Crooks Fluctuation Theorem
Encyclopedia
The Crooks equation
is an equation in statistical mechanics
that relates
the work done on a system during a non-equilibrium transformation to the
free energy difference between the final and the initial state of the
transformation. During the non equilibrium transformation the system is at constant volume and
in contact with a heat reservoir
. The CE is named after the chemist Gavin E. Crooks
(then at
University of California) who discovered it in 1998. The CE
is a special case of the more general fluctuation theorem
.
If we define a generic reaction coordinate of the system as a function
of the Cartesian coordinates of the constituent particles ( e.g. ,
a distance between two particles), we can characterize every point
along the reaction coordinate path by a parameter
, such that 
and
correspond to two ensembles of microstates
(see microstate (statistical mechanics)
)
for which the reaction coordinate is constrained to different
values. A dynamical process where
is externally
driven from zero to one, according to an arbitrary time scheduling,
will be referred as forward transformation , while the time reversal
path will be indicated as backward
transformation. Given these definitions, the CE sets a relation
between the following four quantities:
is an equation in statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
that relates
the work done on a system during a non-equilibrium transformation to the
free energy difference between the final and the initial state of the
transformation. During the non equilibrium transformation the system is at constant volume and
in contact with a heat reservoir
Heat reservoir
In thermodynamics, a heat reservoir, thermal reservoir, or heat bath is a system whose heat capacity is so large that when it is in thermal contact with some other system of interest its temperature remains effectively constant. The heat bath is effectively an infinite reservoir of energy and...
. The CE is named after the chemist Gavin E. Crooks
Gavin E. Crooks
Gavin E. Crooks is an English chemist currently researching in America. He researches in the field of Statistical mechanics, and discovered the Crooks Equation leading to the Crooks fluctuation theorem.-Career:...
(then at
University of California) who discovered it in 1998. The CE
is a special case of the more general fluctuation theorem
Fluctuation theorem
The fluctuation theorem , which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time...
.
If we define a generic reaction coordinate of the system as a function
of the Cartesian coordinates of the constituent particles ( e.g. ,
a distance between two particles), we can characterize every point
along the reaction coordinate path by a parameter
, such that and correspond to two ensembles of microstates(see microstate (statistical mechanics)
Microstate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
)
for which the reaction coordinate is constrained to different
values. A dynamical process where
is externallydriven from zero to one, according to an arbitrary time scheduling,
will be referred as forward transformation , while the time reversal
Time reversal
Time reversal may refer to:* In physics, T-symmetry - the study of thermodynamics and the symmetry of certain physical laws where the concept of time is reversed — ie...
path will be indicated as backward
transformation. Given these definitions, the CE sets a relation
between the following four quantities:
-
, i.e. the joint probability of taking a microstate 
from the canonical ensembleCanonical ensembleThe canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
corresponding to
and of performing the forward transformation to the microstate B corresponding to 
; 
, i.e. the joint probability of taking the microstate 
from the canonical ensemble corresponding to 
and of performing the backward transformation to the microstate 
corresponding to 
;
, where 
is the Boltzmann constant and 
the temperature of the reservoir;
-
, i.e. the work done on the system during the forward transformation (from 
to 
); -
, i.e. the Helmholtz free energyHelmholtz free energyIn thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
difference between the state
and 
, represented by the canonical distribution of microstates having 
and 
, respectively).
The CE
equation reads as follows:
In the previous equation the difference
corresponds to the work dissipated in the forward transformation,
. The probabilities 
and 
become identical when the transformation is performed at infinitely slow speed, i.e. for equilibrium transformations. In such case 
and 
= 0.
Using the time reversal relation
, and grouping together all the trajectories
yielding the same work (in the forward and backward transformation),
we can write the above equation in terms of the work distribution
functions as follows
Note that for the backward transformation, the work distribution function must be evaluated by taking the work with the opposite sign.
The two work distributions for the forward and backward processes cross at
. This fact has been experimentally verified using optical tweezersOptical tweezersOptical tweezers are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force , depending on the refractive index mismatch to physically hold and move microscopic dielectric objects...
for the
process of unfolding and refolding of a small RNARNARibonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....
hairpin and an RNA
three-helix junction http://www.nature.com/nature/journal/v437/n7056/full/nature04061.html
The CE implies the Jarzynski equalityJarzynski equalityThe Jarzynski equality is an equation in statistical mechanics that relates free energy differences between two equilibrium states and non-equilibrium processes...
.-