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The Fluctuation and NonEquilibrium Free Energy Theorems - Theory & Experiment.

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Denis J. Evans, Edie Sevick, Genmaio Wang, David Carberry, Emil Mittag and James Reid

Research School of Chemistry, Australian National University, Canberra, Australia

and

Debra J. Searles

Griffith University, Queensland, Australia

Other collaborators

E.G.D. Cohen, G.P. Morriss, Lamberto Rondoni

(March 2006)

The first statement of a Fluctuation Theorem was given by Evans, Cohen & Morriss, 1993. This statement was for isoenergetic nonequilibrium steady states.

If is total (extensive) irreversible entropy

production rate/ and its time average is: , then

Formula is exact if time averages (0,t) begin from the equilibrium phase . It is true asymptotically , if the time averages are taken over steady state trajectory segments. The formula is valid for arbitrary external fields, .

Evans, Cohen & Morriss, PRL, 71, 2401(1993).

- Show how irreversible macroscopic behaviour arises from time reversible dynamics.
- Generalize the Second Law of Thermodynamics so that it applies to small systems observed for short times.
- Implies the Second Law InEquality .
- Are valid arbitrarily far from equilibrium regime
- In the linear regime FTs imply both Green-Kubo relations and the Fluctuation dissipation Theorem.
- Are valid for stochastic systems (Lebowitz & Spohn, Evans & Searles, Crooks).
- New FT’s can be derived from the Langevin eqn (Reid et al, 2004).
- A quantum version has been derived (Monnai & Tasaki), .
- Apply exactly to transient trajectory segments (Evans & Searles 1994) and asymptotically for steady states (Evans et al 1993)..
- Apply to all types of nonequilibrium system: adiabatic and driven nonequilibrium systems and relaxation to equilibrium (Evans, Searles & Mittag).
- Can be used to derive nonequilibrium expressions for equilibrium free energy differences (Jarzynski 1997, Crooks).
- Place (thermodynamic) constraints on the operation of nanomachines.

Consider a system described by the time reversible thermostatted equations of motion (Hoover et al):

Example:

Sllod NonEquilibrium Molecular Dynamics algorithm for shear viscosity - is exact for adiabatic flows.

which is equivalent to:

(Evans and Morriss (1984)).

- The Liouville equation is analogous to the mass continuity equation in fluid mechanics.
- or for thermostatted systems, as a function of time, along a streamline in phase space:
- is called the phase space compression factor and for a system in 3 Cartesian dimensions
The formal solution is:

Deterministic, time reversible, homogeneous thermostats were simultaneously but independently proposed by Hoover and Evans in 1982. Later we realised that the equations of motion could be derived from Gauss' Principle of Least Constraint (Evans, Hoover, Failor, Moran & Ladd (1983)).

The form of the equations of motion is

a can be chosen such that the energy is constant or such that the kinetic energy is constant. In the latter case the equilibrium, field free distribution function can be proved to be the isokinetic distribution,

In 1984 Nosé showed that if a is determined as the time dependent solution of the equation

then the equilibrium distribution is canonical

We know that

The dissipation function is in fact a generalized irreversible entropy production - see below.

The Loschmidt Demon applies

a time reversal mapping:

Combining shows that

So we have the Transient Fluctuation Theorem (Evans and Searles 1994)

The derivation is complete.

- Isokinetic or Nose-Hoover dynamics/isokinetic or canonical ensemble
- Isoenergetic dynamics/microcanonical ensemble
- or
- (Note: This second equation is for steady states, the Gallavotti-Cohen form for the FT (1995).)
- Isobaric-isothermal dynamics and ensemble.
- (Searles & Evans , J. Chem. Phys., 113, 3503–3509 (2000))

Connection with Linear irreversible thermodynamics

In thermostatted canonical systems where dissipative field is constant,

So in the weak field limit (for canonical systems) the average dissipation function is equal to the “rate of spontaneous entropy production” - as appears in linear irreversible thermodynamics. Of course the TFT applies to the nonlinear regime where linear irreversible thermodynamics does not apply.

If denotes an average over all fluctuations in which the time integrated entropy production is positive, then,

gives the ratio of probabilities that the Second Law will be satisfied rather than violated. The ratio becomes exponentially large with increased time of violation, t, and with system size (since W is extensive).

(Searles & Evans 2004).

If denotes an average over all fluctuations in which the time integrated entropy production is positive, then,

If the pathway is quasi-static (i.e. the system is always in equilibrium):

The instantaneous dissipation function may be negative. However its time average cannot be negative.

For thermostatted systems the NonEquilibrium Partition Identity (NPI) was first proved by Evans & Morriss (1984). It is derived trivially from the TFT.

NPI is a necessary but not sufficient condition for the TFT.

(Evans, Searles and Rondoni 2006, Evans & Searles 2000).

At t=0 we apply a dissipative field to an ensemble of equilibrium systems. We assume that this set of systems comes to a nonequilibrium steady state after a time t. For any time t we know that the TFT is valid. Let us approximate

, so that

Substituting into the TFT gives,

In the long time limit we expect a spread of values for typical values of which scale as consequently we expect that for an ensemble of steady state trajectories,

We expect that if the statistical properties of steady state trajectory segments are independent of the particular equilibrium phase from which they started (the steady state is ergodic over the initial equilibrium states), we can replace the ensemble of steady state trajectories by trajectory segments taken from a single (extremely long) steady state trajectory.

This gives the Evans-Searles Steady State Fluctuation Theorem

(Evans, Searles and Rondoni 2005).

Thermostatted steady state . The SSFT gives

Plus Central Limit Theorem

Yields in the zero field limit Green-Kubo Relations

Note: If t is sufficiently large for SSFT convergence and CLT then is the largest field for which the response can be expected to be linear.

Jarzynski Equality (1997).

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).

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

systems are deterministic and canonical

Crooks proof:

• 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

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

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)

FT shows that entropy-consuming trajectories are observable out to 2-3 seconds in this experiment

Wang, Sevick, Mittag, Searles & Evans, Phys. Rev. Lett. (2002)

Wang et al PRL, 89, 050601(2002).

k1

trapping constant

k0

time

t=0

k1

k0

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

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))

•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.

k1

k0

trapping constant

.

t=0

t=Dk/k

time

undefined as the external field is not time-symmetric

quasi-static, limit

limit is capture

Crooks Relation

Jarzynski Relation

Fluctuation Theorem

(An extended Second Law-like theorem)

NonEquilibrium Partition Identity