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)
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 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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
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.
Ωt ≥ 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):
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 Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
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
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]
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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:
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Phase Space and reversibility
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
The Loschmidt Demon applies a time reversal mapping: Γ=(q,p)→Γ∗=(q,−p)Loschmidt Demon
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
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,
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 Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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 Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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 Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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).
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
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Experimental Confirmation
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
Optical Tweezers Lab
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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).
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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]
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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 Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
-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
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 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.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
-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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
SSFT, Newtonian t=2.5s
ln(Ni/N-i)
Ωtss
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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.
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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
The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment
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)