# 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 September 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.