What is the order of 2D polymer escape transition?
What is the order of 2D polymer escape transition?
Hsiao-Ping Hsu and Kurt Binder
Institut für Physik, Johannes Gutenberg-Universität Mainz
D-55099 Mainz, Staudinger Weg 7, Germany
Leonid I. Klushin
American University of Beirut, Department of Physics, Beirut, Lebanon
Alexander M. Skvortsov
Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia.
(Dated: August 11, 2021)
An end-grafted flexible polymer chain in 3d space between two pistons undergoes an abrupt
transition from a confined coil to a flower-like conformation when the number of monomers in the
chain, N , reaches a critical value. In 2d geometry, excluded volume interactions between monomers
of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear
string of blobs. However, the blob picture raises questions on the nature of this escape transition.
To check the theoretical predictions based on the blob picture we study 2d single polymer chains
with excluded volume interactions and with one end grafted in the middle of a strip of length 2L and
width H by simulating self-avoiding walks on a square lattice with the pruned-enriched-Rosenbluth
method (PERM). We estimate the free energy, the end-to-end distance, the number of imprisoned
monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit
of large N and L but finite L/N , there is a small but finite jump in several average characteristics,
including the order parameter. We also present a theoretical description based on the Landau free
energy approach, which is in good agreement with the simulation results. Both simulation results
and the analytical theory indicate that the 2d escape transition is a weak first-order phase transition.
I. INTRODUCTION
A phenomenon that was called escape transition oc-
curs upon progressive squeezing an end-grafted polymer
chain between two pistons and has attracted great inter-
est [1-16]. At weak deformation the chain is compressed
uniformly into a relatively thick pan-cake conformation.
Beyond certain critical compression, the chain configu-
ration changes abruptly. One part of the chain forms
a stem stretching from the grafting point to the piston
edge, while the rest of the segments form a coiled crown
outside the piston, thus escaping from the region under-
neath the piston. An abrupt change from one state to
another implies a first order transition. Various aspects
of this problem were investigated: The escape transition
of compressed polymer mushrooms in 3d space was inves-
tigated thoroughly by scaling theory [1], numerical cal-
culations [2, 3, 4], and computer modeling under good
solvent [5], and theta solvent [6] conditions. The escape
transition of star polymers was discussed in [7], and the
escape transition of di-block-copolymers was considered
in [8]. The influence of the curvature of the pistons was
investigated in [9, 10], and the effect of adsorption be-
tween the polymer chain and the surface of the piston
was considered in [11]. A comparison between Monte
Carlo simulations and experimental results by atomic
force-electrochemical microscopy was recently presented
in [12]. A rigorous analytical theory for the equilibrium
and kinetic aspects of the escape transition for a Gaus-
sian chain was constructed in [13, 14]. Metastability ef-
fects, negative compressibility and the nonequivalence of
the escape transition in two conjugate ensembles were
analyzed for the same model in [15, 16].
The reason for studying the escape transition is that it
gives the possibility to understand the phenomenon of a
very unconventional phase transition. The concept of a
phase transition requires taking a thermodynamic limit.
For standard low-molecular weight systems as well as for
macromolecular systems in condensed bulk matter finite
size effects are usually negligible. In contrast to that,
phase transitions at the level of a single macromolecule
e.g., the coil-globule transition [17], or polymer adsorp-
tion at an interface [18, 19] - do not have any analogies
in the physics of low molecular mass systems. A sin-
gle macromolecule always consists of a finite number of
monomers N : computer modeling rarely deals with N
larger than 104 so that finite-size effects in the single-
molecule phase transitions are the rule rather than the
exception. The situation is much more complicated in the
case of the escape transition. It was shown [1] that the
escape transition point (critical compression) depends on
the relation between the chain length Na (a is the dis-
tance between neighboring monomers) and the piston ra-
dius L. Therefore, to analyze the escape transition in
the thermodynamic limit, it is necessary to take both
Na → ∞ and L → ∞ but Na/L = const.
The physics of phase transitions is generally known to
be strongly affected by the spatial dimensionality. For
phase transitions at the level of a single macromolecule
the spatial dimensionality is important because excluded
volume effects are especially large for polymeric chains
in two dimensions [20]. An ideal chain without excluded
http://arxiv.org/abs/0704.1857v2
volume interactions retains a Gaussian coil conformation
with the lateral size ∼ N1/2 even when it is confined in a
2d strip. There is a very pronounced difference between
this state and the partially escaped state with a strongly
stretched stem of size L ∼ Na. On the contrary, a 2d
polymer chain with excluded volume interactions con-
fined in a strip is already strongly elongated with the size
∼ Na . It is not clear whether the difference between the
confined and the escaped states is large enough to result
in a phase transition.
To analyze the excluded volume effects of the escape
transition, we study a flexible polymer chain containing
N links of length a (N monomers) grafted in the middle
of a strip of length 2L and width H , under good solvent
conditions. Schematic drawings of a polymer chain in
an imprisoned state and in an escaped state are shown
in Fig. 1. First, we summarize the results of the escape
transition for a 2d Gaussian ideal chain in Sec. II, and
then we give the theoretical predictions based on the blob
picture in Sec. III. In an experimental setup, the escape
transition is driven by changing the piston separation H ,
while in a blob picture, the escape transition is studied
by changing the chain length N or the strip length L at
fixed H which is also the size of a blob. Comparisons
between the escape transition behavior of the Gaussian
chain model and that of the blob model in the thermo-
dynamic limit are given in Sec. IV. In Sec. V we present
our results from Monte Carlo (MC) simulations with
the pruned-enriched Rosenbluth method (PERM) [21] In
Sec. VI we provide a theoretical description based on the
Landau free energy approach which is compared with
MC results. A summary and discussions are given in
Sec. VII. Detailed analyses of the variances of the im-
prisoned monomers are given in the Appendix.
II. ESCAPE FOR A 2D GAUSSIAN CHAIN
For the escape transition of a Gaussian chain, a closed-
form of the exact partition function was obtained earlier
in [14]. The asymptotic form of the free energy F has
two branches:
= Fimp imprisoned state
= Fesc escaped state
. (1)
Here d is the dimensionality of space and a factor of kBT
is absorbed in the free energy throughout the paper. It
was shown by direct numerical comparison that the sim-
ple asymptotic expressions provide a very accurate de-
scription. The two branches meet at the transition point
which is given by
, with d = 2 . (2)
The average lateral forces acting on the grafting point
are obtained by
∂F (N,L,H)
n "blobs"
crown
FIG. 1: Schematic drawings of a flexible polymer chain of
length N grafted in the middle of the strip of length 2L and
width H , in a blob picture: (a) As the chain is imprisoned
inside the strip, it forms a sequence of nb blobs. (b) As the
chain length is larger than the maximum chain length N∗ of
a chain in an imprisoned state, the chain partially escapes
from the strip and forms an escaped state. A escaped state
consists of a “stem” containing N∗ monomers and a “crown”
containing N −N∗ monomers.
which is related to the work required to pull a chain end
into confined space by a unit distance. The average com-
pression force
fH = −
∂F (N,L,H)
is related to the work of compression. It was
shown earlier [14] that the average fraction of impris-
oned monomers is given by the derivative Nimp =
∂F (N,L,H)/∂u of the free energy with respect to the
effective confining potential u = (πa/2H)
The average end-to-end distance per monomer is:
N−1/2 imprisoned state
+N−1/2 (1−Nimp/N)1/2 escaped state
The Landau order parameter s for the escape transition
was introduced in [14] as a stretching degree of the chain
rN/N for the imprisoned state and as the stretching de-
gree of the stem L/nimp for the escaped state. Note
that the order parameter is defined for an instantaneous
configuration with a given end-to-end distance rN and
the number of imprisoned monomers nimp which may
be quite different from the equilibrium average values
R = 〈rN 〉 and Nimp = 〈nimp〉. The Landau free energy is
defined as a function of the order parameter. The Lan-
dau free energy for the imprisoned state is just the free
energy of a coil as a function of its free end position,
which has a standard parabolic form:
Φimp(s) = N
. (6)
At s = simpeq = 0, the system is in equilibrium and the
corresponding equilibrium free energy, i.e., the depth of
the minimum, is Φimp(0) = (N/2d)(πa/H)
2. For the
escaped state,
Φesc(s) =
. (7)
The position of the minimum is at s = sesceq = πa/dH,
and the corresponding depth is Φesc(s
eq ) = πL/H . The
binodal is determined by the condition that the two min-
ima are equally deep, i.e. Φimp(s
eq ) = Φesc(s
eq ), which
leads to the transition point described by Eq. (2). At the
transition point, the average order parameter S =< s >
jumps from simpeq to s
eq . The Landau function allows
one to analyze metastable states and to define the two
lines where either one of the metastable minima vanishes
[16]. Theoretical predictions for the escape transition of
a Gaussian chain are summarized in Table I and shown
in Fig 2. Detailed discussion and comparison with the
prediction from the blob picture for 2d polymer chains
are given in Sec. IV.
III. BLOB PICTURE OF A 2D ESCAPE
A free chain in d = 2, has an average end-to-end dis-
tance given by (a is the distance between neighboring
monomers) [17-20]
RF = aN
3/4 (8)
here the prefactors of order unity are neglected through-
out. Based on the blob picture we have a cigar of blobs
(nb blobs in total) in the confined situation. Thus the
average end-to-end distance is
R = nb(2rb) = nbH (9)
where rb is the blob radius. Within a blob, self-avoiding
walks (SAW) statistics holds, so if g monomers belong to
a blob
H = ag3/4 = 2rb , g = (H/a)
4/3 . (10)
Since every monomer of a chain in an imprisoned state
must be in a blob we furthermore have
N = gnb = nb(H/a)
4/3 , nb = N(H/a)
−4/3 . (11)
This yields, together with Eq. (9), the formula for the
end-to-end distance
R/a = N(H/a)−1/3 (12)
If H is of the same order as RF, Eq. (8), one finds R/a =
N3/4, giving a smooth crossover to mushroom behavior,
as expected.
The free energy excess of the chain in an imprisoned
state (in units of kBT ), relative to an unconfined mush-
room, is simply the number of blobs, nb ,
Fimp = nb = N(H/a)
−4/3 (13)
We now define N = N∗ as the maximum chain length
of an imprisoned chain. Then for N > N∗ the chain con-
sists of a “stem” containing N∗ imprisoned monomers
and an escaped “crown” comprising the rest N − N∗
monomers (Fig. 1b). Thus, we find N∗ from the con-
dition that R becomes equal to L at the transition point,
using Eq. (12)
N∗ = (L/a)(H/a)1/3 . (14)
Since the free energy of an unconfined mushroom is taken
as zero reference point, the “crown” does not contribute
to the excess free energy of the escaped chain, which is
hence due to the stem only:
Fesc = N
∗(H/a)−4/3 = L/H . (15)
The average end-to-end distance of the escaped chain (in
axial direction parallel to the confining boundaries) hence
becomes
Resc = L+ a(N −N∗)3/4 . (16)
Equations (13) and (15) show that the free energy as a
function ofN for fixed H and L, consists of two branches,
i.e., Fimp for the imprisoned state (N < N
∗) and Fesc
for the escaped state (N > N∗), meeting at N = N∗.
The lateral and the compression forces are obtained from
the free energy by using Eqs. (3) and (4). We use the
same definition of the order parameter s as that for the
Gaussian chain, so the average order parameter S
S =< s >=
R/Na imprisoned state
L/N∗a = (H/a)−1/3 escaped state
From Eqs. (12), (14), and (17), we find that the order
parameter does not show any discontinuity at the tran-
sition, but simply stays constant, i.e. S = (H/a)−1/3.
Results of the theoretical predictions are listed in Table I
and also shown in Fig. 2.
IV. COMPARISON OF THE GAUSSIAN AND
BLOB PICTURES
In the thermodynamic limit N → ∞, L → ∞, L/N
remains as a nontrivial variable along with H . In Fig. 2,
theoretical predictions of the Gaussian chain model and
of the blob model are shown by dotted and solid lines,
respectively. The strip width, H , is fixed, and the ratio
L/N is varied. The chain is in an imprisoned state if L
is larger than the corresponding critical value,
the Gaussian chain model or
for the blob model,
and in the escaped state for for L
]. All
curves show only the scaling behavior disregarding nu-
merical coefficients of order one, and the bond length a
is taken as a unit length.
In both models, the free energy per monomer is given
by piecewise linear functions of L/N . For the escaped
~LN( )
f   /NH
−1/2 N
N     /N
FIG. 2: Theoretical predictions for various averaged chain characteristics plotted against L/N at constant strip width H : (a)
the free energy per monomer F/N , (b) the lateral force fL, (c) the compression force per monomer fH/N , (d) the fraction of
imprisoned monomers Nimp/N , (e) the order parameter S, (f) the end-to-end distance per monomer R/N , . Gaussian model
results are shown by dotted lines, blob model results - by solid lines. The chain is in an imprisoned state for L
], and in an escaped state for L
] for the Gaussian chain model [blob picture].
TABLE I: Theoretical predictions for the average values of the free energy per monomer F/N , the lateral force fL, the
compression force per monomer fH/N , the fraction of imprisoned monomers Nimp/N , the order parameter S, and the reduced
end-to-end distance R/Na based on the Gaussian chain model and the blob picture.
Characteristics Imprisoned Escaped
of the chain state state
Gaussian Blob Gaussian Blob
)2 ( a
)4/3 π
fL 0 0
)7/3 π
1 1 2
)1/3 L
)1/3 π
N−1/2
N−1/2
N−1/4
state, the slope of F/N vs. L/N is the same up to numer-
ical coefficients of order 1, and scales as H−1. This slope
has the meaning of the lateral force acting on the grafting
point. It is proportional to the inverse size of a blob which
is defined purely by confinement width H irrespective of
whether excluded volume interactions are present or not.
The free energy per monomer in the imprisoned state is
independent of L/N , and scales as the inverse number of
monomers in one blob, g−1 = (a/H)
, where ν is the
Flory exponent. The transition points in the two models
may be quite far apart since they scale differently with
H . In the setup where L/N is fixed and the piston sepa-
ration is decreasing, the blob model predicts the transi-
tion to happen at lower compression, as compared to the
Gaussian chain. In both models the lateral force jumps
from H−1 to zero at the respective transition point. As
for compression forces, they are strongly affected by ex-
cluded volume interactions (Fig. 2c). The difference in
the plateau values for the imprisoned state reflects the
lower compressibility of the self-avoiding walk as com-
pared to the Gaussian chain. This corresponds directly
to the plateau values of the free energy in Fig. 2a. In
the Gaussian chain model, the compression force jumps
at the transition point by a factor of 2, while the jump
in the blob model is by a factor of 4/3.
Figures 2a - 2c suggest that the behavior of both
models is fundamentally the same and characteristic
of a first-order phase transition. However, the next 3
graphs demonstrate qualitatively different predictions of
the Gaussian and the blob models.
For the Gaussian chain model, a transition from the
confined state to the escaped state is accompanied by a
jump in the average number of imprisoned monomers.
At the transition point, one half of the total number
of monomers are ejected outside to form a crown, see
Fig. 2d. In contrast to that, the blob model predicts a
smooth change without a jump. This directly affects the
behavior of the average order parameter: while in the
Gaussian chain model there is a pronounced jump, the
blob model suggests that the order parameter does not
change at all. The large constant value of S in the blob
model is due to the cigar-shape conformation of the chain
in an imprisoned state which is identical to the shape of
the stem in the escaped state.
The behavior of the average end-to-end distance is also
qualitatively different for the two models. The plateau
value for the imprisoned Gaussian chain, R/N = N−1/2,
is characteristic of the ideal coil, while for the elongated
cigar, this ratio is N -independent. The linear part of the
curves corresponding to the escaped state simply repre-
sents the dominant contribution of the stem to the overall
end-to-end distance, R ≈ L irrespective of the model. At
very low values of L/N corrections due to the crown size
become large, as shown in Fig 2f. The size of the Gaus-
sian chain demonstrates a jump at the transition point
consistent with a strong conformation change accompa-
nying the first-order transition. In the blob model, the
chain size does not have any jump.
It is clear that the predictions of the blob model pre-
sented in Fig. 2 contain some internal contradictions. On
the one hand, the picture of the two branches of the free
energy meeting at some angle suggests a first order tran-
sition. The jumps in the lateral and compression forces
are a simple consequence of that. On the other hand,
nothing dramatic happens to the chain conformation in
the blob picture: the change from a completely confined
state to a state with a small escaped tail is continuous, as
evidenced in Figs 2e and 2f. The presence of discontinu-
ity in the slope of the fraction of imprisoned monomers
suggests that the transition should be classified as second
order.
A crucial question to ask is whether one can identify
two distinct separate states with a bimodal distribution
of some appropriate order parameter. This we address by
employing a Monte-Carlo simulation of 2d self-avoiding
chains undergoing the escape transition.
V. MONTE CARLO SIMULATION
Single polymer chains grafted in the middle of a strip of
length 2L and width H are described by SAWs of N steps
FIG. 3: Schematic drawing of a polymer chain growing as
a self-avoiding walk inside a finite strip and grafted at (x =
0, y = 0). Monomers are allowed to sit on the lattice sites
except for the lattice sites representing the walls {−L ≤ x ≤
L , y = 0} and {−L ≤ x ≤ L , y = H}. The first monomer is
attached with a bond to the grafting site marked by a cross.
Lengths are measured in units of the lattice spacing.
 1000
 0  30  60  90  120  150
N3/4/H
H =  17
H =  33
H =  65
H = 129
1.944 NH-4/3
FIG. 4: (a) Free energy relative to a free chain, F (N,L,H) =
− ln[(Z(N,L,H)/Z0(N)], plotted against N
3/4H−1 for L =
6400 and H = 17, 33, 65, and 129. The dashed curve is
Fimp(N,L,H) = 1.944NH
−4/3 and gives the best fit of the
data.
on a square lattice between two hard walls with distance
H as shown in Fig. 3. Monomers are supposed to sit on
lattice sites but they are forbidden to sit on the two walls,
i.e. {−L ≤ x ≤ L , y = 0} and {−L ≤ x ≤ L , y = H}.
For our simulations we employ PERM and PERMwith k-
step Markovian anticipation as described in Ref. [21, 22].
PERM is a biased chain growth algorithm with popula-
tion control. Polymer chains are built like random walks
by adding one monomer at each step. Thus it has the
advantage of estimating the partition sum and counting
the imprisoned monomers directly.
We simulate 2d SAWs starting at the grafting point of
the strip of length L = 800, L = 1600, 3200, and 6400.
The width of the strip is varied from H = 5 to H = 129.
Depending on the chosen sizes of L andH , the total chain
length is varied from 2500 to 50000 in order to cover the
transition region.
 1000
 10  100  1000
L =   800
L = 1600
L = 3200
L = 6400
2.03 L/H
FIG. 5: The log-log plot of the excess free energy of the
escaped chain, Fesc(N,L,H), plotted against L/H for vari-
ous values of L and H . The dashed line is Fesc(N,L,H) =
2.03L/H and gives the best fit of the data.
 0.02
 0.04
 0.06
 0.08
 0  0.2  0.4  0.6  0.8  1
H = 17
H = 28
H = 33
H = 65
L =   800
L = 1600
L = 3200
L = 6400
2.03 L/(NH)
FIG. 6: Free energy relative to a free chain divided by N ,
F (N,L,H)/N , plotted against L/N for various values of L
and H . The dashed line extrapolated to zero is 2.03L/(NH).
A. Free energy
Let us first discuss the scaling behavior of the free en-
ergy. The partition sum of a free SAW in infinite volume
for N → ∞ scales as
Z0(N) = µ
γ−1 (18)
with µ∞ being the critical fugacity per monomer, and
with γ = 43/32 being a universal exponent [18]. As
chains are still confined in a strip of width H , one
should expect the scaling laws of the excess free energy
including the crossover from the region of wide strips,
RF << H , (RF ∼ Nν is the Flory radius with ν = 3/4
in d = 2), to the region of narrow strips, RF >> H ,
where chains are stretched. Since the length of the strip
is finite, here we are more interested in another expected
crossover behavior of the excess free energy from an im-
prisoned and stretched chain state to an escaped state.
Therefore, we plot the excess free energy F (N,L,H) =
− ln(Z(N,L,H)/Z0(N)) against N3/4H−1 with the pre-
cise estimate of µ∞ = 0.37905228 [23] in Fig. 4. The
partition sum Z(N,L,H) is the total number of possible
configurations of SAW of N steps partially confined in
a strip of length 2L and of width H , which is estimated
directly in the simulation. In Fig. 4, the sharp crossover
behavior from the imprisoned states to the escaped states
is indeed seen as N increases for a fixed value of H . The
excess free energy of the escaped chain is independent of
N , as predicted in a blob picture by Eqs. (13) and (15).
The best fit of the free energy for imprisoned state is
given by
Fimp(N,L,H) ∼ 1.944(2)NH−4/3 . (19)
It is in perfect agreement with the previous estimation
in Ref. [22], where the fugacity per monomer scales as
µH − µ∞ ∼ 0.737H−4/3 and hence the free energy of
the chains of size N in the imprisoned state scales as
Fimp ∼ 0.737µ∞ NH
−4/3 ≈ 1.944NH−4/3. Values of the
excess free energy of the escaped chain, Fesc(N,L,H)
are determined by the horizontal curves shown in Fig. 4.
Results for L = 800, 1600, 3200 and 6400 are shown in
Fig. 5, where we obtain
Fesc(N,L,H) = 2.03(3)L/H . (20)
As H becomes comparable to L, i.e. L/H ∼ O(10),
the data points deviate slightly from the straight line,
indicating that there are further finite size corrections
for small L/H . In order to compare with the theoreti-
cal prediction shown in Fig. 2a, we plot F (N,L,H)/N
against L/N for various values of L and H in Fig. 6.
In the escaped regime, these straight lines extrapolated
to L/N = 0 are indeed described by Eq. (20) very well.
With conventional Monte Carlo simulations it is difficult
to estimate the partition sum precisely. With PERM we
do have very precise estimates of Z(N,L,H), and there-
fore we can obtain the lateral and the compression forces
by differentiating the estimated free energy, Eqs. (19) and
(20), with respect to L and H respectively.
B. Average characteristics of polymer chains
The most straightforward way to understand how the
conformations of the polymer chains change as they un-
dergo the escape transition is to estimate the end-to-end
distance, < x >. In Fig. 7, we plot the end-to-end dis-
tance per monomer< x > /N versus L/N . As long as the
polymer chains are imprisoned the curves are horizontal,
i.e. the degree of chain stretching is constant, and < x >
increases linearly with N as shown in Ref. [22]. As the
width of the strip, H , decreases, the chains are stretched
more. As L/N decreases, we see that there is a jump
 0.15
 0.45
 0  0.15  0.3  0.45  0.6  0.75  0.9
H =  5
H =  9
H = 17
H = 33
L =   800
L = 1600
L = 3200
L = 6400
FIG. 7: Average end-to-end distance divided by N , < x >
/N , plotted against L/N for various values of L and H . The
dashed line is < x > /N = L/N .
 0  0.1  0.2  0.3  0.4  0.5
H =   9
H =  17
H =  33
H =  64
H = 128
FIG. 8: Average fraction of the imprisoned monomers, <
Nimp > /N , plotted against L/N for L = 800 and various
values of H . The jump becomes more prominent as H de-
creases. The ends of the extrapolated straight dashed lines to
L/N = 0 all approach zero.
in each curve to another branch where < x >= L This
means that the chain stretching is abruptly increased so
that the chain reaches the edge of the strip, indicating
that the transition to a partially escaped conformation is
first-order like. This is in contrast to the smooth behav-
ior predicted by the blob picture and shown in Fig. 2f.
A jump-wise change in the chain stretching suggests
a similar change in the average number of imprisoned
monomers, < Nimp >. In Fig. 8, we plot the fraction of
imprisoned monomers < Nimp > /N versus L/N for L =
800. As long as the chain is imprisoned, < Nimp > /N =
1. With the increase in the number of monomers, N ,
each curve indeed develops a jump and the jump becomes
more pronounced for smaller H . For a fixed L and N →
∞, < Nimp > /N → 0 in accord with both theoretical
 0  0.15  0.3  0.45  0.6  0.75  0.9
H =  5
H =  9
H = 17
H = 33
L =   800
L = 1600
L = 3200
L = 6400
FIG. 9: Average order parameter S plotted against L/N for
various values of L and H .
models.
In Fig. 9, we plot the average of the order parameter S,
versus L/N . We see clear jumps between the two states.
For a given value of H , the order parameter behaves as
a step function. The data seem to suggest that the mag-
nitude of the jump decreases with the size of the system,
L. A detailed analysis of the distributions of s will be
presented in the next section. We will see that the ap-
parent decrease in the jump is due to poor sampling of
the escaped state.
C. Transition points
For our simulations the transition points can be deter-
mined by analyzing three quantities: (1) free energy, (2)
variance of the number of imprisoned monomers Nimp,
and (3) variance of the end-to-end distance x. Since at
the transition point the free energy of the imprisoned
state is equal to the free energy of the escaped state,
Eqs. (19) and (20) give the following relation between L,
N and H at the transition point:
= 1.04(2)H1/3 . (21)
This shows that a polymer chain of sizeN can be confined
in a strip by tuning the length L or the width H of the
strip.
The abrupt change scenarios of < x > /N and <
Nimp > /N shown in Fig. 7 and 8 indicate a phase tran-
sition but it is difficult to locate precisely the transition
point.
It is clear that all the chain configurations can be di-
vided into two subsets: imprisoned and escaped. Far
from the transition point, only one subset is important
in defining the average characteristics, but in the vicinity
of the transition point both subsets contribute, as shown
 0.37  0.375  0.38  0.385  0.39  0.395
FIG. 10: Average fraction of the imprisoned monomers
< Nimp > /N (solid line) and two partial contributions,
< Nimp >1 /N due to confined configurations, and < Nimp >2
/N due to escaped configurations (dotted lines) as functions of
L/N near the transition point (L/N)tr = 0.381 for L = 3200
and H = 17.
in Fig. 10. The average < Nimp > can be rewritten as
< Nimp > =
NimpWN (C1) +
NimpWN (C2)
WN (C1) +
WN (C2)
= < Nimp >1 + < Nimp >2 (22)
where C1 (C2) denotes the imprisoned (escaped) con-
figurations, WN (C1) (WN (C2)) are the total weights of
the chain for obtaining the configuration C1 (C2), and
< . . . >α denotes the partial contributions due to the
imprisoned configurations (α = 1) or the escaped config-
uration (α = 2). Similarly,
< N2imp >=< N
imp >1 + < N
imp >2 . (23)
Finally, the variance of Nimp, σ
2(Nimp) =< N
imp > − <
Nimp >
2 can be expressed as follows,
σ2(Nimp) = σ
1(Nimp)+σ
2(Nimp)−2 < Nimp >1< Nimp >2
where σ21,2(Nimp) =< N
imp >1,2 − < Nimp >21,2. It is
shown in the Appendix that the variances of partial con-
tributions to the total number of imprisoned monomers,
σ21(Nimp) and σ
2(Nimp) are much better suited for locat-
ing the transition point than the full variance σ2(Nimp)
because of the very asymmetric behavior of the latter.
We present σ21(Nimp)/N and σ
2(Nimp)/N as functions of
L/N in Fig. 11 for various values of H and L. It is clear
that for a fixed width H , the peaks become sharper as
the length L of the strip increases. The escape transition
points are identified with the positions of the peaks of
σ21(Nimp) and σ
2(Nimp) as determined by a curve fitting.
Values of the transition points, (L/N)tr,1 and (L/N)tr,2
are the same to the third digit for fixed values of L and
H , so the transition point is taken as an average of them,
i.e. (L/N)tr = ((L/N)tr,1 + (L/N)tr,2)/2. Results of
(L/N)tr obtained by this method are listed in Table II
and presented in Fig. 12. A more detailed discussion are
given in the Appendix. A similar method was also used
for determining the transition point from the variance of
the end-to-end distance x. Results are also listed in Ta-
ble IIand presented in Fig. 12. All the values of (N/L)tr
are plotted in Fig. 12 against H1/3. The best fit gives
= 1.025(35)H1/3 (25)
which is consistent with the estimation of Eq. (21) within
the error bar.
VI. LANDAU THEORY: DISTRIBUTION OF
THE ORDER PARAMETER
A. Analytical theory
The approach based on the Landau free energy is per-
fectly suited for analyzing the relevant states in the es-
cape problem, including the metastable states. In the
Landau theory, all the configurations are first subdivided
into subsets associated with a given value of the order
parameter, s , and summation is performed separately
within each subset. The full partition function can be
obtained then by integrating over the order parameter:
Z = exp(−F ) =
ds exp(−Φ(s)) (26)
where Φ(s) is the Landau free energy function, i.e. the
non-equilibrium free energy taken as a function of the
order parameter. In the vicinity of the first order tran-
sition point, the Landau free energy is expected to have
two minima (one stable and the other metastable). Our
analysis will be based on finding the metastable minima
and the associated thermodynamic characteristics. The
proper choice of the order parameter is not always ob-
vious, nor are there any standard recipes for making it.
One criterion is quite clear: the average value of the or-
der parameter should allow one to distinguish between
two phases. For a first-order transition, the average or-
der parameter changes jump-wise. We require that the
properly chosen order parameter changes continuously as
the system evolves from a metastable state, through the
transition state at the top of the barrier, and eventu-
ally falls into the equilibrium minimum. We have shown
earlier [13] that these criteria are satisfied if the order
parameter is defined as the chain stretching in the con-
fined coil state, s = r/(Na) where r is the instantaneous
end-to-end distance of the chain of N monomers, or as
the stretching of the stem only in the flower conforma-
tion: s = L/(na), where n is the number of monomers in
the stem. In analogy to the Gaussian case, the Landau
function consists of two branches that have to be intro-
duced separately. As the chain is in an imprisoned state,
(a) (b)
 1500
 3000
 4500
 6000
 7500
 9000
 0.1  0.2  0.3  0.4  0.5
H=129
L =   800
L = 1600
L = 3200
L = 6400
 1500
 3000
 4500
 6000
 7500
 9000
 0.1  0.2  0.3  0.4  0.5
H=129
L =   800
L = 1600
L = 3200
L = 6400
FIG. 11: Variances of the number of imprisoned monomers divided by N , (a) σ21(Nimp)/N , for the imprisoned state, and (b)
σ22(Nimp)/N , for the escaped state, plotted against L/N . The height of peaks increases with L for a fixed value of H .
TABLE II: Values of the transition points, (L/N)tr, determined from the analysis of the variances σ
1(Nimp) and σ
1(x) for the
imprisoned states, and from the variances σ22(Nimp) and σ
2(x) for the escaped states.
(L/N)tr,Nimp (L/N)tr,x
H L = 800 1600 3200 6400 L = 800 1600 3200 6400
9 0.4790 0.4766 0.4703 0.4649 0.4790 0.4766 0.4723 0.4674
17 0.3831 0.3846 0.3810 0.3754 0.3829 0.3836 0.3810 0.3754
33 0.3027 0.3060 0.3071 0.3036 0.3024 0.3078 0.3070 0.3036
65 0.2363 0.2412 0.2434 0.2455 0.2357 0.2409 0.2433 0.2455
129 0.1816 0.1880 0.1916 0.1932 0.1803 0.1877 0.1914 0.1930
the Landau free energy is directly expressed in terms of
the distribution of the end-to-end distance. There exists
no closed-form formula for such a distribution of confined
chains with excluded volume interactions. However, the
distribution of the gyration radius for 3d chains confined
in a tube was studied analytically and numerically in
[24]. It was proposed that the free energy of a confined
chain with a given gyration radius rg can be presented
as a sum of two terms:
F (rg) = N
Acα +B
, (27)
where c is the segment volume concentration expressed
as a function of the gyration radius and the confinement
geometry, α and δ are linked to the space dimension d
and the Flory exponent ν by α = (νd− 1)−1 and δ =
(1− ν)−1. The first term describes the concentration
effects in the des Cloizeaux [25] form, the second term is
the Pincus [26] scaling form of the stretching free energy,
and A and B are model-dependent numerical coefficients
of order unity. Instead of rg, we use the same ansatz,
Eq. (27), to describe the end-to-end distance distribution
by taking c = Na2/rH , α = 2, and δ = 4. The free
energy of the chain in an imprisoned state as a function
of s is hence given by
Φimp(s) = N
, s ≤ L
. (28)
Since we prefer to keep the basic scaling formula of the
Landau free energy in order to provide predictions in a
simple analytical form, here we are not going to consider
the further logarithmic correction terms as shown in [24].
In the thermodynamic limit, the average value of the
order parameter for the imprisoned state, Simp, is found
by locating the minimum of Φimp(s), i.e. dΦimp(s)/ds =
0 at s = simpeq , and hence
Simp = s
eq = (A/2B)
1/6(a/H)1/3 . (29)
The minimum of the Landau free energy gives the free
energy for the imprisoned state at equilibrium
Fimp = Φimp(Simp) = 3B
N . (30)
 1.5  3  4.5  6  7.5
L =   800, Nimp
L = 1600, Nimp
L = 3200, Nimp
L = 6400, Nimp
L =   800, x      
L = 1600, x      
L = 3200, x      
L = 6400, x      
FIG. 12: Transition points (N/L)tr versus H . The dashed
line is (N/L)tr = 1.025(35)H
1/3 and gives the best fit of the
data.
Compared with Eq. (13), this is indeed the correct scal-
ing of the free energy. The end-to-end distance at equi-
librium is found as
RN = NaSimp = (A/2B)
1/6(H/a)−1/3Na , (31)
which is consistent with the result of the blob model,
Eq. (12).
As the chain is in an escaped state, the formula of the
free energy function is identical to Eq. (28), but corrected
for the fact that only the n monomers that are part of
the stem contribute:
Φesc(s) = n
s−3 +Bs3
, s ≥ L
.(32)
The average value of the order parameter in the escaped
state, Sesc, is found by locating the minimum of Φesc(s)
and is given by
Sesc = s
eq = (A/B)
1/6(a/H)1/3 (33)
Thus, the free energy of the escaped chain at equilibrium
Fesc = Φesc(Sesc) = 2(AB)
The transition point is found from the condition that the
two minima of the Landau free energy function are of
equal depth. Using Eqs. (30) and (34) we get
It is interesting to calculate the size of jumps implied
by the Landau theory in the order parameter, the im-
prisoned monomers and the end-to-end distance at the
N    /Nimpr
(3a/2   )(A/B)   (H/a) L/N
S/Sesceq
5/3                 1/6               −1/3
(3a/2   )(A/B)   (H/a) L/N
5/3                 1/6               −1/3
FIG. 13: Based on the Landau theory, the theoretical predic-
tions of the average values of (a) the fraction of imprisoned
monomers Nimp/N , and (b) the order parameter S, are plot-
ted against L/N . The chain is in an imprisoned state for
L/N > (3a/25/3)(A/B)1/6(a/H)1/3, and in an escaped state
for L/N < (3a/25/3)(A/B)1/6(a/H)1/3.
transition. Using Eqs. (29) and (33) we immediately get
the reduced jump of the order parameter
Sesc − Simp
= 1− 2−1/6 ≈ 0.1091 , (36)
which is independent of H and the coefficients A and
B. For an imprisoned state, < Nimp >= N by defini-
tion, while for the coexisting escaped state with the same
choice of H , L, and N we have only < Nimp >= L/Sesc
monomers. From Eqs. (33) and (35), we obtain the rela-
tive reduction in the number of imprisoned monomers
∆Nimp
N − L/Sesc
≈ 0.055 . (37)
This number has a simple meaning of the fraction of the
chain escaping out of the confinement at the transition
point. It is much smaller than 1/2 in the Gaussian chain
model, but non-zero in contrast to the blob model. Fi-
nally, the reduced jump of the end-to-end distance is ob-
tained by combining Eqs. (31) and (35)
≈ 0.0572 . (38)
Eqs. (36)-(38) show that the sizes of jumps in S, <
Nimp > /N and RN/L are universal quantities. Results
for the average order parameter S and the average frac-
tion of imprisoned monomers < Nimp > /N predicted by
the Landau theory are shown in Fig.13. Comparing with
the numerical results shown in Fig. 8 and Fig.9, we see
that the Landau theory a good qualitative agreement.
The predicted free energy of the chain at equilibrium,
Eqs. (30) and (34) follow the same scaling behavior as
obtained by the MC simulations shown in Eqs. (19) and
(20). This allows us to identify the numerical values of
the constants A and B for our model: A ≈ 1.057 and
B ≈ 0.975.
B. Numerical comparisons
Here we focus on the results of the Landau free energy
of polymer chains partially confined in a strip of width
H = 28 and of length L = 800, 1600, 3200, and 6400.
Since PERM gives the possibility to estimate directly the
partition sum and the properly normalized histograms,
the Landau free energy as a function of s, Φ(N,L,H, s),
is given by
Φ(N,L,H, s) = − ln
P (N,L,H, s)
Z0(N)
where P (N,L,H, s) =
walks δs,s′ is the histogram of
s, and the partition sum of the partially confined chains
can be written as
Z(N,L,H) =
P (N,L,H, s) (40)
in accordance with Eq. (26). In Fig. 14, we plot four
sets of results of the Landau free energy per monomer
Φ(N,L,H, s)/N versus the order parameter s for L =
800, 1600, 3200 and 6400. Since the transition point is
near H1/3, the histograms are obtained for N/L = 3.05,
3.10 and 3.15 for each set. The predicted analytical re-
sults of ΦP (s) = Φimp(s) for the imprisoned state and
ΦP (s) = Φesc(s) for the escaped state, given by Eqs. (28)
and (32) are also shown for comparison. On the left-hand
side of the branch points, due to the finite-size effect, we
see that the excess free energy for the imprisoned state
(the minimum of the curve) at s = s
eq,L converges to
the predicted value (the minimum of the curve ΦP (s)) of
polymer chains confined in an infinite strip at s = simpeq
slowly as L increases but s
eq,L is slightly larger than s
as L → ∞. The difference between those curves corre-
sponding to the different ratio N/L is almost invisible
for a fixed value of L as predicted by Eq. (28). On the
right-hand side of the branch points, we see that only
those curves for L = 800 finally develop a parabola-like
behavior with fluctuations and they are more concave
than those curves predicted by Eq. (32). It shows that
PERM has difficulties to sample configurations in the es-
caped regime as L increases and gives an explanation why
we should not trust the size of those jumps that appear
in Fig. 9 too much. However, one can easily overlook
the existence of two minima in such a delicate situation.
With PERM, at least we are able to give evidence for
this two minimum picture of the first-order like transi-
tion. We also see that additional finite-size correction
terms should be taken into account for the theoretical
predictions in Eqs. (28) and (32).
 0.023
 0.024
 0.025
 0.026
 0.027
 0.25  0.3  0.35  0.4  0.45
N/L=3.05
N/L=3.10
N/L=3.15
L=  800
L=1600
L=3200
L=6400
ΦP(s)/N
FIG. 14: The Landau free energy divided by N ,
Φ(N,L,H, s)/N , plotted against s for various values of L
and H = 28. The predicted Landau free energy functions,
ΦP (s) = Φimp, Eq. (28), in the imprisoned regime and
ΦP (s) = Φesc, Eq. (32), in the escaped regime are also plotted
(dashed lines).
Taking the results for L = 6400 as a reference, we
plot the same data but shift all other curves by some
constants, c0,L = −0.00235, −0.00109, −0.00044 for L =
800, 1600, and 3200 to make the three branch points
for N/L = 3.05, 3.10, and 3.15 coincide with each other
in Fig. 15. According to the prediction by Eq. (28), we
should expect that the four curves for different values of L
overlap with each other in the imprisoned regime. In fact,
it is not the case but the difference between these curves
decreases as L increases, and finally they will converge to
one curve as L becomes very large. In the escaped regime,
surprisingly, we see that those curves corresponding to
different L all overlap with each other for a fixed ratio of
N/L as predicted by Eq. (32). Although the lack of data
for larger L, precludes very strong conclusions, we may
assume that these curves all show the same behavior as
the curve for L = 800, and do further analysis.
In order to determine the transition point and extract
an accurate value for the jump in the order parameter
from simulations, we use two parabolic functions gimp(s)
and gesc(s) to fit the numerical data in the imprisoned
and escaped regimes, respectively:
gimp(s) = a1,L(s− simpeq,L)
2 + c1,L (41)
gesc(s) = a2(s− sesceq,L)2 + c2 + b2
where a1,L, c1,L, a2, c2, b2, s
eq,L, and s
eq,L are deter-
mined by curve fitting, and results are shown in Table III
and Fig. 15. From the condition of equal depth of minima
gimp(s = s
eq,L) = gesc(s = s
eq,L), (43)
 0.023
 0.024
 0.025
 0.25  0.3  0.35  0.4  0.45
N/L=3.05
N/L=3.10
N/L=3.15
L = 6400
L = 3200
L = 1600
L =   800
FIG. 15: The Landau free energy divided by N ,
Φ(N,L,H, s)/N , plotted against s for various values of L and
H = 28. The two minima of Φ(N,L,H, s)/N are determined
by fitting g(s) = gimp(s), Eq. (41), in the imprisoned regime,
and g(s) = gesc(s), Eq. (42), in the escaped regime, going
through those lower points around the two minima, respec-
tively.
 0.06
 0.08
 0.12
 0.14
 0.16
 0.18
 0  0.0003  0.0006  0.0009  0.0012  0.0015
curve fitting
estimation
Eq. (36)
FIG. 16: The reduced jump of the order parameter ∆S/Sesc
plotted against L−1.
we obtain the transition points (N/L)tr = 3.13(2),
3.10(2), 3.09(1), and 3.08(1) for L = 800, 1600, 3200,
and 6400, respectively, which are in perfect agreement
with the results given by free energy, Eq. (21), and the
results given by the variance of the end-to-end distance
and the imprisoned monomers, Eq. (25).
The values for the reduced jump of the order parame-
sesceq,L − s
sesceq,L
obtained by the curve fitting are plotted in Fig 16 against
L−1 together with the direct estimates in the simulations
TABLE III: Results of the coefficients a1,L, c1,L, a2, c2, b2,
eq,L, and s
eq,L for the curve fitting in Fig. 15.
L a1,L c1,L s
eq,L a2 c2 b2 s
800 0.6281 0.02277 0.2945 0.6518 0.0497 -0.0086 0.3510
1600 0.6336 0.02303 0.2986
3200 0.6675 0.02313 0.3007
6400 0.6872 0.02320 0.3017
and the prediction by the analytical theory, Eq. (36). We
see that ∆S/Sesc decreases as L increase. As L → ∞,
it remains finite and the value is slightly larger than the
predicted value by the analytical theory. However, in
view of the numerical uncertainties of our curve fitting
we consider that the predictions of the analytical theory
and the results by the MC simulations agree with each
other quite well.
VII. SUMMARY AND DISCUSSION
In this paper we attack the problem of the 2d-escape
transition by combining several approaches. We first
compare two simple pictures of the transition predicted
for Gaussian chains and by a blob model. This compar-
ison is useful from a general pedagogical point of view
since the two models are in a sense complimentary: each
captures some essential features of the phenomenon while
failing in some other aspects. Both models are attrac-
tive because of their clarity, and although mathemati-
cally simple, lead to non-trivial results including finite-
size effects in a phase transition. The third approach
that was proposed in this paper attempts at incorporat-
ing the excluded volume effects in the framework of the
Landau theory. We were not able to present an exact
theory since it would require a detailed understanding
of the end-to-end distribution of confined self-avoiding
chains. To the best of our knowledge this problem is still
not well explored. The simulations presented allowed us
to evaluate the transition condition Eq. (21) which rep-
resents the binodal line in the (H,L/N) plane. It is of
interest to extend the simulations in order to locate the
spinodal lines where one of the states looses stability, and
to construct the full phase diagram. It is also possible to
explore the properties of metastable states and their life-
times controlled by the barrier heights. It is clear from
the results on the distribution of the order parameter,
Fig. 14, that the PERM algorithm experiences difficul-
ties with sampling the configurations belonging to the
escaped state, especially for long chains. The escaped
branch of the distribution is cut-off quite sharply, which
means that the important set of configurations charac-
terized by larger stretching degree in the stem is vastly
underrepresented. This is a generic problem that one en-
counters when dealing with first-order transitions when
the properties of the phases differ significantly. In our
case, the PERM algorithm based on chain growth tech-
nique is perfectly tuned to generate homogeneous config-
urations of imprisoned chains but fails with strongly in-
homogeneous escaped configurations. It is worth noting
that a naive determination of the jumps in the average
order parameter would have lead one to a wrong con-
clusion that the jump disappears in the thermodynamic
limit. Again, we expect this to be a generic problem when
simulating weak first-order transitions. The most reliable
analysis of the nature of the transition would require a
detailed examination of the order parameter distribution.
Acknowledgements
We are grateful to the Deutsche Forschungsgemein-
schaft (DFG) for financial support: H.-P.H. was sup-
ported under grant NO SFB 625/A3, while L.I.K. and
A.M.S. received partial support under grants NO 436
RUS 113/863/0 and RFBR 05-03-32003-a. H.-P.H.
thanks P. Grassberger and W. Paul for very helpful dis-
cussions.
Appendix
In this appendix, we discuss the finite-size behavior in
the fluctuations in the number of imprisoned monomers
Nimp in more detail. Following the technique of finite-size
scaling analysis for first-order transitions as described
in [27], we write down the probability distribution of the
fraction of imprisoned monomers m = Nimp/N in the
two-state model
P (m) = δ(m−m1)
e(t−ttr)a
e(t−ttr)a + e−(t−ttr)a
(m−m2)
e−(t−ttr)a
e(t−ttr)a + e−(t−ttr)a
.(45)
The first term accounts for the imprisoned state with m
strictly equal to m1 = 1, while the second term describes
the distribution ofm in the escaped state in the Gaussian
approximation with the equilibrium average of m equal
to m2 and dispersion σ0; t is the control parameter, ttr
is its critical value at the transition point, and P (m) is
normalized,
P (m)dm = 1 (46)
At the transition point t = ttr,
P (m) =
δ(m−m1) +
(m−m2)
which obeys the “equal-weight rule”, while for t 6= ttr
the relative weight of the two states is exp [2(t− ttr)a].
The constant a−1 describes the range of t over which
the transition is smeared out. For the Gaussian approx-
imation to be meaningful the dispersion of m in the es-
caped state, σ0, must be small compared to the difference
∆m = m1 − m2. Taking t = L/N and using Eqs. (30),
(32), and (34) of the Landau theory, one expects the fol-
lowing scaling: a−1 ∼ H2/3/L and σ20 ∼ H/L.
Since the probability density is a sum of two contribu-
tions, P (m) = P1(m) + P2(m), the k-th moment of m is
defined by
< mk >=
mkP (m)dm =< mk >1 + < m
k >2 (48)
where < mk >1,2=
mkP1,2(m)dm. Therefore, the first
and second moment are given by
< m >= m1p1 +m2p2 , (49)
< m2 >= m21p1 + (m
2 + σ
0)p2 , (50)
here p1 = e
(t−ttr)a/2 cosh [(t− ttr)a] is the relative weight
of the imprisoned state, and p2 = 1 − p1 is the relative
weight of the escaped state.
Instead of a δ-function singularity at t = ttr, the vari-
ance of the fraction of imprisoned monomers in a finite
system becomes
< m2 > − < m >2= p1p2(∆m)2 + p2σ20 (51)
which shows a smooth asymmetric peak close to t = ttr
of approximate height ∆m2+σ20 . Here ∆m is the relative
reduction in the number of imprisoned monomers at the
transition point, for which the analytical Landau theory
predicts a value of 0.055, see Eq. (37). The first term in
Eq. (51) is symmetric with respect to the transition point
since p1p2 = 1/(4 cosh
2 [(t− ttr)a]). The second term,
however, is asymmetric, as it describes the intrinsic fluc-
tuations in the escaped state. The resultant asymmetry
is clearly seen in Fig. 17. We conclude that the full vari-
ance of m is ill suited for a precise determination of the
transition point.
The situation is quite different if we analyze the vari-
ances calculated with the partial probability densities
P1(m) and P2(m) restricted to the imprisoned (escaped)
configurations. In the simulations, the product Nσ21,2
was calculated. The variance due to the imprisoned con-
figurations only (with m1 = 1) gives a perfectly symmet-
ric curve as a function of the control parameter:
Nσ21(m) = Nm
1p1p2 =
4 cosh2 [(t− ttr)a]
with the peak value ofNtr/4 ≈ LH1/3/4. Numerical data
presented in Fig. 18 supports this prediction with very
high accuracy. The variance due to escaped configura-
tions is somewhat modified by the intrinsic fluctuations
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.37  0.375  0.38  0.385  0.39  0.395
∆Nimp/N
0.1 σ1(m)
0.1 σ2(m)
1-<m>
FIG. 17: The square root of the variance σ1(m) for the im-
prisoned states, σ2(m) for the escaped chains, 1− < m >,
σ(m) of the chain either in an imprisoned state or in an es-
caped state, and the difference ∆σ = σ1(m)− σ2(m) against
L/N .
 1000
 2000
 3000
 0.37  0.375  0.38  0.385  0.39
L = 3200, H = 17
Eq. (53)
A  (L,H)1
  (L,H)1
FIG. 18: Variance due to the imprisoned configuration
multiplied by N , Nσ21(m), plotted against L/N for L =
3200 and H = 17. The solid curve is the best fit of
Eq. (52), Nm21/4cosh
2 [(t− ttr)a], with the height of the peak
A1(L,H) = N(m1)
2/4 ≈ 2099.74, the FWHM Γimp(L,H) ≈
1.7627/a = 0.0035, and the position of the peak ttr,1 =
(L/N)tr,1 = 0.3810. The FWHM are given by the distance
between points on the curve shown at which the correspond-
ing height reaches half height of the peaks (half maximum).
in the escaped state
Nσ22(m) = N(m
2p1p2 + p2σ
≈ Nm22p1p2 =
N(1−∆m)2
4 cosh2 [(t− ttr)a]
but in contrast to Eq. (51), the asymmetric term is al-
ways negligible. Indeed, the coefficient with the sym-
metric term, m22 = (1 − ∆m)2 is close to 1 while both
quantities (∆m)2 and σ20 are quite small. In Fig. 17,
 0.01
 0.02
 0.03
 0.04
 0  0.0003  0.0006  0.0009  0.0012  0.0015  0.0018
H = 129
H =   65
H =   33
H =   17
H =     9
FIG. 19: FWHM Γα(L,H) for the imprisoned state (α = 1)
and for the escaped state (α = 2) against L−1. The dashed
curves are a1,H(H/L) + b1,H(H/L)
2 and give the best fit of
the data. Values of a1,H and b1,H are listed in Table IV.
TABLE IV: Results of the coefficients a1,H , b1,H , a2,H , c2,H
and d2,H for the curve fitting in Fig. 19 and 20.
H a1,H b1,H a2,H c2,H d2,H
9 0.8221 0.7841 0.9940 0.2131 2.2802
17 0.6845 -2.0403 0.9845 0.0961 0.4540
33 0.5600 -2.3531 0.9725 0.0430 0.0570
65 0.4301 -1.3774 0.9545 0.0173 0.0095
129 0.3185 -0.7332 0.9220 0.0070 0.0017
we plot the full dispersion σ(m) (the square root of the
full variance) that includes contributions from all config-
urations, and partial dispersions σ1(m) and σ2(m) due
to imprisoned and escaped configurations separately, the
difference ∆σ = σ1(m) − σ2(m), as well as the aver-
age fraction of escaped monomers, 1− < m >, as func-
tions of L/N . It is clear that the full curve is strongly
asymmetric in contrast to partial dispersion curves, in
good agreement with the theoretical description above.
On the other hand, the curve of ∆σ shows the same be-
haviour as the curve of σ(m) near the transition point,
and the heights of these two peaks correspond to the
half size of the jump ∆m = m1 − m2 = ∆Nimp/N .
By fitting the partial variances Nσ21(m) and Nσ
as functions of L/N according to Eqs. (52) and (53),
respectively, we obtain the full width at half-maximum
(FWHM), Γ(L,H) = 2arccosh(
2)/a, the height of the
peak Aα(L,H) = Nm
α/4, and the transition point
ttr,α = (L/N)tr,α for α = 1 (imprisoned configurations)
and α = 2 (escaped configurations).
One example of the curve fitting for L = 3200 and
H = 17 is shown in Fig. 18. Note that the peak height
and the transition point are related to the theoretical
prediction A1(L,H) ≈ LH1/3/4 with very high accuracy.
(a) (b)
 0.0005
 0.001
 0.0015
 0.002
 0.0025
 0  0.0003  0.0006  0.0009  0.0012  0.0015
H =   9
H =  17
H =  33
H =  65
H = 129
 0.0005
 0.001
 0.0015
 0.002
 0.0025
 0.003
 0  0.0003  0.0006  0.0009  0.0012  0.0015
H =   9
H =  17
H =  33
H =  65
H = 129
FIG. 20: Inverse of the height of the peaks for the imprisoned state (a) A−11 (L,H), and for the escaped state (b) A
2 (L,H),
plotted against L−1. The dashed curves are (a) a2,H(4H
−1/3/L) and (b) c2,H(H/L) + d2,H(H/L)
2, and give the best fit of the
data. Values of a2,H , c2,H , and d2,H are listed in Table IV.
 0.05
 0  0.03  0.06  0.09  0.12  0.15
H =  33
H =  65
H = 129
0.54H/L+0.058
Eq. (37)
FIG. 21: The relative reduction in the number of imprisoned
monomers ∆m, plotted against H/L.
Results of Γα(L,H), Aα(L,H) for α = 1 and for α = 2,
and ttr,α are shown in Fig. 19, 20 and 12. In Fig. 19, we
see that the full widths Γα(L,H) for α = 1 and for α = 2
are overlapped with each other, and Γα → 0 as 1/L → 0
by fitting the data using a1,H(H/L) + b1,H(H/L)
2. In
Fig. 20, the inverse of the height A−11 (L,H) → 0 as
1/L → 0 by fitting the data using a1,H(4H−1/3/L) and
c1,H(H/L)+d1,H(H/L)
2. Since Γα → 0, and A−1α → 0 as
1/L → 0, i.e. a delta function, a sharp phase transition
occurs in the thermodynamic limit. It is a strong indica-
tion [27] that the transition is first-order like. Values of
the coefficients a1,H , b1,H , a2,H , c2,H and d2,H are listed
in Table IV.
The relative reduction in the number of imprisoned
monomers
∆m = m1 −m2
= 2(A
2 )/N
1/2 (54)
Results of ∆m for various values of H and L plotted
against H/L are shown in Fig. 21. We see that there
exist the systematic errors at small H/L. Finally we
obtain ∆m ≈ 0.058 at H/L → 0 by a curve fitting, which
is slightly larger then the prediction, Eq. (37).
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