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The Jarzynski Equation and the Fluctuation Theorem The Jarzynski Equation and the Fluctuation Theorem Kirill Glavatskiy Trial lecture for PhD degree 24 September, NTNU, Trondheim The Jarzynski equation and the fluctuation theorem Fundamental Statiscical physics concepts Recents developments Practical applications Fluctuations 2 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part A.I «Characters in play» Scope of the theorems Main authors A.I 5 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Fluctuation theorem The Second Law of Thermodynamics: Δ S ≥ 0 Macroscopic processes — irreversible Motion of molecules: F = m a Newton's equations — reversible in time 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 Probability ( Δ S ) A.I grows exponentially with time Probability ( -Δ S ) 7 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Fluctuation theorem Garry P. Morriss Debra J. Searles The University of New South Wales, Griffith University, Australia Australia E. G. D. Cohen Giovanni Gallavotti The Rockefeller University, Universita di Roma La Sapienza, USA Italy Gavin E. Crooks and others... A.I Lawrence Berkeley National Lab., USA 8 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part A.II Crash course in statistical mechanics Distribution function Lyapunov exponent A.II 9 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Distribution function Thermodynamic variables are averages of microscopic properties 3 =〈 〉 2 k T E kin { } x1, v1 ; x2, v2 ; xN , vN E i N particles: microscopic configuration k Ek K energy intervals for N particles : distribution function Extensive properties: N K = ∑ = ∑ Etot Ei k Ek Ek i=1 k=1 A.II Ensemble averages: 2 k k 1 K 0 0 0 0 〈 〉 = 1 ∑ E k Ek E k K k=1 10 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Lyapunov exponent ≈ ≠ t 0 t 0 Molecular motion reveals the similar behavior: dynamical systems ≈ t Divirgense of particle's trajectory : t e 0 Lyapunov exponent A.II probabilistic description 11 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Statistical mechanics Link between microscopic and macroscopic properties: 〈 〉 Distribution function i , E kin Lyapunov exponent 0 = Detailed balance PA B PB A A.II 12 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part A.III Thermodynamics and its range of validity Equilibrium systems Fluctuations Non-equilibrium prcesses A.III 13 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Equilibrium system There are configurations with the same distribution function M configurations with the ⋯ same distribution function: 1 2 3 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 1 H i = exp− i Z kT — Gibbs canonical distribution 14 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Fluctuations Large number of molecules: Small number of molecules: All the distributions are incarnated equally often: No way to introduce A.III there is no most probable distribution the state functions 15 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 Aging state 16 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Thermodynamics number of Global equilibrium particles Equilibrium Local equilibrium thermodynamics Fluctuations T , p Non-equilibrium thermodynamics T r , pr Thermodynamics ? Fluctuations A.III Newton's dynamics process rate 17 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part B.I Contents of the theorems Transient Fluctuation theorems Jarzynski equality Crooks Fluctuation theorem B.I 18 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Transient Fluctuation theorems D. J. Evans, E. G. D. Cohen, G. P. Morris, Phys Rev Lett, 71(15), p.2401, 1993 D. J. Evans, D. Searles, Phys Rev E, 50(2), p.1645, 1994 G. Gallavotti and E. G. D. Cohen, Phys Rev Lett, 74(14), p.2694, 1995 G. Gallavotti and E. G. D. Cohen, J. of Stat Phys, 80, p.931, 1995 Dynamical systems Second law vs microscopic reversibility There are two kinds of microscopic trajectories: ordinary trajectories Δ S ≥ 0 0 anti-trajectories Δ S ≤ 0 − P anti ~e t Pordinary dissipation Anti-trajectories are less mechanically stable, then their corresponding trajectories B.I P = e t P− 19 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 f W =∫ f x dx ≥ F irr W V Process average: 1 t W 1 1 K 〚W 〛≡ ∑ W 2 t W 2 j K 1 ⋮ ⋮ ⋮ K t W K W F 〚 − 〛 − 0 1 2 kT = kT x e e = =⋯= same schedule: t t t B.I 1 2 K 〚W 〛≥ F ≠ ≠⋯≠ diffrent work: W 1 W 2 W K 20 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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) − F t R t Reverse (R) 1 2 2 1 = − − Q Detailed balance: [ t ] [ t ] exp F R k T Path function: Any function defined for the process: At and A−t R Fluctuation theorem PF = − − e F W F H P 〚 − 〛 〚 〛 R B.I kT = Jarzynski equality At e F A−t R F 〚 − W 〛 − H e kT =e kT F W 21 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Contents of the theorems Transient Fluctuation theorems P = e t P− Family of theorems Jarzynski equality W F − − H Reduce to 〚 kT 〛= kT e e the common expressions in linear regime Crooks Fluctuation theorem W − F 〚 − H 〛 〚 〛 kT = − B.I A t e F A t R 22 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part B.II Applications and experimental verification B.II 23 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Applications Physical processes Colloids Turbulent flow Biological machines Chemical reactions B.II Energy conversion in ATP 24 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy MD Simulations Relaxation Pt = ln[ ] = Pt =− B.II 25 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Experiments pulling biomolecules: a bead in an optical trap W =∫ f xdx B.II 26 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 folding and unfolding curves coincide 27 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Part B.III Discussions and critique B.III 28 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 The physical meaning of the used quantities 29 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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, J. of Stat Mech: T&E, P09005, 2004 Received: 6 August 2004 Accepted: 30 August 2004 Published: 21 September 2004 «… The communities accepting the Jarzynski equality consists overwhelmingly of chemists and biophysicists, while the physicists have divided opinions ... » B.III E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004 30 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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.
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