# Measuring Thermodynamic Length Page: 4 of 4

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4

likelihood. (See [22] for a clear and concise exposition of

this result.)

Rather than using this free entropy measurement to es-

timate the mean change in the collective variables using

Eq. (14), we will instead show that the Bennett likelihood

is directly related to the thermodynamic divergence. If

we insert the Gibbs ensemble [Eq. (1)] into the log like-

lihood, then in the large sample limit, we find that the

likelihood scales as

[(Ap12) ~ 2K (JS(p1; p2) - 1n2) (16)

where JS(pi;p2) is the Jensen-Shannon divergence, the

mean of the relative entropy of each distribution to the

mean distribution [24].

1 pz 1 q

JS(p; q) = - p2 In i p2 + - q2 ln ,

2 (p2 + q2) 2 (p2 + q2)

(17)

The minimum divergence is zero for identical distribu-

tions and the maximum is ln 2. The square root of the

Jensen-Shannon divergence is a metric between proba-

bility distributions [25]. However, unlike a Riemannian

metric, the Jensen-Shannon metric space is not an in-

trinsic length space. There may not be a mid point b be-

tween points a and c such that d(a, b) + d(b, c) d(a, c)

and consequentially we cannot naturally measure path

lengths. However, on any metric space we can define a

new intrinsic metric by measuring the distance along con-

tinuous paths. The Jensen-Shannon divergence between

infinitesimally different distributions is [26]

1

JS(p; p + dp) . dp1) (18)

If we compare with Eq. (7), we can see that in the con-

tinuum limit[ =/Jd JS

and z= 8 f dJS .

The induced Jensen-Shannon metric is proportional to

the thermodynamic (entropy differential) metric, and

the induced Jensen-Shannon divergence is proportional

to the thermodynamic divergence. Consequentially, the

square root of Jensen-Shannon divergence between two

thermodynamic states gives a lower bound on the ther-

modynamic length of any path between those same

states, and the Jensen-Shannon divergence is a lower

bound to the thermodynamic divergence.

To summarize, we can measure the thermodynamic

length and minimum thermodynamic divergence along

a path in thermodynamics state space by adapting Ben-

nett's method. We perform a series of equilibrium simu-

lations along the path and find the maximum likelihoodfree entropy change [Eq. (15)] and Jensen-Shannon di-

vergence [via Eq. (16)] between neighboring ensembles.

The cumulative Jensen-Shannon metric along the path

provides a lower bound to the thermodynamic length

[Eq. (19)] and a lower bound to the minimum divergence

of the path [via Eq. (9)]. This procedure is then repeated

with finer discretizations of the path, until the estimates

of divergence and length converge.

This research was supported by the Department of En-

ergy, under contract DE-AC02-05CH11231.

* Electronic address: GECrooks@lbl.gov

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Crooks, Gavin E. Measuring Thermodynamic Length, article, September 7, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc926490/m1/4/: accessed April 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.