Far-from-equilibrium measurements of thermodynamic length Page: 1 of 5
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Far-from-Equilibrium Measurements of Thermodynamic Length
Edward H. Feng
College of Chemistry, University of California, Berkeley, Berkeley, California 94720, USA
Gavin E. Crooks
Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Dated: November 5, 2008)
Thermodynamic length is a path function that generalizes the notion of length to the surface
of thermodynamic states. Here, we show how to measure thermodynamic length in far-from-
equilibrium experiments using the work fluctuation relations. For these microscopic systems, it
proves necessary to define the thermodynamic length in terms of the Fisher information. Conse-
quently, the thermodynamic length can be directly related to the magnitude of fluctuations about
equilibrium. The work fluctuation relations link the work and the free energy change during an
external perturbation on a system. We use this result to determine equilibrium averages at inter-
mediate points of the protocol in which the system is out-of-equilibrium. This allows us to extend
Bennett's method to determine the potential of mean force, as well as the thermodynamic length,
in single molecule experiments.
PACS numbers: 05.70.Ln, 05.40.-aModern experimental techniques allow the manipula-
tion of single molecules and the measurement of the ther-
modynamic properties of microscopic systems [1 5]. For
example, Collin et al. [3] recently measured the work
performed on a single RNA hairpin as it was folded
and unfolded using optical tweezers. From these out-
of-equilibrium measurements, they extracted the equi-
librium free energy change using the recently discovered
work fluctuation relations [6 8]. [Eq. (13)] These rela-
tions, which connect the free energy change and the work
done on a system by an external perturbation, remain
valid no matter how far the system is driven away from
thermal equilibrium.
In this Letter, we will demonstrate that free energy is
not the only important quantity that can be extracted
from out-of-equilibrium work measurements; we can also
measure the thermodynamic length [9 16]. Thermody-
namic length is a path function that measures the dis-
tance along a path in thermodynamic state space. This
is in contrast to the free energy change, a state function
which depends only on the initial and final values of the
controllable parameters, and not on the path. Mathe-
matically, the thermodynamic length is defined by a Rie-
mannian metric on the manifold of equilibrium ensem-
bles [17, 18]. Among other useful physical properties, the
thermodynamic length bounds the dissipation of slow,
but finite time transformations [12, 14]. Moreover, the
ability to measure thermodynamic length and free energy
change from out-of-equilibrium measurements indicates
that these equilibrium properties influence the behavior
of driven systems even far-from-equilibrium.
Thermodynamic length was originally defined using
the second derivatives of a thermodynamic potential with
respect to its natural variables [9, 10]. However, this
definition only works for microscopic systems when the
controlled variables are intensive (e.g. temperature) [16].
To circumvent this restriction, herein we will redefinethe thermodynamic length in terms of Fisher informa-
tion [17, 19]. This approach is equivalent to the orig-
inal definition for large systems in the thermodynamic
limit [16, 18], but can also be applied, without restric-
tion, to microscopic systems, or to problems outside of
thermodynamics entirely.
Given a family of probability distributions r(xlA) for
outcomes x that vary smoothly with a collection of pa-
rameters A {A''}, the Fisher information matrix [20, 21]
isLAs(A) A dx &n(xrA) &lnr(xlA)
ak1I &AI(1)
The length of a path A(s) for s E [0, 1] in parameter
space measured using the Fisher metric (also known as
the Fisher-Rao, Rao or entropy differential metric) is [19]- 1/2
I [1 d I'A dAQ)2s
L dA'(s) (A(s)) dds ds.
-0 ((2)
The Fisher matrix I25 acts as a metric tensor and equips
the manifold of parameters with a Riemannian met-
ric [17, 19]. It is also useful to define a related quantity,
the Fisher divergence1 dA1(s) _____
j d z I2(A(s)) d s sThe length and divergence are connected by the relation
J ;> 2 due to the Cauchy-Schwarz inequality.
The Fisher metric can be applied to any family of prob-
ability distributions. Here, we focus on probability dis-
tributions of a system in thermal equilibrium. In the
canonical ensemble [22, 23], the probability of a micro-
state x is)E(x, A)} , (4)
(3)
7r~xIA exp{F(A)
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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/m1/1/: accessed May 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.