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Measuring thermodynamic length

Gavin E. Crooks*

Physical Bioscience Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

(Dated: February 5, 2008)

Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among

other interesting properties, this metric asymptotically bounds the dissipation induced by a finite

time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon diver-

gence, Fisher information and Rao's entropy differential metric. Therefore, thermodynamic length

is of central interest in understanding matter out-of-equilibrium. In this paper, we will consider how

to define thermodynamic length for a small system described by equilibrium statistical mechanics

and how to measure thermodynamic length within a computer simulation. Surprisingly, Bennett's

classic acceptance ratio method for measuring free energy differences also measures thermodynamic

length.

PACS numbers: 05.70.Ln, 05.40.-aINTRODUCTION

Thermodynamic length is a natural measure of the dis-

tance between equilibrium thermodynamic states [1, 2, 3,

4, 5, 6, 7, 8, 9, 10, 11], which equips the surface of thermo-

dynamic states with a Riemannian metric and defines the

length of a quasi-static transformation as the number of

natural fluctuations along that path. Unlike the entropy

or free energy change, which are state functions, the ther-

modynamic length explicitly depends on the path taken

through thermodynamic state space. Thermodynamic

length is of fundamental interest to the generalization of

thermodynamics to finite time (rather than infinity slow)

transformations. Minimum distance paths are geodesics

on the Riemannian manifold and minimize the dissipa-

tion for slow, but finite time transformations [3, 8]. These

insights have been employed to optimize fractional distil-

lation and other thermodynamic processes [12, 13, 14].

The study of thermodynamic length has largely been

restricted to the field of macroscopic, endoreversible ther-

modynamics. However, there are deep connections be-

tween thermodynamic length, information theory and the

statistical physics of small systems far-from-equilibium.

In this paper we will consider the most appropriate def-

inition of thermodynamic length for small systems and

how to measure this distance in a computer simula-

tion. These considerations reveal a surprising connection

between thermodynamic length, Jensen-Shannon diver-

gence and Bennett's acceptance ratio method for free en-

ergy calculations [15]. Bennett's method is an optimal

measure of free energy differences, but it also indirectly

places a lower bound on the thermodynamic length be-

tween neighboring thermodynamic states.

THERMODYNAMIC LENGTH

Consider a physical system, possible microscopically

small, in equilibrium with a large thermal reservoir. Theconfigurational probability distribution is given by the

Gibbs ensemble, [16]p~rjA) 1 -/3H(xA)

1 -A(t)X (x)

-Z(1)

where x is the configuration, t is time, j3 1/kBT is the

reciprocal temperature (T) of the environment in natu-

ral units, (kB is the Boltzmann constant), Z is the par-

tition function, and H is the Hamiltonian of the sys-

tem. This total Hamiltonian is split into a collection of

collective variables X and conjugate generalized forces

A2, 3H A 2(t)X2(x). We use the Einstein convention

that repeated upper/lower indices are implicitly summed.

The sub-Hamiltonians X are time-independent functions

of the configurations, whereas the conjugate variables A

are time dependent and configuration independent. Note

that the conjugate variables include a factor of inverse

temperature.

The A's are the experimentally controllable parameters

of the system and define the accessible thermodynamic

state space. For example, in the isothermal-isobaric en-

semble we have X {U, V} and A {j3, 3p}, where U is

the internal energy, V is the volume and p is the external

pressure. Modern experimental techniques have broad-

ened the range of controllable parameters beyond those

considered in standard thermodynamics. For instance,

optical tweezers can apply a constant force to the ends of

a single DNA molecule. The equilibrium description of

this system includes the extension of the polymer, with

the tension as conjugate variable. In computer simula-

tions we have much greater flexibility. The configuration

functions can be rather arbitrary collective variables de-

lineating high dimensional manifolds of equilibrium ther-

modynamic states.

The partition function that normalizes the probability

distribution, Z, is directly related to the free energy F

(Gibbs potential), the free entropy p (Massieu potential)

and entropy S:lnZ=-3F= =S-- A (X2) (2)

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Crooks, Gavin E. Measuring Thermodynamic Length, article, September 7, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc926490/m1/1/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.