Random polymers

Citation for published version (APA):Hollander, den, W. T. F. (1995). Random polymers. (Katholieke Universiteit Nijmegen. Mathematisch Instituut :report; Vol. 9527). Radboud Universiteit Nijmegen.

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DEPARTMENT OF MATHEMATICS

UNIVERSITY OF NIJMEGEN The Netherlands

RANDOM POLYMERS

Frank den Hollander

Report No. 9527 (July 1995)

DEPARTMENT OF MATHEMATICS UNIVERSITY OF NIJMEGEN T oernooiveld 6525 ED Nijmegen The Netherlands

Random polymers

Frank den Hollander Mathematical Institute University of Nijmegen

Toernooiveld 1, 6525 ED Nijmegen The Netherlands

e-mail: denholla@sci.kun.nl

Abstract

This paper is a mini-review of some recent developments on probabilistic models for polymer chains.'

AMS 1991 subject classifications. 60F05, 6OFIO, 60K35, 60J15. Key words and phrases. Polymer measures, limit theorems, large deviations.

• Invited paper to appear in a special issue of Statistica Neerlandica celebrating the 50-th anniversary of the Dutch Society for Statistics and Operations Research.

o Introduction

A polymer is a long chain of molecules connected by chemical bonds. Polymers occur in a variety of natural and synthetic materials, e.g. rubber, cellulose, soap, polyester and plastics. Sometimes the chains form a tight network, sometimes they float around in a solvent or are absorbed onto the. surface of a substrate.

There is a host of models in physics and chemistry describing characteristic phenom­ena triggered by the polymer structure (both macroscopic and microscopic). For a general overview we refer the reader to the work of Flory (1953, 1969) and de Gennes (1979). Most of the models are by far too complex for a rigorous treatment.

In mathematics, in particular in probability, one studies caricature models of polymers where only the barest effects of excluded volume are incorporated. In this paper we shall describe some of these models. The results obtained sofar are rather modest, but they reveal an interesting mathematical structure as well as some typical technical difficulties, both of which will persist in more complex realistic models. The probabilistic approach has seen major developments only in the last ten years.

A polymer has two characteristic properties:

(1) 'irregular structure' (due to entanglement).

(II) 'stiffness' (due to steric hindrance).

1

To catch these properties, one considers models based on lattice random walks with a self­repellent interaction. The key question addressed is: 'How is the polymer distributed in space?' The easiest version of this question reads: 'What is the distance between the end­points of the chain as a function of its length (as counted by the number of constituent monomers)?' More quantities are worth studying, but in this paper we shall only be con­cerned with the end-to-end distance.

0.1 The basic model

Consider the d-dimensional integer lattice 'll.d. Let

(1)

be simple random walk, i.e., the process Xo = 0, Xn = Y1 + ... + Yn (n ~ 1) where (Yi),>l are LLd. random variables such that Yi = e with probability ia for all e with lei 1. L~t Pn denote its probability law on n-step paths.

We think of Pn as modelling property (I), since it is the uniform distribution on n­step paths starting at the origin. To include also property (II), we use Pn as a reference measure and build in a self-repellent interaction as follows. Pick (3 E [0,00] and define a new probability law Q~ on n-step paths by setting

(2)

Here the sum in the exponential counts the total number of self-intersections of the path until time n, and a penalty exp( - 2(3) is paid for each one of them. zg is the normalization constant, which equals the expected value of the exponential under the reference measure Pn (in physics language: the 'partition sum').

The new measure Q~ is called the n-polymer measure with strength of repellence (3. The path under this measure describes the polymer chain (with the constituent monomers being represented by Xl, ... , Xn and the interaction between the monomers by the penalty). Intuitively, one expects that the self-repellent interaction causes the path to be more 'spread out' than under Pn . The two limiting cases (3 ° and (3 = 00 correspond to the free random walk (Q~ = Pn ) resp. the self-avoiding random walk (Q~ is the uniform distribution on it-step paths without intersection). For (3 E (0,00), equation (2) defines what is called the Domb-Joyce model of 'soft polymers' (Madras and Slade (1993) Section 10.1).

Note that (Q~)n::cO is not a consistent family, i.e., Q~ is not the projection on n-step paths of a stochastic process evolving in time (like Pn ). The reason is that the normalization constant depends on n. This does not mean that the model in (2) is wrong. It simply means that (2) describes a physical situation in which polymers of a fixed length try to arrange themselves in a surrounding space (e.g. a solution or substrate), rather than a situation where polymers are grown in the process. Different models are studied for the latter situation, which we shall not discuss here (see e.g. T6th (prepriut 1994».

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0.2 Mean-squared displacement

One quantity that gives us an idea of how the self-repellence causes the chain to 'spread out' is the mean-squared displacement

Eet. (X~) J X~ dQ~ (3)

(X~ is the square of the Euclidean norm). For fJ = 0 life is easy, since we know that Ep" (X~) = n (n 2 0) in any dimension ('pure diffusive' behavior). For fJ = 00, on the other hand, we have EQl:"(X~) = n2 (n 2 0) in d 1, because weight ~ goes to the two paths running straight left reap. straight tight ('pure ballistic' behavior). However, for fJ = 00 and d 2 2 very little is known about the behavior of (3) (this is the famous self-avoiding walk problem), and for general fJ even less is known.

The following is folklore (based on simulations and heuristic arguments; see Madras and . Slade (1993) Section .1.1):

Conjecture 1 For fJ E (0,00]

(4)

where D = D(fJ, d) > 0 is the amplitude and II = lIe d) > 0 is the exponent. The latter is believed to be independent of fJ and to be given by

d 1 2 3 4

II 1 ! 0.588... ~+

The values in this table should be compared with what we know for the free random walk: lI(d) = ~, D(O, d) = 1 for all d 2 1. Thus, we see that when fJ E (O,~] diffusive behavior occurs only in d 2 4, while in d 1,2,3 the self-repellence qualitatively changes the behavior of the path to make it 'superdiffusive'. In other words, in low dimensions the self-repellence is strong enough to make the polymer spread out faster than the .;n typical for the free random walk, while in high dimensions not. Apparently, in d 2 4 the space is so large that the self-intersections are subdominant (although certainly D(fJ, d) > 1). In physics language: for d 2 4 the Domb-Joyce model is in the same universality class as the free random walk.

REMARKS:

(a) An old conjecture due to Flory says thatl/(d) = maxH, k}. Though this seems correct in d = 1,2 and 2 4, simulations show that 1/(3) < ~ (Madras and Slade (1993) Section 2.2).

(b) Actually, the case d = 4 is critical. One expects that (4) has logarithmic correction terms: EQ~(X~)- Dn(logn)1/4 (n ~ 00) (Brydges, Evans and Imbrie (1992)).

(c) Numerical values are known for D(oo, d) but these are not very accurate (Hara and Slade (1994».

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0.3 What has been proved?

So far Conjecture 1 has been proved only in the following two cases:

(il d 2:: 5: Hara and Slade (1992a,b) prove (4) using a diagrammatic expansion and resummation technique called the 'lace expansion'. Analytically this technique can be carried through to prove that (4) holds for d sufficiently large. But with a computer­assisted proof the dimension can be brought down to 5. The proof is for f3 = 00, but eaSily carries over to finite f3 since the convergence gets better.

(ii) d = 1: Greven and den Hollander (1993) prove iI) using large deviation techniques combined with variational analysis.

The situation for d = 2, 3, 4 is very bad: there are no analytical proofs, not even crud~ esti­mates or inequalities. Thus, for the physically most relevant cases (i.e., solvent or substrate) we are still in the dark.

In Section 1 we shall describe some of the mathematical results known in d = 1. Of course, 1 is a somewhat 'sad' dimension for a polymer to live in, but as a mathematician one has to be modest. (P. Holewijn (VU, Amsterdam) calls this 'polymers for beginners'.) Still, we shall see some interesting phenomena, most of which are bound to survive in higher dimensions. In Section 2 we shall describe variations on the basic model motivated by physically different situations.

1 Theorems for the basic model in d = 1

1.1 A law of large numbers

Theorems 1 and 2 below are taken from Greven and den Hollander (1993).

Theorem 1 For every f3 E (0, (0) there exists O*(f3) E (0,1) such that

lim Q~(I.!.Xn 0*(f3) I > € I Xn > 0) = 0 for all € > O. n-+oo n

The number 0* (f3) is called the asymptotic speed and has the following properties:

(5)

(i) limp.to 0'(f3) 0, liml1too 0*(f3) = 1, (6) (ii) f3 -+ 0*({3) is analytic on (0,00).

Note that Theorem 1 proves (4) with the identification: vel) = 1, D(f3, 1) 0* (f3)2.

The proof of Theorem 1 is rather involved. What one has to do is something like the following (dictated by large deviation theory):

(a) Fix e E [-1,1] and count the number of n-step paths whose end-to-end distance is approximately en.

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(b) For each of the paths in (a) compute the exponential weight factor in (2). (This depends on additional parameters besides (J, associa.ted with the 'local times' of the path at the sites it visits.)

(c) Multiply the results of (a) and (b).

(d) Maximize the product in (c) over (J (and the additional parameters). The maximum is taken at (J = ±O*(f3) (Le., modulo the sign this is the 'most probable' speed of the path). ,

The details of these . steps are tricky and complicated. But let us not dwell on this and formulate the recipe to which (a-d) lead for finding (J*(f3), which is explicit and easy.

For r E lR. and f3 E (0,00), define the N x N-matrix

Ar,p(i,j) er (i+j-I)-P(i+j-l}2 P(i,j) (i,j;:::: 1), (7)

where P is the Markov matrix

(8)

Also define >'(r,f3) is the largest eigenvalue of Ar,p in f2(N). (9)

Theorem 2 Fix f3 E (O,oo). Let r*(f3) E (0,00) be the unique solution 0/ the equation

A(r, (3) = 1. (10)

Then

(11)

The form of the matrix Ar,p, as given in (7), comes out of an underlying large deviation principle and is not particularly illuminating. Roughly, the Markov matrix P handles the statistics of the self-intersections of the free random walk (under the law Pn ), while the exponential factor in front of P handles the effect of the self-repellence, coming in through (2). Condition (10) is a normalization property, and the extra parameter r (appearing beside (3) plays the role of a Lagrange multiplier.

Although the above recipe for 0* (f3) is quite straightforward, it is unfortunately difficult to get a good analytical grip on >'(r, (3). This is mainly due to the quadratic part in the exponential front factor in (7), which embodies the essence of the polymer interaction. How­ever, by appealing to some standard techniques from functional analysis one can at least prove the statements formulated in (6).

Intuitively appealing is the following:

Conjecture 2 f3 --t (J* (f3) is increasing.

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Indeed, as the strength of repellence increases the path should 'spread out' more and therefore .increase its speed. Absurd as it may seem, this property is still open. One easily deduces from (10-11) that Conjecture 2 is equivalent to a 2-nd order differential inequality for >.(r,/3) with mixed derivatives. Unfortunately, this inequality iooks rather forbidding. Apparently, the monotonicity property of the speed lies deeply hidden in the model. This fact is directly related to the observation made ill Section 0.1 that (Q~)n>O is not a consistent family. In fact, there is no natural way to compare Q~ and Q~f for /3 ;f:-/3' (e.g. via 'coupling techniques'). Numerical simulations, however, support Conjecture 2.

1.2 Finer asymptotics: a central limit theorem

Theorem 3 below is taken from Konig (preprint 1994) and extends Theorems 1-2.

Theorem 3 For every /3 E (0,00) there exists (J"*(f3) E (0,00) such that

. (Xn - O*(f3)n I ) J.!..~ Q~ (J"*(f3)v'n :::; C Xn > 0 = N«D, cD for all c E JR, (12)

where N is the normal distribution with mean 0 and variance 1. The number (J"'(/3) is called the asymptotic spread and is given by

(13)

The proof of Theorem 3 comes out of delicate transformation arguments extending the large deviation techniques used to prove Theorems 1-2. The representation for (J"*(f3)2 can be traced back to the curvature in the point O'(f3) of the rate function for the speed (recall (a-d) in Section 1.1), which fits the folklore of large deviation theory.

1.3 Scaling in the weak interaction limit

Theorem 4 below is taken from van der Hofstad and den Hollander (1995) resp. van der Hofstad, den Hollander and Konig (in preparation). It shows that the polymer displays a peculiar type of scaling behavior as the strength of repellence /3 becomes small.

Theorem 4 There exist b*, c* E (0,00) such that as /3 t 0

0* (f3) b* f3i (J"*(f3) -+ c*.

(14)

The result in Theorem 4 is obtained by studying the scaling behavior of A (r, (3) (defined in (9» for r, /3 4- o. It turns out that b*, c* can be expressed in terms of the largest eigenvalue of a certain Sturm-Liouville operator with non-constant coefficients (acting on functions. in L(R+)). This operator catches the full scaling picture of the polymer, not just the speed and the spread but also the statistics of the 'local times'. Numerically one finds b* ~ 1.11 and c ~ 0.7.

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1 The power (33 in (14) shows that the speed is not analytic in (3 = 0. This tells us that

the weakly interacting polymer in d = 1 cannot be understood as a pert~rbation of the free random walk. This confirms what we already saw from the table in Section 0.2: for every (3 E (0,00] the one-dimensional polymer is in a different universality class than the free random walk. (Perturbation expansions only work in d 2: 5, as demonstrated by the' work of Hara and Slade (1992a,b) cited in Section 0.3.) Furthermore, the fact that c* < 1 shows once more that the process has an essential singularity at (3 = 0, since the asymptotic spread of the free random walk is equal to 1 by the classical central limit theorem for i.i.d. increments.

2 Other polymer models

In this section we describe some variations on the 'plain vanilla' model discussed sofar. There are various possible directions, each motivated by a specific physical or chemical application.

2.1 Elasticity

Replace the exponential weight factor in (2) by

exp [-(3 L 1(li - jl)1{x,=Xj}]' O:$i.j$n

i#j

(15)

where I : 1N --+ ffi.+ is some function that is decreasing to zero. Here the penalty paid for a self-intersection decays as the time between the intersections increases. This decay models the effect of 'elasticity' in the polymer. Short loops are more expensive than long loops, because they require the polymer to bend more on itself. We denote the corresponding measure on n-step paths by Q~,J.

Conjecture 3 Let d = 1 and Ip(i) = i-Po For every (3,p E (0,00) there exists O*((3,lp) E [0, 1) such that

For all (3 E (0,00) O*((3,lp) > °

° ilpE(0,1) ifpE[1,00).

(16)

(17)

Thus, p ~ 1 is a critical value where the self-repellence becomes so rapidly decaying with time that the positive speed no longer survives. The short loops are not numerous enough to push the path away at a positive speed. (A natural problem arising from Conjecture 3 is to find the necessary and sufficient condition on I such that 0*((3, f) = 0. I believe I can show that 2::i2:1 I(i) < 00 ===} 0*((3, f) = 0, but the reverse is not true.)

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, The only partial result known to date,is due to Kennedy (1994): if p E (0,1), then for every 6 > 0 there exists flo flo (6) such that

lim Q{J,/p (! Xn > 1 61 Xn > 0) !for all fl ~ flo. n .... oo n n (18)

This shows that in the regime p E (0,1) the speed is sensitive to fl. The proof of (18) uses renormalization group techniques.

In the regime p E (1, (0) a finer conjecture is the following:

Conjecture 4 Let d= 1. For every fl E (0,00), under Q~,Jp as n -t 00 .

Xn is of order n2- p

Xn is of order Vn ifp E (1,~) ifpE [~,oo).

(19)

The second line of (19) is in accordance with the result proved by Brydges and Slade (preprint .1994) that the behavior is diffusive when instead of Ii - jl-P we put n-P in the weight factor.

There are some similar conjectures in d ;::: 2 but no proofs either (Caracciolo et a1. (1994)).

2.2 Repulsion and attraction

This time replace the exponential weight factor in (2) by

(20)

Here appears an additional parameter "( E (0, (0), which gives the path a rewardexp(2"() each time it 'touches itself'. This reward models the effect of the environment on the polymer. Suppose, for instance, that the polymer consists of molecules that do not like to make contact with the molecules of the solution or substrate. Then, by touching itself frequently the polymer can reduce this contact.

Conjecture 5 There exists a critical curve in the (fl,"()-plane where a 'collapse transition' occurs, i.e., below the curve the polymer has a positive speed and above the curve zero speed.

The collapse transition is discussed in Brak, Owczarek and Prellberg (1993). In the collapsed phase the polymer is believed to be subdiffusive, possibly even localized. (I believe I can prove that in d = 1 the critical curve is linear, but the details have not been worked out.)

For d ;::: 2 nothing is known rigorously.

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2.3 Inhomogeneity

This time change not the exponential weight factor in (2) but the reference measure Pn .

Namely, instead of the simple random walk with steps restricted to nearest-neighbor lattice sites, we pick as our reference process a random walk with a larger step size. More specifically, define Xn = Yl + ... + Yn (n ~ 1) as in (1), but now Y; = e with probability (2R + I)-d for all e E [-R, R]d n 'l1,d, with R some positive integer. So, Pn is the uniform distribution on n-step paths whose steps are at most R in the lattice norm.

We think of the new probability law, defined as in (2), as modelling a polymer consisting of different types of monomers, ordered in a random fashion, i.e., an inhomogeneous polymer. The law of the orientation and distance between the monomers is that of the single steps in the path.

Konig (1994, preprint 1994) proves the analogues of Theorems 1-3 (in d = 1) for this finite range model. In principle, a recipe similar to (7-11) and (13) leads to the identification of the asymptotic speed and spread, however, the formulas are considerably more complex and less explicit. Therefore, little is known sofar about the ,B-dependence or about numerical values.

If R > 1, then the one-dimensional model is even .interesting in the limiting case ,B = 00

(the self-avoiding walk). If we write (J*(oo,R) to denote the asymptotic speed for this case, then (J*(oo,R) E (O,R) for all R > 1 (Konig (1993)). The following conjecture is due to Aldous (1986):

1 2 Conjecture 6 (J*(oo,R) ~ 3-3R3 as R -+ 00.

There are no results for d ~ 2, although the behavior should be provably diffusive when d ~ 5, in the spirit of Hara and Slade (I992a,b).

3 Epilogue

The models discussed above are extreme caricatures of reality. Not only do they handle little more than the rudimentary aspects of polymer interaction, we know next to nothing at a rigorous level for the physically most interesting cases d = 2 and 3. Thus, mathematically we are in the dark ages. Still, some interesting results and conjectures start to lead the way.

There are very exciting questions like: 'What does a three-dimensional polymer look like when viewed from the side? Is it roughly spherical or does it rather look bumpy? How big are the holes one can see through?' The answer to these questions lies in the future.

References

[1] D. Aldous, Self-intersections of I-dimensional random walks, Probability Theory and Related Fields 72 (1986) 559-587.

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[2] R. Brak, A.L. Owczarek and T. Prellberg, A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles, Journal of Physics A: Mathematical and General 26 (1993) 4565-4579.

[3] D.C. Brydges, S.N. Evans and J.Z. Imbrie, Self-avoiding walk on a hierarchical lattice in four dimensions, Annals of Probability 20 (1992) 82-124.

[4] D.C. Brydges and G. Slade, The diffusive phase of a model of self-interacting walks. Preprint 1994. To appear in Probability Theory and Related Fields.

[5] S. Caracciolo, G. Parisi and A. Pelissetto, Random walks with short-range interaction and mean-field behavior, Journal of Statistical Physics 77 (1994) 519-543.

[6] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca 1953.

[7] P.J. Flory, Statistiml Mechanics of Chain Molecules, Wiley, New York 1969.

[8) P.G. de Gennes, Smling Concepts in Polymer Physics, Cornell University Press, Ithaca 1979.

[9] A. Greven and F. den Hollander, A variational characterization of the speed of a one­dimensional self-repellent random walk, Annals of Applied Probability 3 (1993) 1067-1099.

[10] T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. 1. The critical behaviour, Communications in Mathematical Physics 147 (1992a) 101-136.

[11] T. Hara and G. Slade, The lace expansion for self-avoiding walk in five or more dimen­sions, Reviews in Mathematical Physics 4 (1992b) 235-327.

[12] T. Hara. and G. Slade, Mean-field behaviour and the lace expansion, in: Probability and Phase Transition (G. Grimmett, ed.), NATO ASI Series C, Volume 420, Kluwer, Dordrecht 1994, pp. 87-122. .

[13] R. van def Hofstad and F. den Hollander, Scaling for a random polymer. Communica­tions in Mathematical Physics 169 (1995) 397-440.

[14] R. van der Hofstad, F. den Hollander and W. Konig, Central limit theorem for a weakly interacting random polymer. In preparation.

[15J T. Kennedy, Ballistic behavior in a ID weakly self-avoiding walk with decaying energy penalty, Journal of Statistical Physics 77 (1994) 565-579.

[16] W. Konig, The drift of a one-dimensional self-avoiding random walk, Probability Theory and Related Fields 96 (1993) 521-543.

[17] W. Konig, The drift of a one-dimensional self-repellent random walk with bounded increments, Probability Theory and Related Fields 100 (1994) 513-544.

[18] W: Konig, A central limit theorem for a one-dimensional polymer measure. Preprint 1994.

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[19] N. Madra.'! and G. Slade, The Self-Avoiding Walk, Birkhiiuser, Boston 1993.

[20] B. T6th, The 'true' self-avoiding walk with bond repulsion on 'lld: limit theorems. Preprint 1994. To appear in Annals of Probability.

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