Far-from-equilibrium measurements of thermodynamic length Page: 3 of 5
This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to Digital Library by the UNT Libraries Government Documents Department.
The following text was automatically extracted from the image on this page using optical character recognition software:
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
Here’s what’s next.
This article can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Article.
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 September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.