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 RondoniOther contributors:Gallavotti, Spohn, Lebowitz, Bonetto, Garrido, Chernov, Ciliberto, Laroche, Segre, Maes, Kurchan, Jarzynski, Crooks...(Jan 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(Σt=A)p(Σt=−A)=exp[At]Σt≡(1t)ds0t∫Σ(s)€

kB

€

Γ(0)

Σ =−βJFe V = dVV∫ σ(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) ⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

=−βAγVt


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.

Ωt ≥ 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=pi/m+CigFe&pi=Fi+DigFe−αSipi:Si=0,1;Sii∑=Nres &qi=pim+iγyi&pi=Fi−iγpyi−αpi &&qi=Fim+iγδ(t)yi−α(&qi−iγyi)


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

∂f(Γ,t)∂t=−∂∂Γg[&Γf(Γ,t)]≡−iLf(Γ,t) dfdt=[∂∂t+&Γγ∂∂Γ]f=−f=−3Nresα


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

α 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 α is determined as the time dependent solution of the equation

then the equilibrium distribution is canonical

&pi=Fi−αpif(Γ)~δ(pi2/2m−3NkBT/2)exp[−F(q)/kBT]∑dαδt=pi2/2m∑()/3NkBT/2()−1⎡⎣⎤⎦/t2f(Γ)~exp[−H0(Γ)/kBT]


We know that

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

ds Ω(Γ(s)0

t

∫ )≡lnf(Γ(0),0)f(Γ(t),0)

⎛⎝⎜

⎞⎠⎟− (Γ(s))δs

0

t

∫

=Ωtt≡Ωt

p(δV Γ (Γ(0),0))p(δV Γ (Γ

*(0),0))=

f(Γ(0),0)δV Γ (Γ(0),0)f(Γ*(0),0)δV Γ (Γ

*(0),0)

=f(Γ(0),0)f(Γ(t),0)

exp − (Γ(s))δs0

t

∫⎡⎣⎢

⎤⎦⎥

=exp[Ωt(Γ(0))]

The Dissipation function is defined as:


Phase Space and reversibility


The Loschmidt Demon applies a time reversal mapping: Γ=(q,p)→Γ∗=(q,−p)Loschmidt Demon


Combining shows that

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

The derivation is complete.

lnp(Ωt=A)p(Ωt=−A)=lnp(δVΓ(Γi(0),0))iΩt,i=A∑p(δVΓ(Γi(0),0))iΩt,i=−A∑=lnp(δVΓ(Γi(0),0))iΩt,i=A∑f(Γi(t),0)f(Γi(0),0)exp(Γi(s))δs0t∫⎡⎣⎢⎤⎦⎥p(δVΓ(Γi(0),0))iΩt,i=A∑=Atlnp(Ωt=A)p(Ωt=−A)=AtTFT


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.

Σ=−JFeV/Tsoi=−JFeV/Tres+O(Fe4)=Ω+O(Fe4)


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 Ω is extensive).

p(Ωt>0)p(Ωt<0)⎡⎣⎢⎤⎦⎥=e−ΩttΩt>0−1=e−ΩttΩt<0>0...Ωt>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.

...Ωt>0Ωt=Ap(Ωt=A)()dA−∞∞∫=Ap(Ωt=A)−Ap(Ωt=−A)()dA0∞∫=Ap(Ωt=A)(1−e−At)()dA0∞∫=Ωt(1−e−Ωtt)Ωt>0≥0,∀t>0Ω(t)=0,∀t(Searles & Evans 2004).


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(−Ωtt)=δAp(Ωt=A)exp(−At)−∞+∞∫=δAp(Ωt=−A)−∞+∞∫=δAp(Ωt=A)−∞+∞∫=1exp(−Ωtt) =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 t. 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,

Ωt+t=ds0τ∫Ω(s)+dsττ+t∫Ω(s)=Ωtss+O(τ)Ωt+t=Ωtss+O(τ/t)Pr(Ωt+t=A)Pr(Ωt+t=−A)=exp[At]=Pr(Ωtss=A+O(t/t))Pr(Ωtss=−A+O(t/t))Ωtsst−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(Ωtss=A)Pr(Ωtss=−A)=exp[At+O(1)]=exp[At],sinceAt=O(t1/2)limt→∞Pr(Ωtss=A)Pr(Ωtss=−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.

Ωt=−βJtVFelim(t→∞Fe2t=c)lnp(βJt=A)p(βJt=−A)⎛⎝⎜⎞⎠⎟=−lim(t→∞Fe2t=c)AVFet,Fe2t=clim(t→∞Fe2t=c)lnp(Jt)=Ap(Jt)=−A⎛⎝⎜⎞⎠⎟=lim(t→∞Fe2t=c)2AJFetsJ(t)2limFe→0JtFe≡−(Fe=0)L(0)=βVδt0∞∫J(0)J(t)Fe=0Fe~t−1/2(Evans, Searles and Rondoni 2005).


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 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)=−AtFeβV−JFe V ≡ dH0ad

dtlnp(Jβt=A)p(Jβt=−A)=−AtFeVlnp(t=A)p(t=−A)=−At−JFe V ≡ dH0ad

dtlnp(Jt=A)p(Jt=−A)=−AtFeβV−JFe V ≡ dI 0ad

dt


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).

f(Γ,0)∼exp[−βH1(Γ)]→f(Γ,t)∼exp[−βH2(Γ)]βΔW(t)≡β[H2(t)−H1(0)]−dsΛ(s)0t∫exp[−βΔΩ]=exp[−βΔA]pF(ΔΩ=B)pR(ΔΩ=−B)=exp[−βΔA−βB]ΔA=A2−A1 ΔA=0≡Jarzynski Equality (1997).


Equilibrium System 1Equilibrium System 2

€

δΓ0 (0)

€

δΓ0 (t)

€

MT : time reversαlmαppinγ

€

δΓ0T (t)

€

δΓ0T (0)

€

Forwαrδ, ΔΩ (t)=B±δB

Crooks Relation: Reverse,ΔW(τ)=−BmdBPrF(ΔW=B)PrR(ΔW=−B)=e−βΔAeβB⇒Jarzynski Relation: e−βΔWF=e−βΔANonEquilibrium Free Energy

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


Experimental Confirmation


ηsphere > ηIntensity gradient of light

Photons impart momenta to the sphere in thedirection of the intensity gradient

change in momentaof rays:net momentum changeon particle:

Optical Trap Schematic


Optical Tweezers Lab


Transient Fluctuations in Brownian Friction

time Verification of integrated TFT using Optical Tweezers.0 1 2 301 p(Ωt<0)p(Ωt>0),exp[−Ωtt]Ωt>0€

Ω t =(tkBT)−1 δsvopt • Fopt(s)0

t

∫

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


Optical Capture of a Brownian Bead. - TFT, NPI

For a sudden isothermal change of strength in an optical trap, the dissipation function is:

Note: as expected, So the TFT becomes:

€

Ω t = 1tβ ds (k1 − k 2)q 0 (s)⋅p 0(s)

m ⎡ ⎣ ⎢

⎤ ⎦ ⎥0

t

∫

=1tβ ds (k1 − k 2)

ddsq 0

2 (s) / 2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥0

t

∫

= 1tβ(k1 − k 2) 1

2[q 0( t) 2 − q 0(0)2 ] Ωt>0,∀tp[(k2−k1)12[q0(t)2−q0(0)2]=B]p[(k2−k1)12[q0(t)2−q0(0)2]=−B]=exp[βB]


0250500750

10001250

-3-2-10 1 2 3 4 5Ωt

Frequency

025050075010001250

0250500750100012500250

50075010001250

A histogram showing the distribution of the dissipation function, , evaluated over 3300 experimental trajectories at times t = 0.002s, t = 0.02s and t = 0.2s after the trap strength is increased from k0 = 1:22 pN/ mm to k1 (kx1 ; ky1)=(2.90, 2.70 pN/ mm). The inset shows the Langevin-predicted distributions, P( ), for similar conditions. The range of the abscissa in the figure and inset are identical.

ΩtΩt


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(Ni/N-i)

Ωt

Cαpture --FT

Inteγrαtion time is 26 mΣ


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 t =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(Ni/N-i)

Ωtss


-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

SSFT, Newtonian t=2.5s

ln(Ni/N-i)

Ωtss


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

Frequency

Work

Non Gaussian distribution ofNonEquilibrium work.


0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500

<exp(−βΔΩ)><exp(−βΔH)>exp(−βΔA)

Numβer of Trαjectories

Exαct result

Jαrzynski Relαtion

Equiliβrium stαt mecη relαtion

Experimentαl Test on NonequiliβriumFree Enerγy Relαtion foropticαl cαpture experiment.exp[−βΔA]=exp[−βΔW]NE

Test of NE WR


Logical Structure of the Fluctuation Theorem

Deterministic thermostat Newton’s EquationsCausality(convenient but unnecessary)

Initial distribution Liouville EquationCentral Limit

TheoremFluctuation TheoremandNonequilibrium free energy theorem

Second Law Inequality, NPI, Integrated FT, Green-Kubo,

(weak fields)Linear irreversible Thermodynamics (weak fields)

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

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