the fluctuation and nonequilibrium free energy theorems - theory & experiment the fluctuation...

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 is total (extensive) irreversible entropy

production rate/ and its time average is: , then

Formula is exact if time averages (0,t) begin from the initial phase , sampled from a given initial distribution . 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

f(Γ(0),0)


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

P xy,t

p(P xy, t )

lnp(P xy,t =A )p( Pxy,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.

• Imply 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 (Mukamel, 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


Maxwell on the Second Law

Hence the Second Law of thermodynamics is continually being violated and that to a considerable extent in any sufficiently small group of molecules belonging to any real body. As the number ofmolecules in the group is increased, the deviations from the meanof the whole become smaller and less frequent; and when the number is increased till the group includes a sensible portion of the body, the probability of a measurable variation from the mean occurring in a finite number of years becomes so small that it may be regarded as practically an impossibility.

J.C. Maxwell 1878


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 for a system in 3 Cartesian dimensions

The formal solution is:

∂f(Γ,t)∂t=−∂∂Γg[&Γf(Γ,t)]≡−iLf(Γ,t)

df(Γ, t)dt

=[∂∂t

+ &Γ(Γ)g∂∂Γ

]f(Γ, t) =−f(Γ, t)(Γ), ∀Γ, t

f(Γ(t), t) =exp[− ds0

t

∫ (Γ(s))]f(Γ(0),0)

(Γ) = −3Nresα (Γ)


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))ds

0

t

∫

=Ωtt≡Ωt

p(δVΓ (Γ(0),0))p(δVΓ (Γ

* (0),0))=

f(Γ(0),0)δVΓ (Γ(0),0)f(Γ* (0),0)δVΓ (Γ

* (0),0)

=f(Γ(0),0)f(Γ(t),0)

exp − (Γ(s))ds0

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


Thomson on reversibility

The instantaneous reversal of the motion of every moving particle of a system causes the system to move backwardseach particle along its path and at the same speed as before…

W. Thomson (Lord Kelvin) 1874(cp J. Loschmidt 1878)


Choose

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

The derivation is complete.

lnp(Ωt=A)p(Ωt=−A)=AtEvans Searles TRANSIENT FLUCTUATION THEOREM

p(δVΓ (Γ(0),0;Ωt(Γ(0)) =A))p(δVΓ (Γ

* (0),0))=

f(Γ(0),0)δVΓ (Γ(0),0)f(Γ* (0),0)δVΓ (Γ

* (0),0)

=f(Γ(0),0)f(Γ(t),0)

exp − (Γ(s))ds0

t

∫⎡⎣⎢

⎤⎦⎥

=exp[Ωt(Γ(0))] =exp[At]

δVΓ (Γ(0),0;Ω t (Γ(0)) = A ± δA)


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 is extensive).

...Ωt>0p(Ωt > 0)p(Ωt < 0)

⎡

⎣⎢

⎤

⎦⎥= e−Ωtt

Ωt>0

−1= e−Ωtt

Ωt<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.Note we can also derive the SLI from the Crooks Equality - later.

...Ωt>0 Ω(t)=0,∀t(Searles & Evans 2004).

Ωt = Ap(Ω t = A)( )dA−∞

∞

∫= Ap(Ω t = A) − Ap(Ω t = −A)( )dA

0

∞

∫= Ap(Ω t = A)(1− e−At )( )dA

0

∞

∫= Ω t (1− e−Ω t t )

Ω t >0≥ 0, ∀ t > 0


The NonEquilibrium Partition Identity (Carberry et al 2004).

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

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

exp(−Ωtt)=dAp(Ωt=A)exp(−At)−∞+∞∫=dAp(Ωt=−A)−∞+∞∫=dAp(Ωt=A)−∞+∞∫=1exp(−Ωtt) =1


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)Steady State ESFT

limt→ ∞

Pr(Ωt

ss =A)

Pr(Ωtss =−A)

=exp[At]


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)2AJFetσJ(t)2limFe→0JtFe≡−L(Fe=0)L(0)=βVdt0∞∫J(0)J(t)Fe=0Fe~t−1/2(Evans, Searles and Rondoni 2005).


The Dissipation Theorem

The initial distribution can be written in a quite arbitrary form, f(Γ,0)=exp[−F(Γ)]dΓ∫exp[−F(Γ)]. Using a Dyso ndecompositio nof propagato ,rs

exp[(A+B)t]=exp[At]+ds0t∫exp[As]Bexp[(A+B)(t−s)]. This is an identity at t= 0 and the LHS and RHS satisfy the same first order differ ential so the

Liouville operato r−iL(Γ) defined as

∂f(Γ,t)∂t=−∂∂Γ•[Γf(Γ,t)]≡−iL(Γ)f(Γ,t) can be written

exp[−(iL)t]=exp[−(iL+Λ)t]=exp[−iLt]−ds0t∫exp[−iLs]Λexp[(−iL)(t−s)].


The Dissipation Theorem -2

The time dependent distribution function can be written as, f(Γ,t)=exp[−ds0t∫(−s)]exp[−(F(Γ(−t))−F(Γ(0)))]f(Γ,0). From th edefinition o f the dissipatio nfunction it follows that, f(Γ,t)=exp[−ds0−t∫Ω(Γ(s))]f(Γ,0). By differentiation and reintegratio nwe can compute the nonlinear respons eof an arbitrary

phase functio nB as B(t)=ds0t∫Ω(0)B(s)]Fe. This is an exceedingly general form o f the Transien tTime Correlation Functio nexpressio nfor

the nonlinear respon .se If the initial distributio nis preserved by th efield free dynamics then

one can sho ,w B(t)=βVds0t∫J(0)B(s)]FeFe. that can be linearized to give Green-Kub o for th elinear respon :se limFe=0B(t)=βVds0t∫J(0)B(s)]Fe=0Fe.

the fluctuation and nonequilibrium free energy theorems - theory & experiment the fluctuation...

Documents

fluctuation theorems

nonequilibrium expressions

equilibrium evans

driven nonequilibrium

types of nonequilibrium

steady states evans

searles mittag

time averages