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# Evans, Mol Phys, 20 ,1551(2003). PowerPoint PPT Presentation

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.

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

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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.

Then we use the NonEquilibrium Partition Identity to obtain the Jarzynski work

Relation:

### 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.

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

QED

### NEFER for thermal processes

Assume equations of motion

Then from the equation for the generalized “work”:

### Generalized “power”

Classical thermodynamics gives

• small system

• 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

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

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)

vopt= 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

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 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

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

•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.