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likelihood. (See  for a clear and concise exposition of
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 .
1 pz 1 q
JS(p; q) = - p2 In i p2 + - q2 ln ,
2 (p2 + q2) 2 (p2 + q2)
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 . 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 
JS(p; p + dp) . dp1) (18)
If we compare with Eq. (7), we can see that in the con-
[ =/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 likelihood
free 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
 F. Weinhold, J. Chem. Phys. 63, 2479 (1975).
 G. Ruppeiner, Phys. Rev. A 20, 1608 (1979).
 P. Salamon and R. S. Berry, Phys. Rev. Lett. 51, 1127
 P. Salamon, J. Nulton, and E. Ihrig, J. Chem. Phys. 80,
 F. Schlgl, Z. Phys. B 59, 449 (1985).
 P. Salmon, J. P. Nulton, and R. S. Berry, J. Chem.
Phys. 82, 2433 (1985).
17 J. D. Nulton and P. Salamon, Phys. Rev. A 31, 2520
 J. Nulton, P. Salamon, B. Andresen, and Q. Anmin, J.
Chem. Phys. 83, 334 (1985).
 H. Janyszek and R. Mrugala, Phys. Rev. A 39, 6515
 R. Mrugala, J. D. Nulton, J. C. Schdn, and P. Salamon,
Phys. Rev. A 41, 3156 (1990).
[Li] D. Brody and N. Rivier, Phys. Rev. E 51, 1006 (1995).
 P. Salamon and J. D. Nulton, Europhys. Lett. 42, 571
 M. Schaller, K. H. Hoffmann, G. Siragusa, P. Salamon,
and B. Andresen, Comp. Chem. Eng. 25, 1537 (2001).
 J. D. Nulton and P. Salamon, J. Non-Equilib. Thermo-
dyn. 27, 271 (2002).
 C. H. Bennett, J. Comput. Phys. 22, 245 (1976).
 H. B. Callen, Thermodynamics and an Introduction to
Thermostatistics (Wiley, New York, 1985), 2nd ed.
 W. K. Wootters, Phys. Rev. D 23, 357 (1981).
 B. Andresen, R. S. Berry, R. Gilmore, E. Ihrig, and
P. Salamon, Phys. Rev. A 37, 845 (1988).
1191 T. M. Cover and J. A. Thomas, Elements of Information
Theory (Wiley, New York, 1991).
 C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).
 J. Burbea and C. R. Rao, J. Multivariate Anal. 12, 575
 M. R. Shirts, E. Bair, G. Hooker, and V. S. Pande, Phys.
Rev. Lett. 91, 140601 (2003).
 P. Maragakis, M. Spichty, and M. Karplus, Phys. Rev.
Lett. 96, 100602 (2006).
 J. Lin, IEEE Trans. Info. Theory 37, 145 (1991).
 J. Endres, D.M. Schindelin, IEEE Trans. Info. Theory
49, 1858 (2003).
 A. Majtey, P. W. Lamberti, M. T. Martin, and A. Plas-
tino, Eur. Phys. J. D 32, 413 (2005).
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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.