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