Far-from-equilibrium measurements of thermodynamic length Page: 3 of 5
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is the work transfered to the system during the for-
ward process [24, 27]. The dissipation Da,b[xa,b]
j3(W,b[Xa,b] AFa,b), the irreversible increase in en-
tropy along the forward trajectory, is proportional to
the difference between the work and the free energy
change. Note that work, free energy change and dissi-
pation are all odd functionals under a time reversal, e.g.
We can express the trajectory ensemble average of an
arbitrary trajectory dependent function F[xa,b], starting
from thermal equilibrium as
K.F[Xa,b a b K Pa,b[xa,b] -F[Xa,b] (15)
and similarly for the conjugate process
Kf[Ib,a]b a Pb,a[b,a] .F[ b,a] (16)
in which F[Ib,a] -F[Xa,b] is an even functional under
A key result in our development links two different
trajectory ensemble averages
K F[Xa,b] )ab
where 0 < a < b < T. Given a protocol AT we can
extract the value of a trajectory ensemble average over a
subinterval Aa,b, as if the system began in equilibrium at
an intermediate time a, by re-weighting the observations
by the exponential of the dissipation from the initial to
This result follows directly from the work fluctuation
relation [Eq. (13)], and the Markovian property of the
Ket ha [tal i]
E-N, J [jfb a]
Ketba [~af[ib a]),a
We truncate the time interval of the trajectory ensemble
average using the stochastic property in Eq. (10), ap-
ply a time reversal with the work fluctuation theorem in
Eq. (13), truncate again, and apply a second time rever-
sal. This result generalizes previous trajectory ensemble
averages of Hummer and Szabo  and Chelli et al. .
We can now use this relation to extract the thermo-
dynamic length from far-from-equilibrium experiments.
The discrete time analogs of the Fisher length and diver-
gence are the cumulative Jensen-Shannon length
s 8 d JS(,(xlAt), (x1At+1)) (19)
and cumulative Jensen-Shannon divergence ,
Jis K 8 JS(7r(xIAt),7r (xAt+1))
Here, JS(pi, p2) is the Jensen-Shannon divergence [29, 30]
between two probability distributions pi and p2,
JS 2 - p2 )ln 1 (x)
p22 [p i(x)+p2(x)]
+ - ) p2(x) In l .z(x
2 X z[p1(x + p2x]
The Jensen-Shannon length is less than the Fisher length
Ljs < , and approaches equality as the step size along
the path decreases .
We can use the contracted trajectory average Eq. (17),
and the canonical probabilities [Eq. (4)] to write the
the Jensen-Shannon divergence between any pair of time
points along the path
JS(rt, t+1) =2 z e Dot ln 1 + cDtt~l (22)
+ 1 + i-vT14 In
+ - + e+ Dtlln)~
as a trajectory average of the dissipation D along the for-
ward and reverse protocols. While JS(rt, rt+1) is defined
in terms of averages over equilibrium probability distri-
butions, it can be related to trajectory ensemble averages
of processes driven arbitrary far-from-equilibrium. The
derivation of Eq. (22) requires the time reversal symme-
try in Eq. (12).
We now encounter an apparent complication. The dis-
sipation Dab - (Wa,b AFb) depends on both the
work and the free energy. Therefore, we must also de-
termine the potential of mean force, the free energy as
a function of A, along the entire path. This problem
of extracting free energy profiles from out-of-equilibrium
work measurements (rather than just the difference in
free energy between the initial and final ensembles) has
attracted recent attention [28, 31, 32]. Here, we will solve
this problem by adapting Bennett's maximum likelihood
method [5, 33 35], which, as we shall see, is intimately
linked to the thermodynamic divergence .
Suppose we have taken measurements of the work dur-
ing N repetitions of a protocol Aab and another N mea-
surements from the conjugate protocol Aba. Each rep-
etition begins in thermal equilibrium with the control
parameter fixed at Aa or Ab. Then the Bennett log-
likelihood that the free energy change AFa,b has a par-
ticular value is 
f(AFa,b) 1 ln
n=1 1+ e /+3Fa
e-Do, a[o,2 a
\ / 0,T
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Feng, Edward H. & Crooks, Gavin E. Far-from-equilibrium measurements of thermodynamic length, article, November 5, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc898172/m1/3/: accessed December 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.