# Far-from-equilibrium measurements of thermodynamic length Page: 3 of 5

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3

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.

Wb,a[Ib,a] -Wa,b[Xa,b].

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)

lab

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

time reversal.

A key result in our development links two different

trajectory ensemble averagesK F[Xa,b] )ab

KED oTb])

(17)

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

intermediate time.

This result follows directly from the work fluctuation

relation [Eq. (13)], and the Markovian property of the

dynamics.K-Do,0[coa]1f[xa,]b (18)

Ket ha [tal i]

E-N, J [jfb a]

b,0Ketba [~af[ib a]),a

K-F[Xa,b a,b

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 [8] and Chelli et al. [28].

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

T- 1

s 8 d JS(,(xlAt), (x1At+1)) (19)

t=oand cumulative Jensen-Shannon divergence [16],

Jis K 8 JS(7r(xIAt),7r (xAt+1))

t-o(20)

Here, JS(pi, p2) is the Jensen-Shannon divergence [29, 30]

between two probability distributions pi and p2,(21)

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 [16].

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 [16].

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 [16]N

f(AFa,b) 1 ln

N

n=1 1+ e /+3Fa(23)

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 September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.