Equilibrium System 1Equilibrium System 2

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dΓ0 (0)

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dΓ0 (τ)

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M T: timereversal

mapping

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dΓ0T (τ)

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dΓ0T (0)

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Forward, ΔW(τ) =B±dB

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).

Crooks proof:

systems are deterministic and canonical

exp[−ΔW(Γ0 )] 0→1= dΓ0∫ f0 (Γ0 ,0)exp[−β[H1(t)−H0 (0)] + dsΛ(s)

0

t

∫ ]

= dΓ0∫ f0 (Γ0 ,0)f1(Γ1

∗,0)dΓ1∗z1

f0 (Γ0 ,0)dΓ0z0

=z1z0

dΓ1∗∫ f1(Γ1

∗,0) =exp[−β(A1 −A0 )]

P0→1(ΔW(Γ0 ) =a)P1→ 0 (ΔW(Γ

1

* ) =−a)=

f0 (Γ0 ,0)dΓ0

f1(Γ1* ,0)dΓ1

*

=exp[ΔW(Γ0 )]z1z0

=exp[a−β(A1 −A0 )]

Jarzynski Equality proof:

Jarzynski and NPI.

exp[−βΔW]F= exp[−βΔWrev −Ωt ] F

=exp[−βΔA] exp[−Ωt ]

=exp[−βΔA], NPI

Take the Jarzynski work and decompose into into its reversible and irreversible parts.Then we use the NonEquilibrium Partition Identity to obtain the Jarzynski workRelation:

Proof of generalized Jarzynski Equality.

For any ensemble we define a generalized “work” function as:

exp[ΔXτ (G)] ≡Pr1(G0 ,δG0 )Z(λ1 )Pr2 (Gτ ,δGτ )Z(λ2 )

=f1(G0 )δG0Z(λ1 )f2 (Gτ )δGτZ(λ2 )

We observe that the Jacobian gives the volume ratio:

∂Gτ

∂G0

=δGτ

δG0

=f1(G0 , 0)

f1(Gτ , τ)

We now compute the expectation value of the generalized work.

exp[−ΔXτ (G)] = dG0f1(G0 )∫f2 (Gτ )δGτZ(λ2 )f1(G0 )δG0Z(λ1 )

=Z(λ2 )Z(λ1 )

dGτf2 (Gτ )∫ =Z(λ2 )Z(λ1 )

If the ensembles are canonical and if the systems are in contact with heat reservoirs at the same temperature

exp[ΔXτ (G)] =

exp[−βH1(G0 )]exp[−βH2 (Gτ )]

f1(Gτ , τ)f1(G0 , 0)

⇒ ΔXτ (G) = β(H2 (Gτ ) − H1(G0 )) − βΔQτ (G0 )

= βΔWτ (G0 ) QED

NEFER for thermal processes

Assume equations of motion

Then from the equation for the generalized “work”:

Generalized “power”

Classical thermodynamics gives

• 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

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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

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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 ΔA=0,and FT and Crooks are “equivalent”

For the drag experiment...

ve

loci

ty

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

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Ωt =WΩt =

1

kBTds

0

t

∫ Fopt (s)• vopt

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. 89, 050601 (2002)

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P Ωt < 0( )P Ωt > 0( )

= exp −Ωt( ) Ωt >0

Histogram of Ωt 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(Ni/N-i)

Ωt

--Capture FT

26 Integration time is mS

Summary Exptl 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 τ =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