The Fluctuation and NonEquilibrium Free Energy Theorems - Theory

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The Fluctuation and NonEquilibrium Free Energy
Theorems- Theory Experiment.
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)
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Fluctuation Theorem (Roughly).
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, .
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Evans, Cohen Morriss, PRL, 71, 2401(1993).
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Why are the Fluctuation Theorems important?

• 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 FTs 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).

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Derivation of TFT (Evans Searles 1994 - 2002)
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)).
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• 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

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Thermostats
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
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Aside - Thermostats and Equilibrium
Consider m thermostats described by the
equations of motion where Einstein
notation is used, , is the
position of the i-th particle in the d-direction,
is the momentum of the ith particle in the
d-direction, and couple
the system with the external field, At
all m-thermostats that violate Gauss
Principle do not generate an equilibrium state
and, among m-thermostats that satisfy Gauss's
Principle to fix the m1 moment of the velocity
distribution, only the conventional Gaussian
isokinetic thermostat (m1) possesses an
equilibrium state.
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If the total system is Hamiltonian
Although ????????for any
Hamiltonian system, in general,
for any subsystem of a Hamiltonian system.
Further, if it is easy to show that if a
subsystem (ie the system of interest SOI)
looses heat at a rate to its
Hamiltonian surroundings (reservoir RES) and if
those surroundings have a heat capacity very much
greater than that of the system of interest, so
that the surroundings can be regarded as being in
thermodynamic equilibrium then, one can
show If the reservoir is thermostatted with a
m-thermostat with a large number of degrees of
freedom, then We see that the phase space
compression factor for the system of interest is
identical in the two cases. This confirms the
fact that the system of interest cannot know
how the heat is ultimately removed from its
vicinity.
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The Dissipation function is defined as
We know that The
dissipation function is in fact a generalized
irreversible entropy production - see below.
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Phase Space and reversibility
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Loschmidt Demon
The Loschmidt Demon applies a time reversal
mapping
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Evans Searles TRANSIENT FLUCTUATION THEOREM
Combining shows that So we
have the Transient Fluctuation Theorem (Evans and
Searles 1994) The derivation is complete.
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FT for different ergodically consistent bulk
ensembles driven by a dissipative field, Fe with
conjugate flux J.

• 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.

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Consequences of the FT
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.
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The Integrated Fluctuation Theorem (Ayton, Evans
Searles, 2001).
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).
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The Second Law Inequality
(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.
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The NonEquilibrium Partition Identity (Carberry
et al 2004).
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.
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Steady state Fluctuation Theorem
(Evans, Searles and Rondoni 2006, Evans Searles
2000).
At t0 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,
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Steady State ESFT
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
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FT and Green-Kubo Relations
(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.
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NonEquilibrium Free Energy Relations
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).
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Evans, Mol Phys, 20,1551(2003).
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Connection between FTs Jarzynski and Crooks.
Definitions
For stochastic systems the initial phase does not
uniquely determine the trajectory, hence the
specification of initial and final phases (0,t).
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Jarzynski Equality proof
systems are deterministic and canonical
Crooks proof
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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.
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Examples
Microcanonical ensemble Canonical ensemble
where
,
and
(Reid et.al. 2005)
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Jarzynski GK and SLI.
If
ie NPI.
then
So if we take the time time weak field limit and
assume a finite decay time for correlations, we
expect Gaussian statistics. This further implies
the FT but there is no need for ergodic
consistency.
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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.
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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 1999. 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.
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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
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Optical Trap Schematic
r
Photons impart momentum to the particle,
directing it towards the most intense part of the
beam.
k lt 0.1 pN/?m, 1.0 x 10-5 pN/Å
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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. kBT4.1 pN nm)
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For the drag experiment . . .
velocity
As DA0, and FT and Crooks are equivalent
Wt gt 0, work is required to translate the
particle-filled trap Wt lt 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)
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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. (2002)
Wang et al PRL, 89, 050601(2002).
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For the Capture experiment . . .
k1
k0
Carberry, Reid, Wang, Sevick, Searles Evans,
Phys. Rev. Lett. (2004)
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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
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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)
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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))
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Experimental 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/ltr2gt. 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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Test of NonEquilibrium Free Energy Theorems for
Optical Capture.
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For the Ramp experiment . . .
k1
k0
trapping constant
.
t0
tDk/k
time
undefined as the external field is not
time-symmetric
quasi-static, limit
limit is capture
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Test of NE WR
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New far-from-equilibrium theoremsin statistical
physics
Crooks Relation
Jarzynski Relation
Fluctuation Theorem (An extended Second Law-like
theorem)
NonEquilibrium Partition Identity