We discuss how the introduction of quenched impurities changes the exponents of a self-avoiding walk on a lattice. We find that γ, the exponent for th...

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Z. Phys. B - Condensed Matter 55, 41-44 (1984)

Zeitschrift ffir Physik B

9 Springer-Verlag 1984

Self-Avoiding Walks on Randomly Diluted Lattices J.W. Lyklema and K. Kremer Institut ftir Festk6rperforschung der Kernforschungsanlage Jiilich, Federal Republic of Germany Received December 23, 1983 We discuss how the introduction of quenched impurities changes the exponents of a self-avoiding walk on a lattice. We find that V, the exponent for the number of walks, does not change. On the other hand the exponent v for the mean square end to end distance does change. This is caused by a singular normalization at P=Pc, which is necessary to compensate for the allowed number of walks on the diluted lattice.

Introduction

The Average Number of SAW's

Recently there has been a considerable interest in the properties of self-avoiding walks (S.A.W.) on randomly diluted lattices [1-8]. These are lattices where a fraction ( l - p ) of the sites is eliminated at random. The disorder is quenched thus one can not interchange the thermal and the disorder average. In this paper we want to study the influence of the dilution on the statistical properties of SAW's on such lattices. We only study the usual SAW, where each configuration has the same probability and not the so-called true SAW [9] where the weight of a configuration depends on the number of previously visited neighbouring sites. The first walk is studied in connection with polymer physics because it describes the statistical properties of a single polymer chain in a good solvent, whereas the second one may be used as a model to study cluster growth phenomena. Considerable confusion exists about the question whether quenched impurities have an influence on the universal behaviour of the walk or not. There are authors [7] who claim that there is no change in the critical exponents at all, while others [3, 10] suggest that there should be a change at Pc, the percolation threshold of the lattice. To analyze this problem in more detail we clarify the difference between the averaging procedure for a normal random walk and a SAW. With this information it is then possible to show which quantities are modified by the dilution.

We study, for a given disorder configuation C, a non returning random walk with nearest neighbour jumps only on a lattice with coordination number q. Because a direct return is forbidden we have only qo = q - 1 positions to jump to. Now we define our one step transition probability from site s' to s, Q(s/s') to be 1/%, independent of the walkers position on the lattice. This, together with the restriction that the walk stops at the moment it tries to return to a previously visited site, defines the SAW on the complete lattice. For the diluted lattice we extend this restriction to the empty sites. With these definitions we can study the statistics of the polymer problem because every configuration of length n carries the same weight. With the assumption that at step n + 1 the walk hasn't returned to a previously visited site, we have for the probability to be at site s at step n + 1 the recurrence relation en+l (s)= E ~(s) Q(s/sn) Pn(sn).

(1)

8n

Here e(s) accounts for the imcompleteness of the lattice, that is e(s) equals 0 or 1 depending on whether site s is empty or not in a particular configuration C. To calculate P,+l(s) we iterate (1) to get the formal solution

P,+ I (s)= ~, e(s) ~(s,)... e(sl) Q(s/s,) (s~}

9 Q(sjs,_ 1)... Q(sl/So) Po(so)

(2)

42

J.W. Lyklema and K. Kremer: Self-Avoiding Walks

with the initial condition Po(so)=e(So). The sum is performed over all self avoiding paths {s~}, which do not depend on the dilution configuration C. Because the transition probability Q(s/s') is a constant we can write this as

P,+ l (s)=qo("+ l) G,+ l (s; C)

(3)

where G,+l(S; C), the number of SAW's of length n + 1 from s o to s on configuration C, is the quantity which is usually studied in Monte Carlo calculations. In this expression the influence of the randomness only shows up in the products of the e(si). Because of the self avoiding character of the walk every site s~ is visited only once and consequently every e(s~) occurs only once in this product. Therefore we can immediately perform the disorder average to get

e(s) e(s,).., e(sl) e(So) = e(s).., e(So) = p , + 2

(4)

the overbar denotes the average over a Bernoulli distribution PB(zi) for every site s~

PB(e) = p c~(e- 1) + (1 - p ) 6(0.

(5)

Here p gives the amount of allowed sites and ( l - p ) the eliminated ones. This factorization does not occur for random walks because there a site can be visited more than once. With this result we can average (2) to get P,+ 1(s) = p , + l ~ Q(s/s,) Q(s,/s,_ 1)-.. Q(sl/so)Po(so) (s.}

= P 2 Q~(s/sn) e. (Sn).

(6)

Sn

Except for the factor p this equation is the same as the one we have for the translational invariant case (p= 1). Thus we can write for the average number of walks starting at s o

we get for the average number of walks On(p) : C(p) qenff(p) n ' - 1.

So it follows that the critical exponent ? does not change by introducing disorder even for P

Other Mean Values Recently [7] this line of reasoning has been used to argue wrongly that no exponent at all changes due to the dilution. In this section we show that due to the self-similarity of the infinite cluster more complicated averages than the one described in the last section indeed have different exponents at Pc. As an example we study the mean square end to end distance ( R 2 ) . For a given disorder configuration C the thermal average is defined as

F s2G.(s, c) y s2e~(s, c) s

2 G,(s, p) = a,(p) = p"+ ~ a,(p = 1).

(7)

(R~(C))-

n [11] O~ = 1) = co q:ff n"- 1

(8)

where c o is a constant which only depends on the lattice type and qeff is the effective coordination number of the walk. If we define the same quantities with dilution as qeff(p)=P'qeff

c(p) =p . c o

s

~ G.(s, C)

2 P.(s, C)

s

s

On the complete lattice one usually writes for large

(10)

The normalization is necessary because ~ P~(s, C)< 1 8

due to the self avoiding character of the walk and the constraints of the disorder configuration C. If we perform the disorder average over Eq. (l l) we encounter the same difficulty. Not every configuration C allows for SAW's of length n. For such situations (11) is not defined. Therefore we have to normalize this expression with the sum over Cn, the configurations on which a SAW of length n exists.

(R2.) = ~ P ( C . ) ( R ~ ) / ~ P(C.). (9)

(11)

$

Cn

C~

(12)

J.W. Lyklema and K. Kremer: Self-AvoidingWalks

43

To study the effect of this average we define

x = 1 - ~ P.(s, C)

(13)

s

with 0 < x < 1 and write for the numerator of (12)

s2 co

P(Cn) Pn(s, Z P.(s, co) $

= Z s 2 Z P(C,)P,(s, s

disorder average are interchanged, this is not allowed (15). In this context it is necessary to mention another approach by Derrida [4] who introduces favourable and unfavourable bonds to account for the dilution. In this transfer matrix method also in the limit of completely "forbidden" bonds, parts of the walk have to exist on such unfavourable bonds. This by definition is a different problem.

Discussion

Cn

dilution the net effect is that the mean square end to end distance is larger on the diluted lattice than on the complete lattice.

To study the dilution effect numerically one needs to do very accurate Monte Carlo simulations. This is normally done by calculating the thermal average (R~) starting at one or more initial sites s o for one particular disorder configuration C. This procedure is repeated several times for different disorder configurations C and with these results the disorder average ( R ,2) is calculated. For an actual Monte Carlo calculation where we study finite SAW's this procedure has the disadvantage that there is a nonzero probability tNt(p) to start at a finite cluster of size t. Here Nt(P) for P>Pc is defined as the probability to have a cluster of size t [15].

(R~2) > (R~(p = 1)).

Nt(p)~exp(_cc(p)t a ).

If we perform the disorder average we have, due to the presence of the product x k P,(s, C), terms like el e2 el e3 =p3 >p4.

(15)

So the disorder average introduces correlations which cause the increase of the numerator of (12) with increasing dilution, but no singular behaviour at Pc is introduced at this point. Because the denominator ~ P(C,) becomes smaller with increasing C~

d-1

(16)

This is easy to understand because on a diluted lattice the effective coordination n u m e r qeff(P) is smaller than on the complete lattice and thus in an n step walk, the walker has to walk further to find a new position. At p~p+~ this effect will become singular. This can be understood as follows. Because we are interested in asymptotic results only, the relevant cluster to study is the infinite cluster. This has a correlation length ~ ( p - p c ) -~. [13, 16] which diverges at Pc. The SAW on such a self-similar cluster then may have different properties compared to the complete lattice. These come from the denominator in (12) because the probability to start a walk at the infinite cluster and thus to have an infinite SAW approaches zero [13, 14, 15] with

pP(p) =p(p -pc) ~.

(17)

Here P(p) is the probability that an occupied site belongs to the infinite cluster. Thus { ~ P ( C , ) } -1 C~

diverges at pc(/~>0) for n ~ o o and this gives rise to a different singular behaviour compared with the p = 1 case, as was found in [3]. This is in clear contradiction with the conclusion that nothing will change due to dilution [7]. This is partly caused by the absence of the denominator of our (12) in this derivation, a procedure which does not apply if one studies polymer statistics. Also the thermal and the

(18)

The total probability to start a n-step walk at a finite size cluster is less than (p > Pc) d--1

Nt(p)dtocexp(_~(p) n a )

(19)

n

since not every n-site cluster allows for an n-step SAW. Clearly this contribution disappears in the limit n--,oe, leaving only the contribution of the infinite cluster, which does not depend on n (17). Thus if we only study short chains, the finite clusters may dominate the behaviour. A solution [31 is to start every walk at the infinite cluster. Thus in practise one has to built clusters so large that also SAW's of the maximal obtainable length can live on it. Near Pc where the correlation length ~ is larger than the maximal chain extension, this procedure is capable of probing the self-similar structure of the infinite cluster. First results of such a Monte Carlo calculation [3] indeed show an increase for the prefactor and no change for the exponent v~-0.59 for P>Pc. For p approaching Pc a crossover scaling to a new exponent vp ~2/3 was found. This result is not yet precise enough because for a very accurate determination of vpo one needs much longer chains closer to Pc than are given in this study. However the remarkable agreement with a modified Ftory formula v w = 3/(2 +ds) should be noticed. Inserting for the fractal dimension ds of the infinite cluster the value 2.5

44 one obtains vpc = 2/3. This of course can be accidental and it is therefore interesting to produce more accurate M o n t e Carlo results b o t h to get better n u m e r i c a l values for the e x p o n e n t a n d to test the usefulness of the F l o r y formula in practical situations. A n o t h e r i m p o r t a n t p o i n t is to test the influence of the finite clusters, this however is a n even more extensive calculation t h a n the first one. At this p o i n t we would like to speculate o n the influence of d i l u t i o n o n the true SAW. As discussed in the int r o d u c t i o n the t r a n s i t i o n probabilities Q(s/s') depend on its e n v i r o n m e n t as opposed to the p o l y m e r walk where Q(s/s')=l/q o is constant. Therefore it is n o t possible to perform the disorder average over the products of e(Si) as in (4) a n d we can n o t say anything definite a b o u t the e x p o n e n t 7. F o r the m e a n square end to end distance this is different. There the influence of the self-similar cluster should also enter in the same way as for the p o l y m e r walk a n d change the e x p o n e n t v. I n c o n c l u s i o n we have shown that the e x p o n e n t ~ of the S A W does n o t change u n d e r the influence of dilution. Also we have f o u n d that the i n t r o d u c t i o n of the disorder n o r m a l i z a t i o n (12), which behaves singular at Pc, causes other exponents to increase [17]. The authors thank W. KinzeI, K. Binder and B. Derrida for useful discussions.

References 1. Hiley, B.J., Finney, J.L., Burke, T.: J. Phys. A10, 197 (1977) 2. Chakrabarti, B.K., Kert~sz, J.: Z. Phys. B - Condensed Matter 44, 221 (1981)

J.W. Lyklema and K. Kremer: Self-AvoidingWalks 3. Kremer, K.: Z. Phys. B - Condensed Matter 45, 149 (1981) 4. Derrida, B.: J. Phys. A - Atoms and Nuclei 15, L 119 (1982) 5. Roy, A.K., Chakrabarti, B.K.: Phys. Lett. 91 A, 393 (1982) 6. Chakrabarti, B.K., Bhadra, K., Roy, A.K., Karmarkar, S.N.: Phys. Lett. 93 A, 434 (1983) 7. Harris, A.B.: Z. Phys. B - Condensed Matter 49, 347 (1983); Kim, Y.: J. Phys. C 16, 1345 (1983) 8. Rexakis, J., Argyrakis, P.: Phys. Rev. B 28, 5323 (1983) 9. See e.g. Amit, D.J., Parisi, G., Peliti, L.: Phys. Rev. B 27, 1635 (1983) 10. Derrida, B.: Les Houches Lectures, 1983 11. de Gennes, P.G.: Scaling concepts in polymer physics. New York: Cornell University Press 1979 12. Noaimi, G.F.AI., Martinez-Mekler, G.C., Velasco, R.M.: preprint Martinez-Mekler, G.C., Moore, M.A.: J. Phys. (Paris) Lett. 42, L413 (1981) 13. Stauffer, D.: Phys. Rep. 54, 1 (1979) 14. Essam, J.W.: Rep. Prog. Phys. 43, 832 (1980) 15. Kunz, H., Souillard, B.: J. Stat. Phys. 19, 77 (1978) 16. Aharony, A., Gefen, Y., Mandelbrot, B., Kirkpatrick, S.: In: Abstracts of International Conference on disordered systems and localization, Rome (1981) 17. After submitting this work we received a preprint from Rareal, R., Toulouse, G., Vannimenus, J.: (to be published, J. de Physique, March 1984) where they also study this problem. They predict through the use of a Flory type of argument a lower value for v on the diluted lattice, in contradiction to our result

J.W. Lyklema K. Kremer Institut f'tir Festk~Srperforschung Kernforschungsanlage Jiilich GmbH Postfach 1913 D-5170 Jiilich 1 Federal Republic of Germany