulinskii, Private communications, 2009 J. M. G. Vilar and J. M. Rubi, PRL 100, , 2008 L. Peliti, J. of Stat Mech: T&E, P05002, 2008 J. Horowitz and C. Jarzynski, PRL 101, , 2008 J. M. G. Vilar and J. M. Rubi, PRL 101, , 2008 A. Imparato and L. Peliti. arxiv:cond-mat/ v1 C. Jarzynski. PRE 73, , 2006 F. Douarche, S. Ciliberto, A. Petrosyan and I. Rabbiosi. EPL, 70(5), 2005, p593 C. Jarzynski. J. Stat. Mech.: Theor. Exp. (2004) P09005 E. G. D. Cohen and David Mauzerall. J. Stat. Mech.: Theor. Exp. (2004) P07006 G. Gallavotti. arxiv:cond-mat/ v1 J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002 C. Jarzynski. PNAS, 98(7), 2001, p3636 G. E. Crooks. PRE, 61(3), 2000, 2361 G. E. Crooks. PRE, 60(3) 1999, p2721 E. G. D. Cohen and G. Gallavotti. J of Stat Phys, Vol. 96, Nos. 5/6, 1999 G. E. Crooks. J of Stat Phys, Vol. 90, Nos. 5/6, 1998 C. Jarzynski. PRE, 56(5), p.5018, 1997 C. Jarzynski. PRL, 78(14), p.2690, 1997 D. J. Evans and D. J. Searles. PRE, 53(6), 1996, p52 G. Gallavotti and E. G. D. Cohen. J. of Stat Phys, 80, p.931, 1995 G. Gallavotti and E. G. D. Cohen. PRL, 74(14), p.2694, 1995 D. J. Evans, D. Searles. PRE, 50(2), p.1645, 1994 D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(21), p.3616, 1993 D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(15), p.2401, 1993 G. N. Bochkov and Yu.E. Kuzovlev. Physica 106A (1981) , Physica 106A (1981)

37 Detailed balance It is convenient to use propabilistic approach in stead of deterministic: x i t j P x t j =x i Newton's equations ma = F are time reversal: principle of detailed balance Probability ( A B ) = Probability ( B A ) state probability to be in state A transient probability to go from state A to state B transient probability to go from state B to state A state probability to be in state B P A x A,t A x A, t A ; x B, t B = x B,t B ; x A,t A P B x B,t B A.II 37 exp HA kt x A,t A ; x B, t B = x B, t B ; x A, t A exp HB kt