The Crooks fluctuation theorem (CFT), sometimes known as 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 CFT is named after the chemist
Gavin E. Crooks (then at University of California, Berkeley) who discovered it in 1998.
The most general statement of the CFT relates the probability of a space-time trajectory
to the time-reversal of the trajectory
. The theorem says if the dynamics of the system satisfies
microscopic reversibility The principle of microscopic reversibility in physics and chemistry is twofold:
* First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with resp ...
, then the forward time trajectory is exponentially more likely than the reverse, given that it produces entropy,
:
If one defines 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), one can characterize every point along the reaction coordinate path by a parameter
, such that
and
correspond to two ensembles of
microstates
A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
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 CFT sets a relation between the following five quantities:
*
, '' i.e. '' the
joint probability
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of taking a microstate
from the
canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat ...
corresponding to
and of performing the forward transformation to the microstate
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
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas 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 energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ener ...
difference between the state
and
, represented by the canonical distribution of microstates having
and
, respectively.
The CFT 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 cases,
and
Using the time reversal relation
, and grouping together all the trajectories yielding the same work (in the forward and backward transformation), i.e. determining the probability distribution (or density)
of an amount of work
being exerted by a random system trajectory from
to
, 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 phenomenon has been experimentally verified using
optical tweezers for the
process of unfolding and refolding of a small
RNA hairpin and an RNA three-helix junction.
The CFT implies the
Jarzynski equality
The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two states and the irreversible work along an ensemble of trajectories joining the same states. It is named after the physicist Chr ...
.
Notes
Non-equilibrium thermodynamics
Statistical mechanics theorems
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