The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Fluctuation and NonEquilibrium Free Energy Theorems
- Theory & Experiment.
Denis J. Evans, Edie Sevick, Genmaio Wang, David Carberry, Emil Mittag and James Reid
Research School of Chemistry, Australian National University, Canberra, Australia
and
Debra J. Searles
Griffith University, Queensland, Australia
Other collaborators
E.G.D. Cohen, G.P. Morriss, Lamberto Rondoni
(March 2006)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Fluctuation Theorem (Roughly).
The first statement of a Fluctuation Theorem was given by Evans, Cohen &
Morriss, 1993. This statement was for isoenergetic nonequilibrium steady
states.
If
  JFe V   dV (r) / k B is total (extensive) irreversible entropy
V
t

production rate/ kB and its time average is:  t  (1 t) ds (s) , then
0
p( t  A)
 exp[At]
p( t  A)
Formula is exact if time averages (0,t) begin from the equilibrium phase (0).
It is true asymptotically t  , if the time averages are taken over steady state
trajectory segments. The formula is valid for arbitrary external fields, . F e
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Evans, Cohen & Morriss, PRL, 71, 2401(1993).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Why are the Fluctuation Theorems important?
•
Show how irreversible macroscopic behaviour arises from time reversible dynamics.
•
Generalize the Second Law of Thermodynamics so that it applies to small systems observed for short times.
•
Implies the Second Law InEquality .
•
Are valid arbitrarily far from equilibrium regime
•
In the linear regime FTs imply both Green-Kubo relations and the Fluctuation dissipation Theorem.
•
Are valid for stochastic systems (Lebowitz & Spohn, Evans & Searles, Crooks).
•
New FT’s can be derived from the Langevin eqn (Reid et al, 2004).
•
A quantum version has been derived (Monnai & Tasaki), .
•
Apply exactly to transient trajectory segments (Evans & Searles 1994) and asymptotically for steady states
(Evans et al 1993)..
•
Apply to all types of nonequilibrium system: adiabatic and driven nonequilibrium systems and relaxation to
equilibrium (Evans, Searles & Mittag).
•
Can be used to derive nonequilibrium expressions for equilibrium free energy differences (Jarzynski 1997,
Crooks).
•
Place (thermodynamic) constraints on the operation of nanomachines.
t  0, t,N
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Derivation of TFT (Evans & Searles 1994 - 2002)
Consider a system described by the time reversible thermostatted equations of motion (Hoover et al):
qi  p i / m  C i Fe
p i  Fi D i Fe  Si pi : Si  0,1;
S
i
 N res
i
Example:
Sllod NonEquilibrium Molecular Dynamics algorithm for shear viscosity - is exact for adiabatic flows.
p
qi  i  iy i
m
p i  Fi  ipyi  p i
which is equivalent to:
qi 
Fi
 i(t)y i  (q i  iyi )
m
(Evans and Morriss (1984)).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Liouville equation is analogous to the mass continuity equation in fluid mechanics.
f(,t)


[f(,t)]  iLf(,t)
t

or for thermostatted systems, as a function of time, along a streamline in phase space:
df(, t)
 &

 [  ()g
]f(, t)  f(, t)(), , t
dt
t

 is called the phase space compression factor and for a system in 3 Cartesian dimensions
()  3N res ()
The formal solution is:
t
f((t), t)  exp[  ds ((s))]f((0), 0)
0
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Thermostats
Deterministic, time reversible, homogeneous thermostats were simultaneously but independently proposed
by Hoover and Evans in 1982. Later we realised that the equations of motion could be derived from Gauss'
Principle of Least Constraint (Evans, Hoover, Failor, Moran & Ladd (1983)).
The form of the equations of motion is
p i  Fi  pi
 can be chosen such that the energy is constant or such that the kinetic energy is constant. In the latter
case the equilibrium, field free distribution function can be proved to be the isokinetic distribution,
f() ~ (p 2i / 2m  3Nk BT / 2)exp[(q) / k BT]
In 1984 Nosé showed that if  is determined as the time dependent solution of the equation
d

dt 
p
2
i

/ 2m / 3Nk B T / 2 1 / 
2
then the equilibrium distribution is canonical f() ~ exp[H 0 ( ) / k B T]
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Dissipation function is defined as:
t
 f((0), 0) 
ds
((s)
)

ln

 f((t), 0)   0 ((s))ds
0
t
 t t  t
We know that
p(V ((0), 0))
f((0), 0)V ((0), 0)

p(V ( * (0), 0)) f( * (0), 0)V ( * (0), 0)

t
f((0), 0)
exp    ((s))ds 
 0

f((t), 0)
 exp[ t ((0))]
The dissipation function is in fact a generalized irreversible entropy production - see below.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Phase Space and reversibility
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Loschmidt Demon
The Loschmidt Demon applies
a time reversal mapping:

  (q,p)   (q,p)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Evans Searles TRANSIENT FLUCTUATION THEOREM
Combining shows that
ln

p( t  A)
 ln
p( t  A)
i  t ,i A

p(V ( i (0), 0))


 ln
p(V ( i (0), 0))
i  t ,i A

p(V ( i (0), 0))
i  t ,i A
exp[ t ((0))]p(V ( i (0), 0))
i  t ,i A
 At
So we have the Transient Fluctuation Theorem (Evans and Searles 1994)
ln
The derivation is complete.
p(t  A)
 At
p(t  A)
*
i
p(V ( (0), 0))
i  t ,i A
i  t ,i  A
 ln

p(V ( i (0), 0))

The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
FT for different ergodically consistent bulk ensembles driven by a dissipative field, Fe
with conjugate flux J.
Isokinetic or Nose-Hoover dynamics/isokinetic or canonical ensemble
p(Jt  A)
ln
 AtFeV
p(Jt  A)
ad
dH 0
JFe V 
dt
Isoenergetic dynamics/microcanonical ensemble
ln
p(Jt  A)
p(Jt  A)
 AtFe V
or
p(t  A)
ln
 At
p( t  A)
ad
dH 0
JFe V 
dt
(Note: This second equation is for steady states, the Gallavotti-Cohen form for the FT (1995).)
Isobaric-isothermal dynamics and ensemble.
p(Jt  A)
ln
 AtFeV
p(Jt  A)
ad
dI
JFe V  0
dt
(Searles & Evans , J. Chem. Phys., 113, 3503–3509 (2000))
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Consequences of the FT
Connection with Linear irreversible thermodynamics
In thermostatted canonical systems where dissipative field is constant,
   J Fe V / Tsoi
  J Fe V / Tres  O(Fe4 )
   O(Fe4 )
So in the weak field limit (for canonical systems) the average dissipation function is equal to the “rate of
spontaneous entropy production” - as appears in linear irreversible thermodynamics. Of course the TFT
applies to the nonlinear regime where linear irreversible thermodynamics does not apply.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Integrated Fluctuation Theorem (Ayton, Evans & Searles, 2001).
If ...  t 0 denotes an average over all fluctuations in which the time integrated entropy production is positive,
then,
 p( t  0) 
 t t

 e
 p( t  0) 
1
 t 0
 e t t
 t 0
0
gives the ratio of probabilities that the Second Law will be satisfied rather than violated. The ratio becomes
exponentially large with increased time of violation, t, and with system size (since  is extensive).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Second Law Inequality
(Searles & Evans 2004).
If ...  t 0 denotes an average over all fluctuations in which the time integrated entropy production is positive,
then,

Ap(
  A p(
  A p(
t  

 A) dA
t
 A)  A p( t  A) dA
t
 A)(1 e  At ) dA

0

0

t

  t (1 e t t )

 t 0
 0,  t  0
If the pathway is quasi-static (i.e. the system is always in equilibrium): (t)  0, t
The instantaneous dissipation function may be negative. However its time average cannot be negative.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The NonEquilibrium Partition Identity (Carberry et al 2004).
exp(t t)  1
For thermostatted systems the NonEquilibrium Partition Identity (NPI) was first proved by Evans &
Morriss (1984). It is derived trivially from the TFT.
exp(t t) 











dA p( t  A)exp(At)
dA p( t  A)
dA p( t  A)  1
NPI is a necessary but not sufficient condition for the TFT.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Steady state Fluctuation Theorem
(Evans, Searles and Rondoni 2006, Evans & Searles 2000) .
At t=0 we apply a dissipative field to an ensemble of equilibrium systems. We assume that this set of
systems comes to a nonequilibrium steady state after a time . For any time t we know that the TFT is
valid. Let us approximate

t   ds (s)  
0
t

ss
ss
ds (s)  t  O() , so that t  t  O( / t)
Substituting into the TFT gives,
ss
Pr( t  A  O( / t))
Pr(t  A)
 exp[At] 
Pr(t  A)
Pr(sst  A  O( / t))
ss
In the long time limit we expect a spread of values for typical values of t which scale as
we expect that for an ensemble of steady state trajectories,
1/2
t
consequently
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Steady State ESFT
lim
t
Pr(sst  A)
ss
Pr(t  A)
 exp[At  O(1)]
 exp[At], since At  O(t1/ 2 )
We expect that if the statistical properties of steady state trajectory segments are independent of the particular
equilibrium phase from which they started (the steady state is ergodic over the initial equilibrium states), we
can replace the ensemble of steady state trajectories by trajectory segments taken from a single (extremely
long) steady state trajectory.
This gives the Evans-Searles Steady State Fluctuation Theorem
ss
lim
t
Pr( t  A)
Pr(sst  A)
 exp[At]
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
FT and Green-Kubo Relations
(Evans, Searles and Rondoni 2005).
Thermostatted steady state t  J t VFe. The SSFT gives
 p(Jt  A) 
2
lim ln 
  lim AVFe t, Fe t  c

( t
( t
 p(Jt  A) 
2
2
F e tc)
F e tc)
Plus Central Limit Theorem
2A J F
 p(J t )  A 
e
lim ln 

lim
2

( t
(
t
tJ ( t)
 p(J t )  A  2
2
F e tc)
F e tc)
Yields in the zero field limit Green-Kubo Relations lim J t Fe  L(Fe  0)
Fe 0

L(0)  V  dt J(0)J(t)
Fe 0
0
Note: If t is sufficiently large for SSFT convergence and CLT then
expected to be linear.
Fe ~ t
1/2
is the largest field for which the response can be
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
NonEquilibrium Free Energy Relations
Jarzynski Equality (1997).
Equilibrium Helmholtz free energy differences can be computed nonequilibrium thermodynamic path
integrals. For nonequilibrium isothermal pathways between two equilibrium states
f(,0  exp[H0 ()]  f(, t  exp[H1 ()]
t
W(t)  H1 (t)  H 0 (0)]   ds (s)
0
implies,
exp[W]  exp[
NB   A1  A0 is the difference in Helmholtz free energies, and if
  0 then JE
Crooks Equality (1999).
p F (W  B)
 exp[  B
p R (W  B)
 KI
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
NonEquilibrium Free Energy
M T : time reversal
Forward, W()  B  dB
mapping
d0 (0)
d0 ()
T
d0 (0)
T
d0 ()
Equilibrium System 1
Crooks Relation:
Reverse, W()  B dB
Equilibrium System 2
PrF (W  B)
A B
e
e  Jarzynski Relation:
PrR (W  B)
W
e
A
F
e
Evans, Mol Phys, 20,1551(2003).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Jarzynski Equality proof:
systems are deterministic and canonical
exp[W( 0 )]
t
  d 0 f0 ( 0 , 0)exp[H1 (t)  H 0 (0)]   ds (s)]
01
0
f1 ( 1 , 0)d 1z1
  d 0 f0 ( 0 , 0)
f0 ( 0 , 0)d 0 z 0

z1


d
f
(
1 1
1 , 0)  exp[(A1  A 0 )]

z0
Crooks proof:
P01 (W( 0 )  a) f0 ( 0 , 0)d 0

P10 (W( *1 )  a) f1 ( 1* , 0)d 1*
 exp[W( 0 )]
z1
 exp[a  (A1  A 0 )]
z0
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Strategy of experimental demonstration of the FTs
• small system
• short trajectory
• small external forces
• single colloidal particle
• position & velocity
measured precisely
• impose & measure small
forces
. . . measure energies, to a fraction of kBT , along paths
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Optical Trap Schematic
r
Photons impart momentum to the
particle, directing it towards
the most intense part of the beam.
Fopt  kr
k < 0.1 pN/m, 1.0 x 10-5 pN/Å

The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Optical Tweezers Lab
quadrant photodiode position detector sensitive to 15 nm, means that
we can resolve forces down to 0.001 pN or energy fluctuations of
0.02 pN nm (cf. kBT=4.1 pN nm)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
velocity
For the drag experiment . . .
vopt = 1.25 m/sec
0
t=0
1 t
t 
ds Fopt (s)gv opt

0
k BT
time
As A=0, t  W
and FT and Crooks are equivalent
t > 0, work is required to translate the particle-filled trap

t < 0, heat fluctuations provide useful work
“entropy-consuming” trajectory
Wang, Sevick, Mittag, Searles & Evans,
“Experimental Demonstration of Violations of the Second Law of Thermodynamics” Phys. Rev. Lett. (2002)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
First demonstration of the (integrated) FT
Pt  0
 exp t   0
t
Pt  0

FT shows that entropy-consuming trajectories are
observable out to 2-3 seconds in this experiment
Wang, Sevick, Mittag, Searles & Evans, Phys. Rev. Lett. (2002)
Wang et al PRL, 89, 050601(2002).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
k0
k1
trapping constant
For the Capture experiment . . .
k1
A  ln
k0
 t=0
k1
k0
time
1
2
2
t  (k1  k 2 ) [q0 (t)  q0 (0) ]
2 
A  0  t  W
1 t &H
W 
ds 

k BT 0

Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Histogram of t for Capture
predictions from
Langevin dynamics
k0 = 1.22 pN/m
k1 = (2.90, 2.70) pN/m
Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
NPI
ITFT
The LHS and RHS of the Integrated Transient Fluctuation Theorem (ITFT) versus time, t. Both sets of data
were evaluated from 3300 experimental trajectories of a colloidal particle, sampled over a millisecond time
interval. We also show a test of the NonEquilibrium Partition Identity.
(Carberry et al, PRL, 92, 140601(2004))
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
4
Capture --FT
3
In tegra tion tim e is 26 mS
2
i
-i
ln(N /N )
1
0
-1
-2
-3

t
-4
-4
-3
-2
-1
0
1
2
3
4
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Experimental Tests of Steady State Fluctuation Theorem
• Colloid particle 6.3 µm in diameter.
• The optical trapping constant, k, was determined by applying the equipartition theorem: k =
kBT/<r2>.
•The trapping constant was determined to be k = 0.12 pN/µm and the relaxation time of the
stationary system was  =0.48 s.
• A single long trajectory was generated by continuously translating the microscope stage in a
circular path.
• The radius of the circular motion was 7.3 µm and the frequency of the circular motion was 4
mHz.
• The long trajectory was evenly divided into 75 second long, non-overlapping time intervals,
then each interval (670 in number) was treated as an independent steady-state trajectory from
which we constructed the steady-state dissipation functions.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
2
SSFT, Ne wtonian, t=0.25s
1.5
1
ln(N i/N -i)
0.5
0
-0.5
-1
-1.5

-2
-0.4
-0.2
0
ss
t
0.2
0.4
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
3
SSFT, Ne wtonian t=2.5 s
2
ln(N i/N -i)
1
0
-1
-2

ss
t
-3
-3
-2
-1
0
1
2
3
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Test of NonEquilibrium Free Energy Theorems for Optical Capture.
400
Non Gaussian distribution of
NonEquilibrium work.
350
300
Workx
Frequency
250
200
150
100
50
0
0
0.2
0.4
0.6
0.8
Work
1
1.2
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
k0
k1
trapping constant
For the Ramp experiment . . .
t=0
.
t=k/k
time
t undefined as the external field is not time-symmetric
1 t ÝH
1 t
W

 ds
 ds kÝs x 2 s
k BT 0
 s 2kBT 0
quasi-static, kÝ 0 limit
P(W )   (W  A)
kÝ
  limit is capture
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Test of NE WR
1
Experimental Test on Nonequilibrium
Free Energy Relation for
optical capture experiment.
0.9
Exact result
0.8
Equilibrium stat mech relation
0.7
0.6
<exp(W)>
<exp(H)>
exp(A)
Jarzynski Relation
0.5
exp[A]  exp[W]
NE
Number of Trajectories
0.4
0
100
200
300
400
500
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
New far-from-equilibrium theorems
in statistical physics
Pf W  a
 exp a exp A
Pr W  a
Crooks Relation
exp W   exp A
Jarzynski Relation

Fluctuation Theorem 
(An extended Second Law-like theorem)
NonEquilibrium Partition Identity

P t  A
P t  A
 exp A
exp t   1