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Continuous area-preserving models for self-interacting polymers
D.C. Khandekar, F.W. Wiegel
To cite this version:
D.C. Khandekar, F.W. Wiegel. Continuous area-preserving models for self-interacting polymers. Jour-
nal de Physique, 1989, 50 (3), pp.263-272. �10.1051/jphys:01989005003026300�. �jpa-00210916�
263
Continuous area-preserving models for self-interacting polymers
D. C. Khandekar (1, *, **) and F. W. Wiegel (2)
(1) Research Centre BiBos, University of Bielefeld, 4800 Bielefeld, FR Germany
(2) Center for Theoretical Physics, Twente University, Enschede 7500 AE, The Netherlands
(Reçu le 5 juillet 1988, accepté sous forme définitive le 4 octobre 1988)
Résumé. 2014 Nous obtenons les distributions de probabilité dans deux modèles de polymères
formés de courbes fermées du plan entourant une aire fixe. Dans le premier modèle, les
monomères interagissent de façon quadratique ; dans le second, les courbes sont contraintes à faire un nombre de tours constant autour de l’origine.
Abstract. 2014 We derive probability distributions for two models for polymers constrained to a
plane, both of which consider closed loops which enclose a fixed area. In the first model the
monomers interact with each other with a quadratic interaction ; in the second model the loops
are also constrained to have a fixed winding number around the origin of the plane.
J. Phys. France 50 (1989) 263-272 ler FÉVRIER 1989,
Classification
Physics Abstracts 36.20Ey - 05.40+j
1. Introduction.
Topological problems continue to generate much interest in theoretical physics. Their importance is being recognized, for example, in the context of quantum mechanics, optics and polymer science. In spite of this there are very few exactly solvable models, which were
reviewed in [1].
Recently, Breretôn and Butler [2] suggested a simple way to take entanglement constraints into account, in the context of polymer physics. They specifically consider a single polymer in
a plane, which they represent by a discrete random walk of N steps, each of length f. The steric and topological effects of the other polymers in the system are approximated by
the constraint that this walk encloses a fixed (algebraic) area. The Brereton-Butler model can
be studied both numerically, as in [2], and analytically, as shown by the authors in [3] and [4].
In our view the Brereton-Butler model, although somewhat crude, deserves further consideration as it is simple and amenable to explicit calculation. In this paper we consider the
(*) Alexander von Humboldt fellow.
(**) Permanent address : Theoretical Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005003026300
model, in the continuous limit, with interactions. The first case corresponds to repeating units
which interact with each other through a quadratic self-interaction which can be attractive or
repulsive. Second, we solve the model with an
«interaction » which is equivalent to another topological constraint, more explicitely we solve the model in which the polymer configur-
ations : (i) enclose a fixed algebraic area A and (ii) have a given entanglement index
n around a fixed point, say the origin.
The present paper, therefore, constains the first solvable example in which self-interaction appears in the context of a topological constraint. It also illustrates explicitly how to take into
account two or more topological constraints (a problem hinted at in an early publication [5]).
Although we formulate the problems in the context of polymers we expect that the results might turn out to be applicable to a wider class of systems. For example, self-interaction with
an area constraint, in imaginary time, can be used to study disordered systems in magnetic
fields [6]. In many cases topological constraints can be interpreted to correspond to
movements in magnetic fields [1], therefore most of these models can be used to study the quantum dynamics of charged particles in magnetic fields (provided time is allowed to take
imaginary values).
Although the present paper is concemed with the physics of polymers, problems regarding
the distribution of the area of a plane random walk have been studied by mathematicians for
quite some time. Indeed, it was Paul Lévy who for the first time introduced the concept of the stochastic area, i.e. the area of the region of the plane enclosed by the Brownian curve and the chord connecting the origin with the terminal point of the Brownian motion, averaged
over all Brownian curves [7]. This is an example of the often-noticed phenomenon of a
mathematician developing concepts which find physical applications only after many years
(half a century in this case). The interested reader is also referred to work of Spitzer [8] who
studied winding angle distributions, and to the monograph by Itô and McKean [9]. Of course,
the present work studies more general situations in the context of polymer science.
To make our models even more realistic one should of course include the effects of excluded volume. This would lead to the study of self-avoiding random walks with various
topological constraints. Although analytical work is now prohibitive, the method of the renormalization group leads of approximate results due to Puri et al. [10], Rudnick and Hu [11] and Duplantier and Saleur [12].
2. Area distribution for a polymer with quadratic self interaction.
The configuration sum for a polymer model whose repeating units interact with each other
through a quadratic interaction and which is constrained to enclose a fixed area
A, can be obtained from the generating function :
The positive sign before the quadratic interaction applies to the case of a repulsive pair
interaction between the repeating units while the negative sign holds for attractive
interactions. The quantity f2 0 2essentially determines the strength of the self interaction. We
shall calculate the generating function explicitly for attractive interactions, from which the
results for repulsive interactions can be found by analytic continuation no - i f2(). We shall
perform the integral (2.1 ) over all configurations with the initial monomer at r’ =
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(x’, y’ ) and the final monomer at r" = (x ", y" ). Since the area is meaningful only for closed loops we shall eventually set r’ = r", which can be set = 0 in view of the translational invariance of the model.
The explicit evaluation of (2.1) starts by rewriting the second term in the exponential in the
form
Further, representing the delta function in (2.1) as a Fourier integral one finds
The second exponent in the integrand has a non-local character. The path integration for such
a problem is much easier if a transformation enables one to represent the non-local factors as an average over local terms. Using the identity
with
equation (2.4) becomes
Note that the path integration in (2.7) indeed involves a local
«Lagrangian » (2.8).
This Lagrangian is formally identical to the Lagrangian of a charged particle in (1) an
external quadratic potential ; (2) a constant electric field ; (3) a constant magnetic field perpendicular to the x, y plane. This problem has been studied in the literature in the context of path integrals, cf. [13] and [14]. In order to use this earlier work one transforms the vector r
to 9 by means of
Substitution into (2.8) gives
The fourth and fifth term on the right hand side, when integrated over v, give a contribution
where
At this point we use equation (2.16) of [13] and identify fi with 1, t with
N, m with with and w with to find
where
In order to evaluate the (unnormalized) probability distribution for the area we put
lq
=r’
=r"
=0 in (2.7) and use (2.13) with the following result for the Fourier transform
Although this is just an intermediate result in the polymer context, the last equation is an explicit formula for the density matrix of a charged
«two-time
»oscillator in a magnetic field,
in imaginary time, which is quoted in [6] as an unpublished result of Sa-Yakanit.
When (2.17) is substituted into (2.3) one finds (note that there is no N’ dependence
anymore)
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We could not evaluate this integral explicitly because of the g dependence of ,f2’, cf. (2.14).
The probability distribution of the area is given by
In the limit f2 0 --+ 0 this reduces to equations (21, 22) of reference [3] or to equation (3.1) of
reference [4].
Further we might also mention that (2.13) enables one to find an explicit formula for the Fourier transform (2.4) of the full generating function (i.e. arbitrary ’1). First note that the
term multiplying ’1) in (2.7) depends only on the position N’ of the specific repeating unit.
Therefore, 2(g ) can be immediately written as
Next, note that both G (r ", N 1 r, N’ ) and G(r, N’ 1 r’, 0 ) are essentially Gaussians in r by
virtue of equation (2.13). Therefore, the integrations in equation (2.20) involve two Gaussian integrations and can be done to arrive at a closed form expression for the Fourier transform of the generating function. However, since in this paper we do not discuss the statistical mechanical properties of these systems we shall not give the explicit expression for the generating function.
Finally, we consider the case of a repulsive quadratic self-interaction, in which we put
where f2l is real and positive. With these substitutions (2.19) gives
where
It can be seen from (2.17) that the integral over u converges only when N 1 f2’l 1 7r for imaginary values of ,l2’. Imaginary values of f2’ occur only for 0 - 9 - go; in this case N 1 n’ 1 will be smaller than ir if N nl - 7T B/2. Hence for a long polymer with quadratic repulsion the validity of (2.22) is restricted to very weak forces.
As has been pointed out earlier a model involving purely quadratic interactions is rather crude in the sense that the molecule either tends to form a globule (when the interaction is
attractive) or tends to spread out (if the interaction is repulsive). One can approach the
realistic situation more closely if the attractive interactions dominate at large monomer separations while at short distances the monomers predominantly repel. This means one might add an additional term
to the exponent in (2.1 ) . However, when such a term is included, the resulting path integral
can no longer be evaluated analytically. To a reasonable approximation, however, if we approximate V 2 by
the problem can be studied analytically to some extent. Note, however, that (2.25) is not as strongly repulsive as (2.24).
To evaluate the path integral (Zl ) associated with the action functional
in the presence of an area contraint, one first sums over all those paths which not only enclose
a fixed area A, but also preserve the value of the integral
The actual path integral is then obtained by summing over all values of u. Explicity this means
By representing the second delta function by its Fourier representation it is easy to see that
where Z (A, nÓ) is the probability distribution (2.18) ; the quantity fii is related to
fio by the relation
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3. Two constraints : area and winding number.
In this section we discuss a case in which two constraints play a role. The first one is identical
to the constraint of section 2 and limits the polymer configurations to those which have a fixed
algebraic area A. The second constraint fixes the entanglement index (0) of the polymer configuration with respect to some fixed point in the plane. Taking this point as the origin of polar coordinates r and 0 with 0 r oo and 0 8 2 TT these two constraints read
The configuration sum can be expressed by
where the dot denotes differentiation with respect to v . The normalization of Q will be
determined shortly by a separate argument.
The introduction of the constraint on the entanglement index makes the plane multiply-
connected. Hence the path integral has to be performed on paths in this multiply-connected
space. In these situations one can construct a universal covering space in which the path integral is unique and well defined. A new way of doing such path integrals on a covering
space was developed by the authors in a previous publication [15].
The covering space in this case is constructed by distinguishing various values of 9 which differ by 2 7r and thereby give a different entanglement index ; this leads to
-
oo « 0 + oo. Hence the paths in the covering space are now characterized by functions r ( v ) and 0 ( v ) with 0 v N, 0 r + oo, - oo 8 + oo. Also, note that in covering
space the paths have the property
where the integer n is the winding number (n
=0 for paths which do not wind around the
origin. Positive values of n signify anti-clock wise winding while negative values denote
clockwise winding).
Again we represent the first delta function in (3.3) as a Fourier integral, which gives
where
Note that the factor which appeared inside the path integral in (3.3), is
now independent of the path and has therefore be brought outside the path integral in (3.5).
The last term £4/16 r2 in (3.7) deserves some comments. It was shown by several authors that
an extra term appears in the Lagrangian when a path integral is transformed to curved coordinates. A full analysis of the case of polar coordinates can be found in [14] ; an application in [15]. It is also shown in the review [14] how the formal factor fi r ( v ) is
v
cancelled by a similar factor due to the asymptotic formula for the Bessel functions, leaving only the factor (r’ r" )-112. Introducing the transformed angular variable
the Lagrangian takes the form
This Lagrangian is quadratic in the .p coordinate, hence the corresponding integration is straightforward, leading to
where M is defined by
and is still a functional of r ( v ).
Further, the quantities fjJ" and fjJ’ can be evaluated by their defining equation (3.8) and are given by
The next step in this calculation is to use the one-dimensional analog of (2.5)
with
to obtain
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The last integral is the Greens function for a particle moving in a quadratic potential plus an inverse-square potential. This Greens function has been worked out by Khandekar and Lawande (details can be found in [14]) ; the result is
Combining (3.17) with (3.15) and (3.7) one finds for closed configurations (r’ = r" - ro, 8 "
=0’ = 8 0) the normalized probability P,, (A) dA that the entanglement index will equal
n, and the enclosed area is between A and A + dA :
1
