The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment.

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

andDebra J. Searles

Griffith University, Queensland, Australia

Other collaboratorsE.G.D. Cohen, G.P. Morriss, Lamberto Rondoni

(March 2006)


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 is total (extensive) irreversible entropy

production rate/ and its time average is: , then

Formula is exact if time averages (0,t) begin from the equilibrium phase . It is true asymptotically , if the time averages are taken over steady state trajectory segments. The formula is valid for arbitrary external fields, .

p( t A)

p( t A)exp[At]

t (1 t) ds0

t

(s)

kB

(0)

JFe V dVV (r) / kB

t

Fe


Evans, Cohen & Morriss, PRL, 71, 2401(1993).

P xy,t

p(P xy, t )

lnp(P xy,t A)

p(P xy ,t A)

AVt


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


Derivation of TFT (Evans & Searles 1994 - 2002)

Consider a system described by the time reversible thermostatted equations of motion (Hoover et al):

Example:

Sllod NonEquilibrium Molecular Dynamics algorithm for shear viscosity - is exact for adiabatic flows.

which is equivalent to:

(Evans and Morriss (1984)).

qi p i / m C i Fe

p i Fi D i Fe Sipi : Si 0,1; Sii N res

qi p i

m iy i

p i Fi ipyi p i

qi

Fi

m i(t)y i ( q i iyi )


The Liouville equation is analogous to the mass continuity equation in fluid mechanics.

or for thermostatted systems, as a function of time, along a streamline in phase space:

is called the phase space compression factor and for a system in 3 Cartesian dimensions

The formal solution is:

f(, t)

t

[ f(, t)] iLf(, t)

df(, t)

dt[

t

&()g

]f(, t) f(, t)(), , t

f((t), t) exp[ ds0

t

((s))]f((0),0)

() 3Nres()


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

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,

In 1984 Nosé showed that if is determined as the time dependent solution of the equation

then the equilibrium distribution is canonical

p i Fi pi

f() ~ ( p i2 / 2m 3NkBT / 2)exp[ (q) / kBT]

ddt

p i2 / 2m / 3NkBT / 2 1 / 2

f() ~ exp[ H0() / kBT]


We know that

The dissipation function is in fact a generalized irreversible entropy production - see below.

ds ((s)0

t

) lnf((0),0)

f((t),0)

((s))ds

0

t

t t t

p(V ((0),0))

p(V (* (0),0))

f((0),0)V ((0),0)

f(* (0),0)V (* (0),0)

f((0),0)

f((t),0)exp ((s))ds

0

t

exp[t ((0))]

The Dissipation function is defined as:


Phase Space and reversibility


The Loschmidt Demon applies a time reversal mapping:

(q,p) (q, p)

Loschmidt Demon


Combining shows that

So we have the Transient Fluctuation Theorem (Evans and Searles 1994)

The derivation is complete.

lnp(t A)

p(t A)At

Evans Searles TRANSIENT FLUCTUATION THEOREM

lnp(t A)

p(t A)ln

p(V ( i (0),0))i t ,iA

p(V ( i (0),0))

i t ,i A

ln

p(V ( i (0),0))i t ,iA

p(V ( i

* (0),0))i t ,iA

ln

p(V ( i (0),0))i t ,iA

exp[ t ((0))]p(V ( i (0),0))

i t ,iA

At


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

Isoenergetic dynamics/microcanonical ensemble

or

(Note: This second equation is for steady states, the Gallavotti-Cohen form for the FT (1995).)

Isobaric-isothermal dynamics and ensemble.

(Searles & Evans , J. Chem. Phys., 113, 3503–3509 (2000))

lnp(Jt A)

p(Jt A) AtFeV JFe V

dH0ad

dt

lnp(Jt A)

p(Jt A) AtFeV ln

p(t A)

p( t A) At JFe V

dH0ad

dt

lnp(Jt A)

p(Jt A) AtFeV JFe V

dI 0ad

dt


Consequences of the FT

Connection with Linear irreversible thermodynamics

In thermostatted canonical systems where dissipative field is constant,

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.

J FeV / Tsoi

J FeV / Tres O(Fe

4 )

O(Fe

4 )


The Integrated Fluctuation Theorem (Ayton, Evans & Searles, 2001).

If denotes an average over all fluctuations in which the time integrated entropy production is positive, then,

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).

... t 0

p(t 0)

p(t 0)

e t t

t 0

1

e t t

t 0 0


The Second Law Inequality

If denotes an average over all fluctuations in which the time integrated entropy production is positive, then,

If the pathway is quasi-static (i.e. the system is always in equilibrium):

The instantaneous dissipation function may be negative. However its time average cannot be negative.

... t 0

(t) 0, t

(Searles & Evans 2004).

t Ap(t A) dA

Ap(t A) Ap(t A) dA

0

Ap(t A)(1 e At ) dA

0

t (1 e t t )

t 00, t 0


The NonEquilibrium Partition Identity (Carberry et al 2004).

For thermostatted systems the NonEquilibrium Partition Identity (NPI) was first proved by Evans & Morriss (1984). It is derived trivially from the TFT.

NPI is a necessary but not sufficient condition for the TFT.

exp( t t) dA p( t A)exp( At)

dA p( t A)

dA p( t A)

1

exp( t t) 1


Steady state Fluctuation Theorem

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

, so that

Substituting into the TFT gives,

In the long time limit we expect a spread of values for typical values of which scale as consequently we expect that for an ensemble of steady state trajectories,

t ds0

(s) ds

t

(s) tss O() t t

ss O( / t)

Pr(t A)

Pr(t A)exp[At]

Pr(t

ss A O( / t))

Pr(tss A O( / t))

tss t 1/2

(Evans, Searles and Rondoni 2006, Evans & Searles 2000).


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

limt

Pr(t

ss A)

Pr(tss A)

exp[At O(1)]

exp[At], since At O(t1/2 )

limt

Pr(t

ss A)

Pr(tss A)

exp[At]

Steady State ESFT


FT and Green-Kubo Relations

Thermostatted steady state . The SSFT gives

Plus Central Limit Theorem

Yields in the zero field limit Green-Kubo Relations

Note: If t is sufficiently large for SSFT convergence and CLT then is the largest field for which the response can be expected to be linear.

t Jt VFe

lim( tFe

2 tc)

lnp(Jt A)

p(Jt A)

lim

( tFe

2 tc)

AVFe t, Fe2 t c

lim( tFe

2 tc)

lnp(J t ) A

p(J t ) A

lim

( tFe

2 tc)

2A JFe

tJ( t )2

limFe 0

J t Fe L(Fe 0)

L(0) V dt0

J(0)J(t) Fe 0

Fe ~ t 1/2

(Evans, Searles and Rondoni 2005).


NonEquilibrium Free Energy Relations

Equilibrium Helmholtz free energy differences can be computed nonequilibrium thermodynamic path integrals. For nonequilibrium isothermal pathways between two equilibrium states

implies,

NB is the difference in Helmholtz free energies, and if then JE KI

Crooks Equality (1999).

0

Jarzynski Equality (1997).

f(,0exp[ H0 ()] f(, texp[ H1()]

W(t) H1(t) H0 (0)] ds(s)0

t

exp[ W] exp[

A1 A0

pF (W B)

pR (W B)exp[ B


Equilibrium System 1 Equilibrium System 2

d0 (0)

d0 ()

M T: time reversal

mapping

d0T ()

d0T (0)

Forward, W() B dB

Crooks Relation:

Reverse, W() B dB

PrF (W B)

PrR (W B)e AeB Jarzynski Relation: e W

Fe A

NonEquilibrium Free Energy

Evans, Mol Phys, 20,1551(2003).


Crooks proof:

systems are deterministic and canonical

exp[ W(0 )]01

d0 f0 (0 ,0)exp[ H1(t) H0 (0)] ds(s)0

t

]

d0 f0 (0 ,0)f1(1

,0)d1z1

f0 (0 ,0)d0z0

z1

z0

d1 f1(1

,0) exp[ (A1 A0 )]

P01(W(0 ) a)

P1 0 (W(1

* ) a)

f0 (0 ,0)d0

f1(1*,0)d1

*

exp[W(0 )]z1

z0

exp[a (A1 A0 )]

Jarzynski Equality proof:


• single colloidal particle• position & velocity measured precisely• impose & measure small

forces

• small system• short trajectory• small external forces

Strategy of experimental demonstration of the FTs

. . . measure energies, to a fraction of , along paths

kBT


Optical Trap Schematic

Photons impart momentum to the particle, directing it towards the most intense part of the beam.

r

k < 0.1 pN/m, 1.0 x 10-5 pN/Å

Fopt kr


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)


As A=0,and FT and Crooks are equivalent

For the drag experiment . . . v

elocity

time

0

t=0

vopt = 1.25 m/sec

Wang, Sevick, Mittag, Searles & Evans, “Experimental Demonstration of Violations of the Second Law of Thermodynamics” Phys. Rev. Lett. (2002)

t > 0, work is required to translate the particle-filled trap

t < 0, heat fluctuations provide useful work

“entropy-consuming” trajectory

t W

t

1

kBTds

0

t

Fopt (s)gvopt


Wang et al PRL, 89, 050601(2002).

First demonstration of the (integrated) FT

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)

P t 0 P t 0

exp t t 0


For the Capture experiment . . .

Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)

k0k1

timet=0trapping constant

A lnk1k0k0

k1

A 0 t W

t (k1 k2 )1

2[q0 (t)2 q0 (0)2 ]

W

1

kBTds

0

t

&H


Histogram of t for Capture

k0 = 1.22 pN/m

k1 = (2.90, 2.70) pN/m

predictions from Langevin dynamics

Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)


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))

ITFT

NPI


-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

ln(N

i/N-i)

t

Capture --FT

Integration time is 26 mS


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.


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.4 -0.2 0 0.2 0.4

SSFT, Newtonian, t=0.25s

ln(N

i/N-i)

t

ss


-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

SSFT, Newtonian t=2.5s

ln(N

i/N-i)

t

ss


Test of NonEquilibrium Free Energy Theorems for Optical Capture.

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2

Workx

Fre

quen

cy

Work

Non Gaussian distribution ofNonEquilibrium work.


For the Ramp experiment . . .

trapping constant

timet=0 t=k/k

k0k1

.

t

W 1kBT

ds0

t Ý H

s 1

2kBTds Ý k s x2 s

0

t

undefined as the external field is not time-symmetric

quasi-static, limit

limit is capture

Ý k

Ý k 0

P(W ) (W A)


0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500

<exp( W)>

<exp( H)>

exp( A)

Number of Trajectories

Exact result

Jarzynski Relation

Equilibrium stat mech relation

Experimental Test on NonequilibriumFree Energy Relation for

optical capture experiment.

exp[ A] exp[ W] NE

Test of NE WR


New far-from-equilibrium theoremsin statistical physics

Crooks Relation

Jarzynski Relation

Fluctuation Theorem(An extended Second Law-like theorem)

NonEquilibrium Partition Identity

exp W exp A

P t A P t A

exp A

exp t 1

Pf W a Pr W a

exp a exp A

The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment.

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