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. 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
F Temperature of the initial equilibrium state − W − H 〚 kT 〛= kT Usage of the temperature of the surroundings for every irreversible path e e makes no physical sense
A rigorous derivation is possible only for «linear regime», which is already known
B.III Essentially reversible isothermal experiments were performed True irreversible processes, have so far not been considered experimentally
31 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Vilar arguments
J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601, 2008
L. Peliti, J. of Stat Mech: T&E, P05002, 2008 J. Horowitz and C. Jarzynski, Phys. Rev. Lett. 101, 098901, 2008 (Comment) J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 101, 098902, 2008 (Reply)
JE is not general: there are systems, where it does not hold 〈 −W 〉= 〈 −W 〉= − F Harmonic oscilator e 1 Jarzynski: e e
∂ ∂ The Jarzynski definition of the work is not general: =∫ H W dt ∂ ∂ is not necessarily the (generalized) coordinate t א Parameter Hamiltonian is defined up to an arbitrary time-dependent function
JE holds, but not between the work and free energy Z t 〈exp[−∫ f xdx ]〉≠〈exp[−∫ dH x ,t]〉= ≠exp [− F t ] Z 0 Z
B.III The experiments confirm JE beacuse of specialy chosen conditions Yet, the agreement is good, maily close to relatively slow perturbations
32 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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 canonical average N N 〚 〛= 1 ∑ 〈 〉= 1 ∑ A Ai A Ai Ai N i=1 N i=1
B.III diversity is not only due to initial configuration does ρ = 1 for an irreversible process?
33 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Kulinskii arguments
recent communications The definition of the work is misleading { } microscopic energy: H micro x1, v1 ; x2, v2 ; x N , vN =∫ = NB! equality for any process: microscopic work: W micro dH H molecules do not know about heat! { } macroscopic configuration: H macro T , =∫ ∗ macroscopic work: W macro force dx
W ≠W Because of averaging over microscopic degrees of freedom micro macro we loose information: Q ∂ ∂ −W − = H Jarzynski: 〚 macro 〛= F and W ∫ dt e e ∂ t ∂
JE does not hold for a simple process Irreversible adiabatic expansion of ideal gas into vacuum W =0 no work on the system V F ~ln 2 increase of entropy and free energy V 1 〚 −W 〛 − F e ≠e 34 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy The End
The story just begins, doesn't it?
35 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy 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, 2007. Ritort. Nonequilibrium fluctuations in small systems: From physics to biology. arXiv, cond-mat:0705.0455 Articles: V. L. Kulinskii, Private communications, 2009 J. M. G. Vilar and J. M. Rubi, PRL 100, 020601, 2008 L. Peliti, J. of Stat Mech: T&E, P05002, 2008 J. Horowitz and C. Jarzynski, PRL 101, 098901, 2008 J. M. G. Vilar and J. M. Rubi, PRL 101, 098902, 2008 A. Imparato and L. Peliti. arXiv:cond-mat/0706.1134v1 C. Jarzynski. PRE 73, 046105, 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/0301172v1 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) 443-479, Physica 106A (1981) 480-520
36 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Detailed balance
It is convenient to use propabilistic approach in stead of deterministic:
= xi t j P x t j xi
Newton's equations ma = F are time reversal: principle of detailed balance
Probability ( A → B ) = Probability ( B → A )
state probability transient probability transient probability state probability to be to go to go to be in state A from state A to state B from state B to state A in state B = P A x A ,t A x A ,t A ; x B ,t B xB ,t B ; x A ,t A P B xB ,t B
A.II H H exp− A x ,t ; x ,t = x ,t ; x ,t exp− B k T A A B B B B A A k T
37 The Jarzynski equation and the Fluctuation theorem K. Glavatskiy