Biomedical Engineering Reference
In-Depth Information
work will always exceed the free energy for a driven process, a fraction of trajecto-
ries within the distribution ρ
will have work values less than Δ
G
. If the system
parameters are changed rapidly, the system is driven far from equilibrium and the
distribution ρ
(
W
)
will be broad. On the other hand, if the system parameters are
changed quasi-statically, the distribution ρ
(
W
)
approaches a delta function centered
at the free energy Δ
G
of the transformation. Consider now a system that is initially
at thermal equilibrium. An increasing force is applied that drives the system to a new
state. Then, the system reaches a new equilibrium and is driven backward in time
(decreasing force) at the same rate to the original state. Accumulating many realiza-
tions of this process produces probability distributions ρ
for
(
W
)
of the
work done on the system along the forward and reverse processes. Crooks (1999)
showed that the relative probability of a work value
W
(
W
)
and ρ
rev
(
W
)
=
w
during the forward pro-
cess to the probability of a work value
W
w
during the reverse process is given
by the following detailed fluctuation theorem as
=
−
ρ
for
(
W
)
)
=
exp
[(
W
−
Δ
G
)
/
k
B
T
]
(3.107)
(
−
ρ
rev
W
W
)=
W
)
This relationship indicates that the unique work value at whichρ
for
(
ρ
rev
(
−
is equal to the free energy,
W
=
Δ
G
. Noting that the dissipated work is
W
d
=
W
−
Δ
G
,
we can also express the Crooks (1999) theorem as
ρ
for
(
W
d
)
)
=
exp
[
W
d
/
k
B
T
]
(3.108)
(
−
ρ
rev
W
d
Rearranging Equation 3.107 and integrating
W
over all work values [
−
∞
,
∞] yields
the following integral fluctuation theorem:
ρ
for
ρ
rev
(
W
)
exp
[
−
W
/
k
B
T
]
d
W
=
(
−
W
)
exp
[
−
Δ
G
/
k
B
T
]
d
W
exp
[
−
W
/
k
B
T
]
=
exp
[
−
Δ
G
/
k
B
T
]
(3.109)
Equation 3.109 is Jarzynski's nonequilibrium work relation Jarzynski (1997). This
relation states that the ensemble average over the Boltzmann-weighted work is equiv-
alent to the Boltzmann-weighted free energy. Application of these theorems is con-
tingent on a proper definition of the work done on the system of interest. We will
briefly address this topic for two commonly encountered cases in the next section.
3.6.2 W
ORK AND
F
REE
E
NERGY IN
F
ORCED
T
RANSITIONS
The
thermodynamic work W
performed on a system follows as the integral over the
time rate of change of the Hamiltonian covering an observation time from 0 to
t
f
:
t
f
∂
H
(
x
(
t
)
,
t
)
=
W
d
t
(3.110)
∂
t
0
Search WWH ::
Custom Search