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Kirill Glavatskiy
The Jarzynski Equation and
the Fluctuation Theorem
Trial lecture for PhD degree
24 September, NTNU, Trondheim
2The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
The Jarzynski equation and the fluctuation theorem
Fundamentalconcepts
Practical applications
Recents developments
Statiscical physics
Fluctuations
3The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Scope
the Jarzynski equality
the Fluctuation theorem
… 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
«
»
4The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Outline
The Jarzynski equality and the Fluctuation theorem
The contents of the theorems
Applications and experimental verification
Discussions and critique
General introduction
«Characters in play»
Crash course in statistical mechanics
Thermodynamics and its range of validity
5The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Part A.I
«Characters in play»
Scope of the theorems
Main authors
A.I
6The 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
Work ≥ Δ Energy
depends on the process path
f(Work) = f(Δ Energy)
Christopher Jarzynski
«Nonequilibrium Equality for Free Energy Differences»
University of Maryland, Assoc. Prof. Chemistry and Biochemistry
Physical Review Letters, 78(14), p.2690, 1997A.I
7The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Fluctuation theorem
grows exponentially with time
Probability ( Δ S )
Probability ( -Δ S )
«Probability of Second Law Violations in Shearing Steady States»
The Australian National University, Prof., Research School of Chemistry
Evans et al, Physical Review Letters, 71(15), p.2401, 1993
Denis J. Evans
Δ S ≥ 0 The Second Law of Thermodynamics:
Macroscopic processes — irreversible
Motion of molecules:
Newton's equations — reversible in time F = m a
A.I
8The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Fluctuation theorem
Giovanni GallavottiE. G. D. Cohen
Garry P. Morriss Debra J. Searles
Gavin E. Crooks
The Rockefeller University, USA
Universita di Roma La Sapienza, Italy
Lawrence Berkeley National Lab., USA
The University of New South Wales, Australia
Griffith University, Australia
and others...A.I
9The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Part A.II
Crash course in
statistical mechanics
Distribution function
Lyapunov exponent
A.II
10The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
0 20 k 0 k10
k Ek
Distribution function
k T =⟨E kin⟩32
Thermodynamic variables are averages of microscopic properties
{x1, v1 ; x2, v2 ; xN , vN }
N particles: microscopic configuration
E i
K energy intervals for N particles : distribution function
E tot = ∑i=1
N
E i = ∑k=1
K
k Ek Ek
⟨ E ⟩ = 1K ∑
k=1
K
k Ek E k
Ensemble averages:
Extensive properties:
A.II
11The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Lyapunov exponent
t ≈e t 0
p r o b a b i l i s t i c d e s c r i p t i o n
t≈0 t≠0
Molecular motion reveals the similar behavior: dynamical systems
Divirgense of particle's trajectory :
Lyapunov exponent
A.II
12The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Statistical mechanics
Distribution function
Lyapunov exponent
Detailed balance
Link between microscopic and macroscopic properties:
i ,⟨ E kin⟩
PA B=PB A
0
A.II
13The 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
14The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Equilibrium system
i=1Z
exp− H i
kT — Gibbs canonical distribution
Meaningfull only for systems
⋯1 2 3 M-1 M
M configurations with the same distribution function:
There are configurations with the same distribution function
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
With large number of molecules, N
With no external perturbations
A.III
15The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Fluctuations
Large number of molecules:
Small number of molecules:
All the distributions are incarnated equally often:
there is no most probable distribution No way to introduce
the state functionsA.III
16The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Non-equilibrium processes
Relaxation
Steady statesTime-dependent conservative
Non-conservative
Transition between steady states
Aging state
Microscopic configuration evolves in time: non-equilibrium fluctuations
A.III
17The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Newton's dynamics
Thermodynamics
process rate
number of
particles
Non-equilibriumthermodynamics
Local equilibrium
T r , pr
Global equilibrium
Equilibrium thermodynamics
T , pFluctuations
Thermodynamics ?
Fluctuations
A.III
18The 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
19The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Transient Fluctuation theoremsD. J. Evans, E. G. D. Cohen, G. P. Morris, Phys Rev Lett, 71(15), p.2401, 1993D. J. Evans, D. Searles, Phys Rev E, 50(2), p.1645, 1994G. Gallavotti and E. G. D. Cohen, Phys Rev Lett, 74(14), p.2694, 1995G. 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:
0
Panti
Pordinary
Anti-trajectories are less mechanically stable, then their corresponding trajectories
ordinary trajectoriesΔ S ≥ 0
anti-trajectoriesΔ S ≤ 0
P P −
= e t
~e− t
dissipation
B.I
20The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Jarzynski equalityC. Jarzynski, Phys Rev Lett, 78(14), p.2690, 1997C. Jarzynski, Phys Rev E, 56(5), p.5018, 1997
P
VW
W rev= F
W irr=∫ f xdx ≥ Ff
〚W 〛≡ 1K ∑
1
K
W j
〚e− W
kT 〛=e− F
kT
Process average:
10x
2
⋮
W 1
W 2
W K
⋮
1t
2t
K t
⋮
1t =2t =⋯=K t same schedule:
W 1 ≠W 2 ≠⋯≠W K diffrent work:〚W 〛≥ FB.I
21The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
W
F
R
F
Crooks Fluctuation theoremG. E. Crooks, J. of Stat Phys, 90, p.1481, 1998G. E. Crooks, Phys Rev E, 60(3), p.2721, 1999G. E. Crooks, Phys Rev E, 61(3), p.2361, 2000
1 2F t
2 1R −t
Path ensemble: Initial thermal equilibrium (canonial distribution)
The process, perturbing from equilibrium
Direction of the process: Forward (F)
Reverse (R)
[F t ]=[R−t ] exp− Qk T Detailed balance:
〚A t e−
W − F H
kT 〛F= A−t
〚 〛
〚 〛R
〚e−
WkT〛=e
− F H
kT
Jarzynski equality
PF PR−
=
Fluctuation theorem
e
B.I
Path function: At A−t Any function defined for the process: and
22The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Contents of the theorems
Transient Fluctuation theorems
Jarzynski equality
Crooks Fluctuation theorem
Family of
theorems
P P −
= e t
〚e− W
kT 〛=e−
F H
kT
B.I〚A t e
−W − F H
kT 〛F= A−t〚 〛R
Reduce to the common expressions
in linear regime
23The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Part B.II
Applicationsand
experimental verification
B.II
24The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Applications
Physical processes
Biological machines
Colloids Turbulent flow
Energy conversion in ATP
Chemical reactions
B.II
25The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
MD Simulations
Relaxation
ln[P t =P t =− ] =
B.II
26The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Experiments
pulling biomolecules: a bead in an optical trap
B.II
W =∫ f xdx
27The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Experiments
B.II
WJE1 ↔ WJE2
WJE1 ↔ ΔF
Expectations:
Conditions: 40 folding-unfolding cycles
7 datasets with different molecules
Reversible work: slow rate
folding and unfolding curves coincide
Prerequisites: small number of molecules
both, Eq and Neq regimes
J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002
28The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Part B.III
Discussions
and critique
B.III
29The 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?
The physical meaning of the used quantitiesB.III
30The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Debates
E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004Received: 23 June 2004Accepted: 29 June 2004Published: 13 July 2004
C. Jarzynski, J. of Stat Mech: T&E, P09005, 2004Received: 6 August 2004Accepted: 30 August 2004Published: 21 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
31The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Cohen arguments
B.III
E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004C. 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
Essentially reversible isothermal experiments were performed True irreversible processes, have so far not been considered experimentally
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
A rigorous derivation is possible only for «linear regime», which is already known
〚e−
WkT〛=e
− F H
kT Temperature of the initial equilibrium state
Usage of the temperature of the surroundings for every irreversible path
makes no physical sense
32The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Vilar arguments
B.III
J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601, 2008
L. Peliti, J. of Stat Mech: T&E, P05002, 2008J. 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
Harmonic oscilator ⟨e−W ⟩=1 ⟨e−W ⟩=e− F Jarzynski:
The Jarzynski definition of the work is not general:Parameter א is not necessarily the (generalized) coordinate
Hamiltonian is defined up to an arbitrary time-dependent function
W =∫ dt ∂∂ t
∂ H∂
JE holds, but not between the work and free energy
⟨exp [−∫ f x dx ]⟩≠⟨exp [−∫ dH x , t ]⟩= Z t Z 0
≠exp [− FZ t ]
The experiments confirm JE beacuse of specialy chosen conditions Yet, the agreement is good, maily close to relatively slow perturbations
33The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
My arguments
B.III
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
〚 A〛= 1N ∑
i=1
N
Ai ⟨ A⟩= 1N ∑
i=1
N
Ai Ai
canonical average
does ρ = 1 for an irreversible process? diversity is not only due to initial configuration
34The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Kulinskii arguments
The definition of the work is misleading
JE does not hold for a simple process
W =∫ dt ∂∂ t
∂ H∂
H micro{x1, v1 ; x2, v2 ; x N , vN }
W micro=∫ dH = Hmicroscopic energy: microscopic work: NB! equality for any process:
molecules do not know about heat!
macroscopic configuration: H macro{T ,}
W macro=∫ force ∗ dxmacroscopic work:
Because of averaging over microscopic degrees of freedom we loose information: Q
〚e−W macro〛=e− F
W micro≠W macro
Jarzynski: and
Irreversible adiabatic expansion of ideal gas into vacuum
W =0
F ~lnV 2
V 1
〚e−W 〛≠e− F
no work on the system
increase of entropy and free energy
recent communications
35The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
The End
The story just begins, doesn't it?
36The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Bibliography Books & Reviews:
Articles:
Kvasnikov. Thermodynamics and Statistical Physics. Editorial, 2002Landau, Lifshitz. Statistical Physics. Pergamon Press, 1980Rumer, Ryvkin. Thermodynamics, Statistical Physics and Kinetics. Nauka, 1972
Bustamante, Liphardt, Ritort. The Nonequilibrium Thermodynamics of Small Systems, Physics Today, p43, 2005Evans, Searles. The Fluctuation Theorem, Advances in Physics, 51(7), p1529, 2002Jarzynski. 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
V. L. Kulinskii, Private communications, 2009J. M. G. Vilar and J. M. Rubi, PRL 100, 020601, 2008L. Peliti, J. of Stat Mech: T&E, P05002, 2008J. Horowitz and C. Jarzynski, PRL 101, 098901, 2008 J. M. G. Vilar and J. M. Rubi, PRL 101, 098902, 2008A. Imparato and L. Peliti. arXiv:cond-mat/0706.1134v1C. Jarzynski. PRE 73, 046105, 2006F. Douarche, S. Ciliberto, A. Petrosyan and I. Rabbiosi. EPL, 70(5), 2005, p593C. Jarzynski. J. Stat. Mech.: Theor. Exp. (2004) P09005E. G. D. Cohen and David Mauzerall. J. Stat. Mech.: Theor. Exp. (2004) P07006G. Gallavotti. arXiv:cond-mat/0301172v1J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002C. Jarzynski. PNAS, 98(7), 2001, p3636G. E. Crooks. PRE, 61(3), 2000, 2361G. E. Crooks. PRE, 60(3) 1999, p2721E. G. D. Cohen and G. Gallavotti. J of Stat Phys, Vol. 96, Nos. 5/6, 1999G. E. Crooks. J of Stat Phys, Vol. 90, Nos. 5/6, 1998C. Jarzynski. PRE, 56(5), p.5018, 1997C. Jarzynski. PRL, 78(14), p.2690, 1997D. J. Evans and D. J. Searles. PRE, 53(6), 1996, p52G. Gallavotti and E. G. D. Cohen. J. of Stat Phys, 80, p.931, 1995G. Gallavotti and E. G. D. Cohen. PRL, 74(14), p.2694, 1995D. J. Evans, D. Searles. PRE, 50(2), p.1645, 1994D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(21), p.3616, 1993D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(15), p.2401, 1993G. N. Bochkov and Yu.E. Kuzovlev. Physica 106A (1981) 443-479, Physica 106A (1981) 480-520
37The Jarzynski equation and the Fluctuation theorem K. Glavatskiy
Detailed balance
P A xA , t A x A , t A ; xB , t B=xB , t B ; xA , t A P B xB , t B
exp− H A
k T xA , t A ; xB , tB= xB , tB ; xA , t A exp− H B
k T
Probability ( A → B ) = Probability ( B → A )
It is convenient to use propabilistic approach in stead of deterministic:
Newton's equations ma = F are time reversal: principle of detailed balance
xi t j Px t j=xi
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
A.II