We compute the distribution of the work done in stretching a Gaussian
polymer, made of N monomers, at a finite rate. For a one-dimensional polymer
undergoing Rouse dynamics, the work distribution is a Gaussian and we
explicitly compute the mean and width. The two cases where the polymer is
stretched, either by constraining its end or by constraining the force on it,
are examined. We discuss connections to Jarzynski's equality and the
fluctuation theorems.Comment: 5 pages, 2 figure

Abhishek Dhar

F. Ritort

English

arXiv

Crossref

Work distribution functions in polymer stretching experiments

We compute the distribution of the work done in stretching a Gaussian polymer, made of N monomers, at a finite rate. For a one-dimensional polymer undergoing Rouse dynamics, the work distribution is a Gaussian and we explicitly compute the mean and width. The two cases where the polymer is stretched, either by constraining its end or by constraining the force on it, are examined. We discuss connections to Jarzynski's equality and the fluctuation theorems

Dhar, Abhishek

Indian Academy of Sciences

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Work distribution functions in polymer stretching experiments
Abhishek Dhar1, ∗
1Raman Research Institute, Bangalore 560080
(Dated: February 2, 2008)
We compute the distribution of the work done in stretching a Gaussian polymer, made of N
monomers, at a finite rate. For a one-dimensional polymer undergoing Rouse dynamics, the work
distribution is a Gaussian and we explicitly compute the mean and width. The two cases where the
polymer is stretched, either by constraining it’s end or by constraining the force on it, are examined.
We discuss connections to Jarzynski’s equality and the fluctuation theorems.
PACS numbers: 05.40.-a, 05.70.Ln
I. INTRODUCTION.
Classical thermodynamics does not give us all the de-
tails about a nonequilibrium process. For example con-
sider a nonequilibrium process during which we perform
work W on a system kept in contact with a heat bath
at some fixed temperature T . The system starts from an
equilibrium state described by the temperature T and
some other parameter, say λ (e.g. volume). During the
process the parameter changes from its initial value λi to
a final value λf . At the end of the process the system
need not be in equilibrium but will eventually relax to
an equilibrium state described by T and λf . The second
law then tells us that
W ≥ ∆F (1)
where ∆F is the difference of free energy between the
two equilibrium states. The equality holds if the pro-
cess is reversible. For an irreversible process what other
information can one extract from a measurement of the
work done ? First note that for specified initial and final
values of the parameter λ and a fixed path λ(t) connect-
ing them, the work done will not have a unique value.
Every time we repeat the process we will get a different
work done because: (a) the initial microscopic state we
start from may be different and (b) for a given initial
microscopic state the time evolution is not unique since
the system is in contact with a heat bath. Thus we will
get a probability distribution for the work done and it is
of interest to examine the properties of this distribution.
Recently there has been a lot of interest on issues related
to properties of such distribution functions. Two very
interesting results involving universal properties of these
distributions have been proposed.
The first is a very surprising exact equality obtained by
Jarzynski1 which states that:
〈e−βW 〉 = e−β∆F (2)
where the average is over the work distribution function.
This result seems to hold very generally and should be
compared with the inequality in Eq. 1 that one obtains
from usual thermodynamics.
The second set of results are obtained when one looks
at the probability distribution of various non-equilibrium
quantities (including W ) such as, for example, the en-
tropy production. In this case some new fluctuation theo-
rems have been proposed2,3,4. These theorems were origi-
nally derived for deterministic systems but have also been
proved for stochastic systems5,6,7 These theorems look at
the ratio of the probabilities of positive to negative en-
tropy production during a nonequilibrium process and
thus give some measure of “second law violations” which
can be significant if one is looking either at small systems
or at small time intervals. There are two versions of the
fluctuation theorem, the steady state fluctuation theorem
(SSFT) and the transient fluctuation theorem (TFT). In
the former case one looks at a system in a nonequilib-
rium steady state and the average entropy production
rate is examined. In the transient fluctuation theorem a
system is initially prepared in thermal equilibrium and
one looks at the entropy produced in a finite time τ .
An important point to note is that the definition of en-
tropy production in small (non-thermodynamic) systems
and in a nonequilibrium situation is somewhat ad-hoc
and various definitions have been used. A number of
authors8,9,10,11,12,13have looked, both theoretically and in
experiments, at fluctuations of quantities such as work,
power flux, heat absorbed, etc. during a nonequilibrium
process. In an interesting work it was shown by Crooks6
that the Jarzynski equality and the TFT are connected.
Finding universal properties of various nonequilibrium
distribution functions is of obvious interest. At the same
time the explicit forms of the distributions for different
systems is clearly of interest too. Infact for systems such
as polymers these are experimentally accessible14 and it
seems plausible that they can give information on the
dynamics of the system. There has not been much work
in this direction. In a recent paper15 Speck and Seifert
have shown that in the limit of slow driving the work
distribution becomes a Gaussian. They consider systems
(which could be nonlinear) evolving through Langevin
dynamics. Apart from this work, most other explicit cal-
culations of nonequilibrium distribution functions have
considered single particle systems.
In this paper we consider the well-known model of a
flexible polymer whose motion is governed by Rouse dy-
namics. We look at the work done when the polymer is
stretched at a finite rate. The work distribution func-
2
tions in different ensembles ( constant force and constant
extension) are computed explicitly and the dependence
of the distributions on switching rates and system sizes
is examined. We discuss our results in the context of
Jarzynski’s equality and the fluctuation theorems (TFT).
II. DISTRIBUTION OF WORK IN THE
CONSTANT EXTENSION ENSEMBLE.
We consider a one-dimensional Gaussian polymer
whose energy is given by:
H =
∑
l=1,N+1
kl
2
(yl − yl−1)
2 (3)
with y0 = 0 and yN+1 = α(t) which is a specified function
of time. For the moment we take the spring constants kl
to be arbitrary. We assume the following Rouse dynamics
for the chain:
ẏl = −
kl
γ
(yl−yl−1)+
kl+1
γ
(yl+1−yl)+ηl l = 1, 2...N (4)
which can be written in matrix notation as
dy
dt
= −
1
γ
Ay +
1
γ
h(t) + η (5)
where yT = {y1, y2....yN}, η
T = {η1, η2, ...ηN}, h
T =
{0, 0, ...kN+1α(t)} and the noise satisfies 〈ηl(t)〉 = 0 and
〈ηl(t)ηm(t
′)〉 = 2/(βγ)δlmδ(t − t
′) . The matrix A is
tridiagonal with elements Al,l = (kl + kl+1), Al,l+1 =
−kl+1, Al,l−1 = −kl. The general solution of this equa-
tion is given by:
y(t) = G(t)y(0)+
∫ t
0
dt′G(t−t′)
h(t′)
γ
+
∫ t
0
dt′G(t−t′)η(t′)
(6)
where G(t) = e−At/γ . The work done, as defined by
Jarzynski, is then given by:
WJ =
∫ τ
0
∂H
∂α
α̇dt
= kN+1
∫ τ
0
(α − yN)α̇dt
=
kN+1
2
(α2(τ) − α2(0)) −
∫ τ
0
dtḣT y.
This work is equal to the true mechanical work done by
the external force which we will call W (thus in this case
W = WJ ). We now plug in the solution for yN from
Eq. 6 to get
W =
kN+1
2
(α2(τ) − α2(0)) −
∫ τ
0
dtḣT [G(t)y(0)
+
∫ t
0
dt′G(t − t′)
h(t′)
γ
+
∫ t
0
dt′G(t − t′)η(t′)] (7)
Since W is linear in y(0) and η both of which are Gaussian
variables, it follows that the distribution of W will also
be Gaussian. We then only need to find the mean and
the second moment which we now obtain. We first state
a few results on equilibrium properties of the Gaussian
chain. The equilibrium free energy of the chain is given
by
Z(α) =
∫
dy1dy2...dyNe
−βH where
H =
N+1∑
i=1
ki
2
(yi − yi−1)
2
=
1
2
yTAy − hT y +
1
2
kN+1α
2 (8)
This leads to the following equilibrium free-energy of the
polymer (apart from α-independent constants).
F (α) =
1
β
ln(Z) =
1
2
kN+1α
2 −
1
2
hTA−1h (9)
=
1
2
k̄α2
where 1/k̄ = 1/k1 + 1/k2 + ...1/kN+1. The mean posi-
tions of the particles and their fluctuations can be easily
computed at any force and are given by:
〈y〉 = A−1h (10)
〈〈(y − 〈y〉)(yT − 〈yT 〉)〉 =
1
β
A−1 (11)
Mean and fluctuations of the work done:
From Eq. 7 we get for the mean work done:
〈W 〉 =
kN+1
2
(α2(τ) − α2(0)) −
∫ τ
0
dtḣT [G(t)〈y(0)〉
−
1
γ
∫ τ
0
dt
∫ t
0
dt′ḣTG(t − t′)h(t′) (12)
We do integration by parts so as to express everything
in terms of the rates ḣ. Using the equilibrium results in
Eq. (9,10) we finally get:
〈W 〉 = F (α(τ)) − F (α(0))
+
∫ τ
0
dt
∫ t
0
dt′ḣT (t)A−1G(t − t′)ḣ(t′) (13)
The fluctuations of the work 〈(W −〈W 〉)2〉 = σ2 is given
by:
σ2 =
∫ τ
0
dt
∫ τ
0
dt′ḣT (t)G(t)〈[y(0) − 〈y(0)〉][y(0) − 〈y(0)〉]T 〉
×G(t′)ḣ(t′) +
∫ τ
0
dt1
∫ t1
0
dt′1
∫ τ
0
dt2
∫ t2
0
dt′2ḣ
T (t1)G(t1 − t
′
1)
×〈η(t′1)η
T (t′2)〉G(t2 − t
′
2)ḣ(t2)
3
-2 0 2
W
0
0.5
1
1.5
P(
W
)
r=0.1
r=1
r=10
r=100
FIG. 1: Distributions of the work done in pulling a short
polymer (N = 1) at different rates r = τR/τ .
Using Eq. (11) and the relation 〈η(t)ηT (t′)〉 =
2/(βγ)δ(t − t′)I this simplifies to:
σ2 =
1
β
∫ τ
0
dt
∫ τ
0
dt′ḣT (t)G(t)A−1G(t′)ḣ(t′) +
4
βγ
×
∫ τ
0
dt1
∫ t1
0
dt2
∫ t2
0
dt′2ḣ
T (t1)G(t1 − t
′
2)G(t2 − t
′
2)ḣ(t2)
Finally using the relation
∫ t2
0 dt
′
2G(t1 − t
′
2)G(t2 − t
′
2) =
γA−1
2 [G(t1 − t2) − G(t1)G(t2)] we get
σ2 =
2
β
∫ τ
0
dt
∫ t
0
dt′ḣT (t)A−1G(t − t′)ḣ(t′) (14)
The distribution of work done is then:
P (W ) =
1
(2πσ2)1/2
e−
(W−〈W〉)2
2σ2 (15)
As expected for a Gaussian process we find that
〈W 〉 − ∆F = βσ2/2. (16)
Note that in the present polymer model both the equi-
librium free energy and the average work are indepen-
dent of temperature while the width of the distribution
σ2 depends linearly on temperature. It is easily ver-
ified that the work distribution satisfies the Jarzynski
equality 〈e−βW 〉 = e−β∆F . On the other hand, with
the present definition of work, the fluctuation theorem
is not satisfied. However if we define the “dissipated
work” Wdiss = W − ∆F then the distribution of Wdiss,
P̃ (Wdiss) satisfies the fluctuation theorem:
P̃ (Wdiss)
P̃ (−Wdiss)
= eβWdiss .
Since Wdiss gives some measure of deviation from a qua-
sistatic and adiabatic process it seems reasonable to think
of it as an entropy production term which is what usually
appears in the fluctuation theorems.
0 10 20 30 40
W
0
0.1
0.2
0.3
0.4
0.5
P(
W
)
r=0.1
r=1
r=10
r=100
FIG. 2: Distributions of the work done in pulling a long poly-
mer (N = 100) at different rates r = τR/τ .
Some special cases: Let us consider the case where
all spring constants are equal kl = k and the let us assume
that the polymer is pulled at a constant rate so that
α(t) = at/τ . The effective spring constant is k/(N + 1)
and so the free energy of the polymer is kα
2
2(N+1) . From
Eq. 14 we get the spread in the work done:
σ2 =
2a2γ
βτ
[k2A−2 +
γ
kτ
k3A−3(e−Aτ/γ − 1)]NN (17)
where [...]NN denotes a matrix element. The average
work can be obtained from Eq. 16. In the two limits of
very slow (τ → ∞) and very fast (τ → 0) processes, we
get:
σ2 =
2k2a2γA−2NN
β
1
τ
≈
4a2γN
βπ2τ
(Slow) (18)
=
ka2N
β(N + 1)
(Fast) (19)
For an instantaneous pulling process the work done is
simply W = kN+1[(α(τ) − yN )
2 − (α(0) − yN )
2]/2 and
the result in Eq. 19 can be directly obtained.
It is instructive to plot the work distributions for differ-
ent pulling rates. To see the effect of the polymer length
on the distribution we consider two cases: (i) A short
polymer with N = 1, k = 1, γ = 0.1, a = 1, β = 1 and
(ii) a long polymer with N = 100, k = 1, γ = 0.1, a =
10, β = 1 (in arbitrary units). In each case the param-
eter values are chosen so that the change in equilibrium
free energy given by ∆F = ka
2
2(N+1) is of the order of 1/β .
The pulling rate has to be compared with the relaxation
time of the polymer which is given by τR = γ/λsm where
λsm = 4k sin
2( π2(N+1) ) is the smallest eigenvalue of A.
For large N we get τR = γN
2/(kπ2). For the cases (i)
and (ii) we numerically evaluate Eq. 17 for pulling rates
r = τR/τ = 0.1, 1, 10, 100. The resulting distributions
are plotted in Fig. 1 and Fig. 2. For the long polymer the
probability of negative work realizations is quite small.
We can increase their probability by increasing the tem-
4
perature which broadens the distributions while keeping
the mean unchanged.
III. DISTRIBUTION OF WORK IN THE
CONSTANT-FORCE ENSEMBLE.
Next we compute the work distribution in the
constant-force ensemble. Instead of constraining the end
of the polymer we apply a time-dependent force f(t) on
it. The time-dependent Hamiltonian of the system is now
given by:
H =
∑
l=1,N
kl
2
(yl − yl−1)
2 − f(t)yN (20)
and the equations of motion are again:
dy
dt
= −
1
γ
Ay +
1
γ
h(t) + η
with hT (t) = [0, 0, ...f(t)]. In this case we note that
the the generalized work WJ , as defined by Jarzynski,
is given by WJ = −
∫ τ
0
dtyN ḟ(t)dt and is not equal to
the true mechanical work done on the system which is
W =
∫ τ
0
dtf(t)ẏN . In this paper we will compute the
distribution of the true mechanical work W . We again
find that W has a Gaussian distribution with the follow-
ing mean and variance:
〈W 〉 =
1
γ
∫ τ
0
dthT (t)h(t) −
1
γ2
∫ τ
0
dt
∫ t
0
dt′hT (t)
×G(t − t′)Ah(t′) −
1
g
∫ τ
0
dthT (t)G(t)h(0)
σ2 =
2
βγ
∫ τ
0
dthT (t)h(t) −
2
βγ2
∫ τ
0
dt
∫ t
0
dt′hT (t)
×G(t − t′)Ah(t′)
For f(0) = 0, we get 〈W 〉 = β2 σ
2 which means that P (W )
satisfies the fluctuation theorem. It is then natural to
again ask if W is some measure of entropy production.
If this was so then W should vanish for an adiabatic
process. To check this we first express 〈W 〉 in terms of
the rate ḣ(t). We get
〈W 〉 =
1
2
[hT (τ)A−1h(τ) − hT (0)A−1h(0)]
−hT (τ)
∫ τ
0
dtA−1G(τ − t)ḣ(t) +
∫ τ
0
dt
∫ t
0
dt′ḣT (t)
×A−1G(t − t′)ḣ(t′)
Hence for an adiabatic process we get 〈W 〉 = −∆G(f)
where G(f) = −hTA−1h/2 is the polymer free energy in
the constant force ensemble. Thus in this case 〈W 〉 is not
zero for an adiabatic process and so it is not an obvious
measure of entropy production even though it does satisfy
the fluctuation theorem.
Interestingly, since σ2 = 2〈W 〉/β, thus even for an
adiabatic process, the work-distribution does not tend to
a δ function as one might naively expect. However in the
thermodynamic limit N → ∞ the width approaches zero
as σ ∼ 1/N1/2 so in this limit the usual expectation is
indeed satisfied.
IV. CONCLUSIONS.
In conclusion, in this paper we have computed explic-
itly the distribution of work done when a polymer is
stretched at a finite rate. We examine different ensem-
bles and look at different definitions of work. As has been
noted by earlier authors, for different definitions of the
work, the corresponding distributions can have quite dif-
ferent properties. In the constant extension ensemble the
generalized work WJ is the same as the true mechanical
work W and Jarzynski’s identity is satisfied. In this case,
the fluctuation theorem is satisfied by a different quantity
Wdiss which does seem like a quantity which gives some
measure of entropy production. In the constant force en-
semble, WJ is different from the true work W which does
not satisfy the Jarzynski identity. On the other hand the
distribution of W satisfies a fluctuation-theorem like re-
lation. However in this case we find that it is not possible
to identify the work as a measure of entropy production.
As a practical use, the Jarzynski identity has been pro-
posed as an efficient method for computing equilibrium
free energy profiles from nonequilibrium measurements,
both in simulations and experiments16,17. For the speci-
cific case of polymers, we expect that a combination of
simulations and our exact results on the work distribu-
tion, should lead to better estimates on efficiency and of
errors18,19,20 (and how they depend on rates and system
sizes) involved while using the nonequilibrium methods
in free energy computations.
For non-Gaussian models of polymers such as, for ex-
ample, semiflexible polymers, the work distribution func-
tion is likely to be non-Gaussian. For small pulling rates
one again expects a Gaussian distribution. It will be
interesting to compute such distributions explicitly and
study their dependences on rates, system sizes and other
parameters such as the rigidity of the polymer.
∗ Electronic address: dabhi@rri.res.in
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Work distribution functions in polymer stretching experiments

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