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Evans, Mol Phys, 20 ,1551(2003). Jarzynski Equality proof:. systems are deterministic and canonical. Crooks proof:. Jarzynski and NPI. Take the Jarzynski work and decompose into into its reversible and irreversible parts.

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


Jarzynski equality proof
Jarzynski Equality proof:

systems are deterministic and canonical

Crooks proof:


Jarzynski and npi
Jarzynski and 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 work

Relation:


Proof of generalized jarzynski equality
Proof of generalized Jarzynski Equality.

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

We observe that the Jacobian gives the volume ratio:


We now compute the expectation value of the generalized work
We now compute the expectation value of the generalized work.

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

QED


Nefer for thermal processes
NEFER for thermal processes work.

Assume equations of motion

Then from the equation for the generalized “work”:


Generalized power
Generalized “power” work.

Classical thermodynamics gives


• small system work.

• short trajectory

• small external forces

Strategy of experimental demonstration of the FTs

• single colloidal particle

• position & velocity measured precisely

• impose & measure small forces

. . . measure energies, to a fraction of , along paths


Optical trap schematic
Optical Trap Schematic work.

r

Photons impart momentum to the particle, directing it towards the most intense part of the beam.

k < 0.1 pN/m, 1.0 x 10-5 pN/Å


Optical tweezers lab
Optical Tweezers Lab work.

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)


v work.opt= 1.25mm/sec

0

t=0

time

For the drag experiment...

velocity

As DA=0,

and FT and Crooks are “equivalent”

Wt > 0, work is required to translate the particle-filled trap

Wt < 0, heat fluctuations provide useful work

“entropy-consuming” trajectory

Wang, Sevick, Mittag, Searles & Evans,

“Experimental Demonstration of Violations of the Second Law of Thermodynamics”Phys. Rev. Lett. (2002)


First demonstration of the integrated ft
First demonstration of the ( work.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)


Histogram of work.Wt for Capture

predictions from Langevin dynamics

k0 = 1.22 pN/mm

k1 = (2.90, 2.70) pN/mm

Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004)


NPI work.

ITFT

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))


Summary exptl tests of steady state fluctuation theorem
Summary Exptl Tests of Steady State Fluctuation Theorem work.

•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.


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