NonEquilibrium Free Energy Relations and experiments - Lecture 3 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). Jarzynski Equality (1997).

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

Crooks proof: systems are deterministic and canonical Jarzynski Equality 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

Connection between FTs Jarzynski and Crooks. For stochastic systems the initial phase does not uniquely determine the “trajectory”, hence the specification of initial and final phases (0,t). Definitions:

Then Crooks: Evans_SearlesFT: Reid et al.: and: Reid et.al.: which gives a formal relationship between Crooks (therefore Jarzynski) and Evans and Searles FT. Summary of generalized NEFERs

Examples: Microcanonical ensembleCanonical ensemble where and, (Reid et.al. 2005) Summary of generalized NEFERs for common ensembles

NEFER for thermal processes Assume equations of motion Then from the equation for the generalized “work”:

Generalized “power” Classical thermodynamics gives

Comments on van Zon & Cohen heat function If then the phase space compression factor and the dissipation function are exactly related by the equation So when van Zon and Cohen introduce the heat function Q, for a single particle obeying the inertialess Langevin equation for a particle in an optical trap And thus when van Zon and Cohen show They show that for colloids whose underlying dynamics is Newtonian, GCFT does not hold.

Conflicting views on the Fluctuation Theorem “Sometimes change is a result of an illusory quest for novelty. It is quite possible to pursue blind alleys in physics, roads through an imaginary landscape, which lead nowhere.... So far there is no indication that something like pairing, or a Fluctuation Theorem, holds for a system with realistic nonequilibrium boundary conditions” p236, Time Reversibility, Computer Simulation and Chaos, W. G. Hoover, World Scientific “The TFT HAS to be satisfied, since it is in a way an identity...I feel that the verification of the TFT is almost more a check on the experiments than on the theorem, because it HAS to hold. Nevertheless it is very nice that you can do this!” E.G.D. Cohen, private correspondence, 30 June 2001.

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

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 pN/Å

Optical Tweezers Lab quadrant photodiode position detector sensitive to 15 nm, means that we can resolve forces down to pN or energy fluctuations of 0.02 pN nm (cf. k B T=4.1 pN nm)

As  A=0, and FT and Crooks are “equivalent” For the drag experiment... velocity time 0 t=0 v opt = 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

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, (2002)

For the Capture experiment... Carberry, Reid, Wang, Sevick, Searles & Evans, Phys. Rev. Lett. (2004) k0k0 k1k1 time t=0 trapping constant k0k0 k1k1

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:

Histogram of  t for Capture k 0 = 1.22 pN/  m k 1 = (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, (2004)) ITFT NPI

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 = k B T/. The trapping constant was determined to be k = 0.12 pN/µm and the relaxation time of the stationary system was  =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.