- Email: [email protected]
Invariance relations for random walks on square-planar lattices
Chemical Physics Letters 406 (2005) 38–43 www.elsevier.com/locate/cplett
Invariance relations for random walks on square-planar lattices Roberto A. Garza-Lo´pez a
a,*,1
, John J. Kozak
b
Laboratory for Molecular Sciences, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA 91125, USA b Department of Chemistry, DePaul University, Chicago, IL 60604-2287, USA Received 14 January 2005; in final form 17 February 2005 Available online 17 March 2005
Abstract We outline a systematic procedure for obtaining exact analytic expressions for the mean walklength Ænæ for a random walker transiting on a d = 2 square-planar lattice with a single deep trap subject to periodic boundary conditions. As demonstrated in our earlier work onP hexagonal lattices [Chem. Phys. Lett. 371 (2003) 365], the procedure depends on generalizing MontrollÕs P first invariance relation (n = 1) to nth order. The central result of this work is the following exact analytic expression for (n) for square-planar lattices: (" ) # n1 X X 1 i n n1 ðnÞ ¼ 4 5 3 þ 12ðn 2Þ3 þ 3 : 3 N 3! i¼0 2005 Elsevier B.V. All rights reserved.
1. Introduction In this study, we outline a systematic procedure for obtaining exact analytic expressions for the mean walklength Ænæ of a random walker transiting on a d = 2 square-planar lattice. In particular, we determine Ænæ as a function of the size N of a lattice subject to periodic boundary conditions and having a single, deep trap. The classic studies of Montroll and Weiss [2–5] over 40 years ago led to amazingly accurate, asymptotic expressions for the N-dependence of the mean walklength Ænæ. For a d = 2 square-planar lattice of N sites with a simple deep trap, they found that
*
Corresponding author. Fax: +1 909 607 7726. E-mail address: [email protected] (R.A. Garza-Lo´pez). 1 Present address: Department of Chemistry, Pomona College, Claremont, CA 91711, USA. 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.078
N 1 hni ¼ A1 N ln N þ A2 N þ A3 þ A4 ; N 1 N
ð1Þ
where 1 A1 ¼ ; A2 ¼ 0:195062532; p A3 ¼ 0:116964779; A4 ¼ 0:05145660: Later, den Hollander and Kastelyn [6] corrected the coefficient A4 and determined that A4 = 0.484 065 704. With this set {Ai} of values, errors in Ænæ when compared with numerically exact values of Ænæ (see [7,8]) are essentially negligible for square-planar lattices N > 25. A systematic algebraic method for obtaining exact analytic expressions for Ænæ had also been reported years ago. In [7,8], where an exact algorithm for d-dimensional walks on finite and infinite lattices with traps was elaborated, one could develop analytic expressions for Ænæ. Taking advantage of the fact that flows to a given lattice
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
site are simply related to those from the four nearestneighbor sites, and given an exact relationship between a corner site i and its two neighboring sites (i 1) hnii hnii1 ¼ 2:
ð2Þ
one could work out algebraically the exact analytic expressions for the first few lattices; see Table 1. As is perhaps evident, there does not seem to be an obvious generalization to arbitrarily large N that springs from the results presented in Table 1. So, the issue is not that exact analytic expressions for Ænæ as a function of N cannot be obtained via straightforward algebraic methods; the issue is that to date there is no general program that can be followed to obtain the N-dependence of Ænæ for arbitrarily large lattices N. It is this program we laid out in [1] for hexagonal lattices, and extend here to square-planar lattices. The approach developed in our earlier study [1] was to return to an insight already available in the classic studies [2–5]. What Montroll and Weiss showed exactly was that the mean number of steps required for trapping from a site nearest neighbor to a single deep trap is given by hni ¼ N 1;
ð3Þ
where N is the total number of lattice sites. In effect, this result may be regarded as an invariance relation which subsequent studies have shown is valid for all d-dimensional lattices of uniform valency (connectivity) [9]. For the particular case of random walks on d = 2 hexagonal lattices, it was noticed subsequently [10] that walks from sites that are second-nearest-neighbors to the trap satisfy hni ¼ 3½ðN =2Þ 1:
ð4Þ
The question addressed in [1] was whether one could develop relations similar to Eqs. (3) and (4), invariance relations that described the average number of steps required for trapping from all sites that are third-nearestneighbor to the trap, fourth-nearest-neighbor to the trap, etc. The hope was that there would be a similarity of analytic structure in the expressions obtained, so that one could develop a general analytic expression for walks from all sites which were nth nearest neighbor sites to the trap. A first realization of this program for hexagonal lattices was the central result of that study (Eq. (12) of [1]), which we rewrite here in the equivalent form
Table 1 The values of Ænæ in the ÔN representationÕ for the 3 · 3 and 5 · 5 square-planar lattices N
Ænæ
3·3
hni ¼ ð2Þð5ÞNð2Þð3Þ N 1
5·5
2
hni ¼
ð2Þð3Þð31ÞN ð2Þð5Þð17Þ ð5Þ
N1
X
(" ðnÞ ¼ 3
n1 X
# i
n
2 N 2 þ 3ðn 2Þ2
39
n1
þ2
) ;
i¼0
ð5Þ P where P (1) is the first nearest-neighbor sum, Eq. (3), and (2) is Eq. (4). In this Letter, we follow the procedure laid down in [1] to determine the invariance relations for d = 2, square-planar lattices, subject to periodic boundary conditions with a single deep trap. However, as we shall show, more comprehensive results can be obtained for square-planar lattices since the deep trap can be positioned at the centro-symmetric site of the lattice; in slightly different language, the fact that the square-planar lattice is self-dual simplifies the complexity of the problem.
2. Invariance relations The invariance relations for square-planar lattices are determined as follows. First, numerically exact values for the mean walklength Ænæ for all sites i = 1, . . . , N of a given square-planar lattice are determined using the exact algorithm presented in [7,8] and reviewed in [9]. Then, all one-step, two-step, . . . , n-step walks to the centrosymmetric trap are tabulated and the resulting sumP mand (n) is determined for a sequence of squareplanar lattices. The results of this procedure for a family of lattices are presented in Table 2. P These data for (n) can be represented by the following set of equations: X ð1Þ ¼ 4ðN 1Þ; ð6Þ X X X X X X
ð2Þ ¼ 4ð4N 8Þ;
ð7Þ
ð3Þ ¼ 4ð13N 41Þ;
ð8Þ
ð4Þ ¼ 4ð40N 176Þ;
ð9Þ
ð5Þ ¼ 4ð123N 683Þ;
ð10Þ
ð6Þ ¼ 4ð372N 2504Þ;
ð11Þ
ð7Þ ¼ 4ð1129N 8837Þ:
ð12Þ
P Up to and including four-step walks, i.e., (4), the analytic expression that captures the above data, viz., Table 2 and Eqs. (6)–(9) is given by (" ) # n1 X X 1 i n n1 ðnÞ ¼ 4 3 N ð5 3 þ 12ðn 2Þ3 þ 3Þ : 3! i¼0 ð13Þ
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
40
Table 2 Summand values for the nth-nearest-neighbor flows for a walker on a square-planar lattice with a single (deep) trap, and subject to periodic boundary conditions P P P P P P P N (1) (2) (3) (4) (5) (6) (7) 9 25 49 81 121 169 225
32 96 192 320 480 672 896
112 368 752 1264 1904 2672 3568
1136 2384 4048 6128 8624 11,536
3296 7136 12,256 18,656 26,336 35,296
The structural similarity between this expression and the one reported earlier for hexagonal lattices, Eq. (5), is evident. For n > 4 step walks, a new feature appears, viz. cycles with one or more loops. We have determined analytic expressions for the first three correction terms nX o Dð5Þ ¼ 2 ð1Þ þ 24 1 ; ð14Þ Dð6Þ ¼ 2
nX
o ð2Þ þ 24 8 ;
62,896 110,512 170,032 241,456 324,784
185,936 330,448 511,088 727,856 980,752
the 12 immediately adjacent sites of the neighboring four unit cells. Let us write the sum of the four first-nearest neighbor sites to the trap as r1 ¼ hnð1Þi þ hnð1Þi þ hnð1Þi þ hnð1Þi and the sum of the Æn(i)æ for the second nearest-neighbor sites to the trap as r2 ¼ hnð2Þi þ hnð2Þi þ hnð2Þi þ hnð2Þi:
ð15Þ
X 5 5 Dð7Þ ¼ 2 ð3Þ þ 2 ð4! 1Þ þ ðn 4Þðn 3Þ ; 2 n P 7:
9568 21376 37,120 56,800 80,416 107,968
ð16Þ
When the analytic expression, Eq. (13), is coupled with the results above, Eqs. (14)–(16), all the data in Table 2 can be recovered exactly. P To summarize, the analytic expression for (n) is valid for all (n, N). The expressions for correction terms are exact for n = 5, 6, 7. Although we have not obtained a general expression for D(n), the above results suggest that D(n) can be approximated by X DðnÞ ðn 4Þ: ð17Þ
Using this notation, the first invariance relation can be written X ð1Þ ¼ r1 and the second invariance relation, which gives the sum total of all trajectories to the central trap that can be reached in a two-step walk, as X ð2Þ ¼ r1 þ 2r2 : Thus, using the earlier results, Eqs. (6) and (7), we have the following pair of equations: ð1Þr1 þ ð0Þr2 ¼ 4ðN 1Þ; ð1Þr1 þ ð2Þr2 ¼ 4ð4N 8Þ:
The worth of this approximation will be quantified in the following section.
In matrix form, this can be written P 1 0 r1 ð1Þ 4ðN 1Þ ¼ P ¼ ; 1 2 r2 ð2Þ 4ð4N 8Þ
3. Analytic expression for Ænæ
from which one can determine explicitly the individual ri as a function of N, and the overall average walklength can be written down explicitly
We now outline a program for calculating analytic expressions for Ænæ as a function of N for square-planar lattices. As noted earlier, progress can be made by taking advantage of the symmetry of the lattice or, more precisely, organizing the sites of the lattice in terms of symmetry classes. This procedure was introduced in [7,8] to contract the vector space in calculating numerically exact values of the walklengths from a given site i. To illustrate the method using the simplest possible example, consider the 3 · 3 lattice diagrammed in Fig. 1. All sites 1 are symmetry equivalent, as are all sites 2. To make explicit the consequences of imposing periodic boundary conditions, we have diagrammed as well
Fig. 1. Site specification for the 3 · 3 square-planar lattice accounting for symmetry.
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
hni ¼
1 1 ðr1 þ r2 Þ ¼ ð10N 18Þ: N 1 N 1
Consider now a more robust example, the 7 · 7 square-planar lattice diagrammed in Fig. 2 with a deep trap positioned at site 25. The corresponding values of the ri, each of which designates a set of symmetry-equivalent sites, are given by r1 ¼ hnð18Þi þ hnð24Þi þ hnð26Þi þ hnð32Þi;
are results previously reported in [8]. Taking these results into account, the set of defining equations X ð1Þ ¼ r1 ; X X X
r2 ¼ hnð17Þi þ hnð19Þi þ hnð31Þi þ hnð33Þi;
X
r3 ¼ hnð11Þi þ hnð23Þi þ hnð27Þi þ hnð39Þi; r4 ¼ hnð10Þi þ hnð12Þi þ hnð20Þi þ hnð34Þi þ hnð40Þi
41
X
ð2Þ ¼ 2r2 þ r3 ; ð3Þ ¼ 2r1 þ 3r4 þ r6 ; ð4Þ ¼ 4r2 þ 6r3 þ 6r5 þ r6 þ 4r7 ; ð5Þ ¼ 10r1 þ r3 þ 11r4 þ 12r6 þ 5r7 þ 10r8 ; ð6Þ ¼ r1 þ 30r2 þ 26r3 þ 6r4 þ 24r5 þ 20r6 þ 26r7 þ 15r8 þ 20r9 ;
þ hnð38Þi þ hnð30Þi þ hnð16Þi; X
r5 ¼ hnð9Þi þ hnð13Þi þ hnð41Þi þ hnð37Þi;
ð7Þ ¼ 66r1 þ 14r2 þ 30r3 þ 73r4 þ 42r5 þ 62r6
r6 ¼ hnð4Þi þ hnð28Þi þ hnð46Þi þ hnð22Þi;
þ 52r7 þ 55r8 þ 70r9 ;
r7 ¼ hnð3Þi þ hnð5Þi þ hnð21Þi þ hnð35Þi þ hnð47Þi þ hnð45Þi þ hnð29Þi þ hnð15Þi; r8 ¼ hnð2Þi þ hnð6Þi þ hnð14Þi þ hnð42Þi þ hnð48Þi þ hnð44Þi þ hnð36Þi þ hnð8Þi; r9 ¼ hnð1Þi þ hnð7Þi þ hnð49Þi þ hnð43Þi: Following from MontrollÕs work, we know that X ð1Þ ¼ r1 ¼ 4ðN 1Þ: The Ôcorner relationÕ r9 ¼ r8 =2 þ 8 and the conservation condition r8 ¼ 4r5 r4 16
reduces to a set of six equations in six unknowns. In matrix form 32 3 2 P 3 2 r2 ð2Þ 2 1 0 0 0 0 P 7 7 7 60 6 6 0 3 0 1 0 76 r3 7 6 ð3Þ 7 6 76 7 6 P 7 6 64 6 0 6 1 4 76 r4 7 6 ð4Þ 7 76 7 ¼ 6 P 7 6 60 6 7 6 7 1 1 40 12 5 7 76 r5 7 6 ð5Þ 7 6 76 7 6 P 7 6 4 30 26 19 124 20 26 54 r6 5 4 ð6Þ 5 P 14 30 17 402 62 52 r7 ð7Þ 3 2 16N 32 7 6 44N 156 7 6 7 6 7 6 160N 704 7 ¼6 6 452N 2532 7: 7 6 7 6 4 1484N 9772 5 4252N 34204 Inversion leads to values of the ri from which the overall Ænæ for the 7 · 7 lattice can be determined. The result is
1
2
3
4
5
6
7
hni ¼
ð2Þ3 ð120569ÞN ð7Þ2 ð13Þð31Þ
8
ð3Þð5Þð11Þ ð2Þð13Þð31Þ
:
ð18Þ
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
4. Discussion
43
44
45
46
47
48
49
The method presented in this paper allows one to obtain analytic expressions for the N-dependence of Ænæ. As is evident, rather than involving sophisticated analytic calculations or combinatorics, the method is technically
Fig. 2. Square-planar 7 · 7 lattice, with a deep trap positioned at site 25.
N 1
Comparison of this result with those in Table 1 for the 3 · 3 and 5 · 5 lattices reinforces the fact that deducing a generalization for Ænæ as a function of N is not obvious, if not impossible, if one focused only on the analytic representation of the Ôend resultÕ.
42
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
rather straightforward, viz., one uses an exact algorithm [7–9] to calculate numerically exact values of the mean walklength from each site of a given square-planar lattice, determines the total number of one-step, twostep, . . . , n-step walks to the trap and then, by recognizing patterns in the resulting expressions (invariance relations), seeks analytic representations of the results. Since the numerical expressions obtained involve integers, recognizing the patterns generated becomes a straightforward exercise in number theory. It is this which distinguishes our approach from the much more elegant methods introduced by Montroll and Weiss. We remark in passing that the above method was also used recently by one of us (with V. Balakrishnan) to obtain the exact analytic solution for the mean time to absorption for a random walker on the Sierpinski gasket [11] and on the Sierpinski tower [12]. In these studies, the task was lightened considerably by taking advantage of the self similarity of the underlying fractal lattices. We now comment on two further points. First, since the D(n) have been determined only up to n = 7, it is important to assess how accurate the results would be if one either neglected completely D(n) or used the approximation, Eq. (17), for D(n). To examine this point we compare the exact ri with the ri determined using these two approximations. Focusing on the 7 · 7 lattice for which exact results have been presented in the previous section,Pwe calculate the ri using only the expression (13) for (n) to construct the underlying matrix equation.P Then, we calculate ri using the exact expression (n), Eq. (13), augmented by the (approximate) correction factor, Eq. (17). A comparison of the results obtained is presented in Table 3. Approximation I gives a result with an error of 4.06% relative to the exact result equation (18), while the use of Approximation II gives a result with 1.68% error. For comparison, use of the Montroll–Weiss asymptotic result, Eq. (1), gives an error of 0.02%. So, numerically, the asymptotic estimate is superior to the results calculated using Approximations I and II. The second issue to be discussed concerns the D(n). P As noted, these corrections to (n) arise because when n > 4, there occur for the first time n-step trajectories to the trap that have cycles. To illustrate this for the 7 · 7 lattice, listed in Table 4 are the cycles found in a six-step walk for which the penultimate step in the walk is site 18 (see Table 4). As is seen, of the 20 trajectories in this set, 16 involve one intermediate intersection (crossing) and four (in bold face) involve two. Overall, removing all trajectories for which an intermediate site is the P deep trap, there are 860 trajectories which comprise (6) for the 7 · 7 lattice, of which 80 are cycles and the rest are chains. That is, already for the 7 · 7 lattice, 9.3% of the set of six-step trajectories are cycles. So, returning to Table 3, Approximation I, which neglects all cycles,
Table 3 Comparison of exact versus approximate values of the ri(n) for the 7 · 7 square-planar Table 4
r1 r2 r2 (decimal) r3 r3 (decimal) r4 r4 (decimal) r5 r5 (decimal) r6 r6 (decimal) r7 r7 (decimal) r8 r8 (decimal) r9 r9 (decimal) Ænæ (fraction) Ænæ (decimal)
Exact
Approximation I
Approximation II
192
192
192
3144 13
4668072 19747
5240408 19747
241.8461538
236.3939839
265.3774244
3488 13
5513600 19747
4368928 19747
268.3076923
279.2120322
221.2451512
7376 13
10651184 19747
10861360 19747
567.3846154 304 304.0
539.3823872
550.0258267
403472 1519
426128 1519
265.6168532
280.5319289
3872 13
7540448 19747
6909920 19747
297.8461538
381.8528384
349.9225199
7920 13
12537360 19747
13397872 19747
609.2307692
634.8994784
678.4763255
8224 13
10013408 19747
10981344 19747
632.6153846
507.0850256
556.1018889
4216 13
5164680 19747
5648648 19747
324.3076923
261.5425128
286.0509444
931 13
83068 13
85127 1209
71.61538462
68.70802316
70.41108354
Lattice.
Table 4 One intersection and two intersection (bold) cycles for a n = 6 step walk passing through the nearest-neighbor site 18 to a deep trap positioned at site 25 on the 7 · 7 lattice diagrammed in Fig. 2 11 ! 10 ! 3 ! 4 ! 11 ! 18 ! 25 11 ! 12 ! 5 ! 4 ! 11 ! 18 ! 25 11 ! 4 ! 3 ! 10 ! 11 ! 18 ! 25 11 ! 18 ! 17 ! 10 ! 11 ! 18 ! 25 19 ! 18 ! 17 ! 10 ! 11 ! 18 ! 25 11 ! 4 ! 5 ! 12 ! 11 ! 18 ! 25 11 ! 18 ! 19 ! 12 ! 11 ! 18 ! 25 17 ! 18 ! 19 ! 12 ! 11 ! 18 ! 25 17 ! 16 ! 9 ! 10 ! 17 ! 18 ! 25 17 ! 18 ! 11 ! 10 ! 17 ! 18 ! 25 19 ! 18 ! 11 ! 10 ! 17 ! 18 ! 25 17 ! 10 ! 9 ! 16 ! 17 ! 18 ! 25 17 ! 24 ! 23 ! 16 ! 17 ! 18 ! 25 17 ! 16 ! 23 ! 24 ! 17 ! 18 ! 25 17 ! 18 ! 11 ! 12 ! 19 ! 18 ! 25 19 ! 18 ! 11 ! 12 ! 19 ! 18 ! 25 19 ! 20 ! 13 ! 12 ! 19 ! 18 ! 25 19 ! 12 ! 13 ! 20 ! 19 ! 18 ! 25 19 ! 26 ! 27 ! 20 ! 19 ! 18 ! 25 19 ! 20 ! 27 ! 26 ! 19 ! 18 ! 25
leads to an error of 4% when 9.3% of the total number of six-step trajectories is removed, as well as all cycles which arise in seven-step walks, eight-step walks, . . . , nstep walks. Although the Montroll–Weiss asymptotic estimate is numerically superior, the above trajectory analysis gives a deeper insight into the ÔmicroscopicÕ differences between the exact and approximate solutions to the random walk problem. The relevance of the above results on cycles to the problem of avoiding versus self-avoiding
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
walks and to problems in polymer physics will be explored in subsequent work. Acknowledgments R.A.G.L. gratefully acknowledges The American Chemical Society Petroleum Research Fund, PRF#39961-UFS for their support while on sabbatical at Caltech. This work has been supported by a grant from The James Irvine Foundation granted to Pomona College. The authors also thank Juan J. Jime´nez Go´mez for his help in entering some of the data for this paper.
43
References [1] R.A. Garza-Lo´pez, J.J. Kozak, Chem. Phys. Lett. 371 (2003) 365. [2] E.W. Montroll, Proc. Symp. Appl. Math. Am. Math. Soc. 16 (1964) 193. [3] E.W. Montroll, G.H. Weiss, J. Math. Phys. 6 (1965) 167. [4] E.W. Montroll, J. Math. Phys. 10 (1969) 753. [5] G.H. Weiss, Aspects and Applications of the Random Walk, North-Holland, New York/Amsterdam, 1994. [6] W.Th. den Hollander, P. Kasteleyn, Physica A 112 (1982) 523. [7] C.A Walsh, J.J. Kozak, Phys. Rev. Lett. 47 (1981) 1500. [8] C.A. Walsh, J.J. Kozak, Phys. Rev. B 26 (1982) 4166. [9] J.J. Kozak, Adv. Chem. Phys. 115 (2000) 245. [10] P.A. Politowicz, J.J. Kozak, Phys. Rev. B 28 (1983) 5549. [11] J.J. Kozak, V. Balakrishnan, Phys. Rev. E 65 (1–5) (2002) 021105. [12] J.J. Kozak, V. Balakrishnan, Int. J. Bifurc. Chaos 12 (2002) 2379.
Invariance relations for random walks on square-planar lattices Roberto A. Garza-Lo´pez a
a,*,1
, John J. Kozak
b
Laboratory for Molecular Sciences, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA 91125, USA b Department of Chemistry, DePaul University, Chicago, IL 60604-2287, USA Received 14 January 2005; in final form 17 February 2005 Available online 17 March 2005
Abstract We outline a systematic procedure for obtaining exact analytic expressions for the mean walklength Ænæ for a random walker transiting on a d = 2 square-planar lattice with a single deep trap subject to periodic boundary conditions. As demonstrated in our earlier work onP hexagonal lattices [Chem. Phys. Lett. 371 (2003) 365], the procedure depends on generalizing MontrollÕs P first invariance relation (n = 1) to nth order. The central result of this work is the following exact analytic expression for (n) for square-planar lattices: (" ) # n1 X X 1 i n n1 ðnÞ ¼ 4 5 3 þ 12ðn 2Þ3 þ 3 : 3 N 3! i¼0 2005 Elsevier B.V. All rights reserved.
1. Introduction In this study, we outline a systematic procedure for obtaining exact analytic expressions for the mean walklength Ænæ of a random walker transiting on a d = 2 square-planar lattice. In particular, we determine Ænæ as a function of the size N of a lattice subject to periodic boundary conditions and having a single, deep trap. The classic studies of Montroll and Weiss [2–5] over 40 years ago led to amazingly accurate, asymptotic expressions for the N-dependence of the mean walklength Ænæ. For a d = 2 square-planar lattice of N sites with a simple deep trap, they found that
*
Corresponding author. Fax: +1 909 607 7726. E-mail address: [email protected] (R.A. Garza-Lo´pez). 1 Present address: Department of Chemistry, Pomona College, Claremont, CA 91711, USA. 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.078
N 1 hni ¼ A1 N ln N þ A2 N þ A3 þ A4 ; N 1 N
ð1Þ
where 1 A1 ¼ ; A2 ¼ 0:195062532; p A3 ¼ 0:116964779; A4 ¼ 0:05145660: Later, den Hollander and Kastelyn [6] corrected the coefficient A4 and determined that A4 = 0.484 065 704. With this set {Ai} of values, errors in Ænæ when compared with numerically exact values of Ænæ (see [7,8]) are essentially negligible for square-planar lattices N > 25. A systematic algebraic method for obtaining exact analytic expressions for Ænæ had also been reported years ago. In [7,8], where an exact algorithm for d-dimensional walks on finite and infinite lattices with traps was elaborated, one could develop analytic expressions for Ænæ. Taking advantage of the fact that flows to a given lattice
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
site are simply related to those from the four nearestneighbor sites, and given an exact relationship between a corner site i and its two neighboring sites (i 1) hnii hnii1 ¼ 2:
ð2Þ
one could work out algebraically the exact analytic expressions for the first few lattices; see Table 1. As is perhaps evident, there does not seem to be an obvious generalization to arbitrarily large N that springs from the results presented in Table 1. So, the issue is not that exact analytic expressions for Ænæ as a function of N cannot be obtained via straightforward algebraic methods; the issue is that to date there is no general program that can be followed to obtain the N-dependence of Ænæ for arbitrarily large lattices N. It is this program we laid out in [1] for hexagonal lattices, and extend here to square-planar lattices. The approach developed in our earlier study [1] was to return to an insight already available in the classic studies [2–5]. What Montroll and Weiss showed exactly was that the mean number of steps required for trapping from a site nearest neighbor to a single deep trap is given by hni ¼ N 1;
ð3Þ
where N is the total number of lattice sites. In effect, this result may be regarded as an invariance relation which subsequent studies have shown is valid for all d-dimensional lattices of uniform valency (connectivity) [9]. For the particular case of random walks on d = 2 hexagonal lattices, it was noticed subsequently [10] that walks from sites that are second-nearest-neighbors to the trap satisfy hni ¼ 3½ðN =2Þ 1:
ð4Þ
The question addressed in [1] was whether one could develop relations similar to Eqs. (3) and (4), invariance relations that described the average number of steps required for trapping from all sites that are third-nearestneighbor to the trap, fourth-nearest-neighbor to the trap, etc. The hope was that there would be a similarity of analytic structure in the expressions obtained, so that one could develop a general analytic expression for walks from all sites which were nth nearest neighbor sites to the trap. A first realization of this program for hexagonal lattices was the central result of that study (Eq. (12) of [1]), which we rewrite here in the equivalent form
Table 1 The values of Ænæ in the ÔN representationÕ for the 3 · 3 and 5 · 5 square-planar lattices N
Ænæ
3·3
hni ¼ ð2Þð5ÞNð2Þð3Þ N 1
5·5
2
hni ¼
ð2Þð3Þð31ÞN ð2Þð5Þð17Þ ð5Þ
N1
X
(" ðnÞ ¼ 3
n1 X
# i
n
2 N 2 þ 3ðn 2Þ2
39
n1
þ2
) ;
i¼0
ð5Þ P where P (1) is the first nearest-neighbor sum, Eq. (3), and (2) is Eq. (4). In this Letter, we follow the procedure laid down in [1] to determine the invariance relations for d = 2, square-planar lattices, subject to periodic boundary conditions with a single deep trap. However, as we shall show, more comprehensive results can be obtained for square-planar lattices since the deep trap can be positioned at the centro-symmetric site of the lattice; in slightly different language, the fact that the square-planar lattice is self-dual simplifies the complexity of the problem.
2. Invariance relations The invariance relations for square-planar lattices are determined as follows. First, numerically exact values for the mean walklength Ænæ for all sites i = 1, . . . , N of a given square-planar lattice are determined using the exact algorithm presented in [7,8] and reviewed in [9]. Then, all one-step, two-step, . . . , n-step walks to the centrosymmetric trap are tabulated and the resulting sumP mand (n) is determined for a sequence of squareplanar lattices. The results of this procedure for a family of lattices are presented in Table 2. P These data for (n) can be represented by the following set of equations: X ð1Þ ¼ 4ðN 1Þ; ð6Þ X X X X X X
ð2Þ ¼ 4ð4N 8Þ;
ð7Þ
ð3Þ ¼ 4ð13N 41Þ;
ð8Þ
ð4Þ ¼ 4ð40N 176Þ;
ð9Þ
ð5Þ ¼ 4ð123N 683Þ;
ð10Þ
ð6Þ ¼ 4ð372N 2504Þ;
ð11Þ
ð7Þ ¼ 4ð1129N 8837Þ:
ð12Þ
P Up to and including four-step walks, i.e., (4), the analytic expression that captures the above data, viz., Table 2 and Eqs. (6)–(9) is given by (" ) # n1 X X 1 i n n1 ðnÞ ¼ 4 3 N ð5 3 þ 12ðn 2Þ3 þ 3Þ : 3! i¼0 ð13Þ
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
40
Table 2 Summand values for the nth-nearest-neighbor flows for a walker on a square-planar lattice with a single (deep) trap, and subject to periodic boundary conditions P P P P P P P N (1) (2) (3) (4) (5) (6) (7) 9 25 49 81 121 169 225
32 96 192 320 480 672 896
112 368 752 1264 1904 2672 3568
1136 2384 4048 6128 8624 11,536
3296 7136 12,256 18,656 26,336 35,296
The structural similarity between this expression and the one reported earlier for hexagonal lattices, Eq. (5), is evident. For n > 4 step walks, a new feature appears, viz. cycles with one or more loops. We have determined analytic expressions for the first three correction terms nX o Dð5Þ ¼ 2 ð1Þ þ 24 1 ; ð14Þ Dð6Þ ¼ 2
nX
o ð2Þ þ 24 8 ;
62,896 110,512 170,032 241,456 324,784
185,936 330,448 511,088 727,856 980,752
the 12 immediately adjacent sites of the neighboring four unit cells. Let us write the sum of the four first-nearest neighbor sites to the trap as r1 ¼ hnð1Þi þ hnð1Þi þ hnð1Þi þ hnð1Þi and the sum of the Æn(i)æ for the second nearest-neighbor sites to the trap as r2 ¼ hnð2Þi þ hnð2Þi þ hnð2Þi þ hnð2Þi:
ð15Þ
X 5 5 Dð7Þ ¼ 2 ð3Þ þ 2 ð4! 1Þ þ ðn 4Þðn 3Þ ; 2 n P 7:
9568 21376 37,120 56,800 80,416 107,968
ð16Þ
When the analytic expression, Eq. (13), is coupled with the results above, Eqs. (14)–(16), all the data in Table 2 can be recovered exactly. P To summarize, the analytic expression for (n) is valid for all (n, N). The expressions for correction terms are exact for n = 5, 6, 7. Although we have not obtained a general expression for D(n), the above results suggest that D(n) can be approximated by X DðnÞ ðn 4Þ: ð17Þ
Using this notation, the first invariance relation can be written X ð1Þ ¼ r1 and the second invariance relation, which gives the sum total of all trajectories to the central trap that can be reached in a two-step walk, as X ð2Þ ¼ r1 þ 2r2 : Thus, using the earlier results, Eqs. (6) and (7), we have the following pair of equations: ð1Þr1 þ ð0Þr2 ¼ 4ðN 1Þ; ð1Þr1 þ ð2Þr2 ¼ 4ð4N 8Þ:
The worth of this approximation will be quantified in the following section.
In matrix form, this can be written P 1 0 r1 ð1Þ 4ðN 1Þ ¼ P ¼ ; 1 2 r2 ð2Þ 4ð4N 8Þ
3. Analytic expression for Ænæ
from which one can determine explicitly the individual ri as a function of N, and the overall average walklength can be written down explicitly
We now outline a program for calculating analytic expressions for Ænæ as a function of N for square-planar lattices. As noted earlier, progress can be made by taking advantage of the symmetry of the lattice or, more precisely, organizing the sites of the lattice in terms of symmetry classes. This procedure was introduced in [7,8] to contract the vector space in calculating numerically exact values of the walklengths from a given site i. To illustrate the method using the simplest possible example, consider the 3 · 3 lattice diagrammed in Fig. 1. All sites 1 are symmetry equivalent, as are all sites 2. To make explicit the consequences of imposing periodic boundary conditions, we have diagrammed as well
Fig. 1. Site specification for the 3 · 3 square-planar lattice accounting for symmetry.
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
hni ¼
1 1 ðr1 þ r2 Þ ¼ ð10N 18Þ: N 1 N 1
Consider now a more robust example, the 7 · 7 square-planar lattice diagrammed in Fig. 2 with a deep trap positioned at site 25. The corresponding values of the ri, each of which designates a set of symmetry-equivalent sites, are given by r1 ¼ hnð18Þi þ hnð24Þi þ hnð26Þi þ hnð32Þi;
are results previously reported in [8]. Taking these results into account, the set of defining equations X ð1Þ ¼ r1 ; X X X
r2 ¼ hnð17Þi þ hnð19Þi þ hnð31Þi þ hnð33Þi;
X
r3 ¼ hnð11Þi þ hnð23Þi þ hnð27Þi þ hnð39Þi; r4 ¼ hnð10Þi þ hnð12Þi þ hnð20Þi þ hnð34Þi þ hnð40Þi
41
X
ð2Þ ¼ 2r2 þ r3 ; ð3Þ ¼ 2r1 þ 3r4 þ r6 ; ð4Þ ¼ 4r2 þ 6r3 þ 6r5 þ r6 þ 4r7 ; ð5Þ ¼ 10r1 þ r3 þ 11r4 þ 12r6 þ 5r7 þ 10r8 ; ð6Þ ¼ r1 þ 30r2 þ 26r3 þ 6r4 þ 24r5 þ 20r6 þ 26r7 þ 15r8 þ 20r9 ;
þ hnð38Þi þ hnð30Þi þ hnð16Þi; X
r5 ¼ hnð9Þi þ hnð13Þi þ hnð41Þi þ hnð37Þi;
ð7Þ ¼ 66r1 þ 14r2 þ 30r3 þ 73r4 þ 42r5 þ 62r6
r6 ¼ hnð4Þi þ hnð28Þi þ hnð46Þi þ hnð22Þi;
þ 52r7 þ 55r8 þ 70r9 ;
r7 ¼ hnð3Þi þ hnð5Þi þ hnð21Þi þ hnð35Þi þ hnð47Þi þ hnð45Þi þ hnð29Þi þ hnð15Þi; r8 ¼ hnð2Þi þ hnð6Þi þ hnð14Þi þ hnð42Þi þ hnð48Þi þ hnð44Þi þ hnð36Þi þ hnð8Þi; r9 ¼ hnð1Þi þ hnð7Þi þ hnð49Þi þ hnð43Þi: Following from MontrollÕs work, we know that X ð1Þ ¼ r1 ¼ 4ðN 1Þ: The Ôcorner relationÕ r9 ¼ r8 =2 þ 8 and the conservation condition r8 ¼ 4r5 r4 16
reduces to a set of six equations in six unknowns. In matrix form 32 3 2 P 3 2 r2 ð2Þ 2 1 0 0 0 0 P 7 7 7 60 6 6 0 3 0 1 0 76 r3 7 6 ð3Þ 7 6 76 7 6 P 7 6 64 6 0 6 1 4 76 r4 7 6 ð4Þ 7 76 7 ¼ 6 P 7 6 60 6 7 6 7 1 1 40 12 5 7 76 r5 7 6 ð5Þ 7 6 76 7 6 P 7 6 4 30 26 19 124 20 26 54 r6 5 4 ð6Þ 5 P 14 30 17 402 62 52 r7 ð7Þ 3 2 16N 32 7 6 44N 156 7 6 7 6 7 6 160N 704 7 ¼6 6 452N 2532 7: 7 6 7 6 4 1484N 9772 5 4252N 34204 Inversion leads to values of the ri from which the overall Ænæ for the 7 · 7 lattice can be determined. The result is
1
2
3
4
5
6
7
hni ¼
ð2Þ3 ð120569ÞN ð7Þ2 ð13Þð31Þ
8
ð3Þð5Þð11Þ ð2Þð13Þð31Þ
:
ð18Þ
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
4. Discussion
43
44
45
46
47
48
49
The method presented in this paper allows one to obtain analytic expressions for the N-dependence of Ænæ. As is evident, rather than involving sophisticated analytic calculations or combinatorics, the method is technically
Fig. 2. Square-planar 7 · 7 lattice, with a deep trap positioned at site 25.
N 1
Comparison of this result with those in Table 1 for the 3 · 3 and 5 · 5 lattices reinforces the fact that deducing a generalization for Ænæ as a function of N is not obvious, if not impossible, if one focused only on the analytic representation of the Ôend resultÕ.
42
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
rather straightforward, viz., one uses an exact algorithm [7–9] to calculate numerically exact values of the mean walklength from each site of a given square-planar lattice, determines the total number of one-step, twostep, . . . , n-step walks to the trap and then, by recognizing patterns in the resulting expressions (invariance relations), seeks analytic representations of the results. Since the numerical expressions obtained involve integers, recognizing the patterns generated becomes a straightforward exercise in number theory. It is this which distinguishes our approach from the much more elegant methods introduced by Montroll and Weiss. We remark in passing that the above method was also used recently by one of us (with V. Balakrishnan) to obtain the exact analytic solution for the mean time to absorption for a random walker on the Sierpinski gasket [11] and on the Sierpinski tower [12]. In these studies, the task was lightened considerably by taking advantage of the self similarity of the underlying fractal lattices. We now comment on two further points. First, since the D(n) have been determined only up to n = 7, it is important to assess how accurate the results would be if one either neglected completely D(n) or used the approximation, Eq. (17), for D(n). To examine this point we compare the exact ri with the ri determined using these two approximations. Focusing on the 7 · 7 lattice for which exact results have been presented in the previous section,Pwe calculate the ri using only the expression (13) for (n) to construct the underlying matrix equation.P Then, we calculate ri using the exact expression (n), Eq. (13), augmented by the (approximate) correction factor, Eq. (17). A comparison of the results obtained is presented in Table 3. Approximation I gives a result with an error of 4.06% relative to the exact result equation (18), while the use of Approximation II gives a result with 1.68% error. For comparison, use of the Montroll–Weiss asymptotic result, Eq. (1), gives an error of 0.02%. So, numerically, the asymptotic estimate is superior to the results calculated using Approximations I and II. The second issue to be discussed concerns the D(n). P As noted, these corrections to (n) arise because when n > 4, there occur for the first time n-step trajectories to the trap that have cycles. To illustrate this for the 7 · 7 lattice, listed in Table 4 are the cycles found in a six-step walk for which the penultimate step in the walk is site 18 (see Table 4). As is seen, of the 20 trajectories in this set, 16 involve one intermediate intersection (crossing) and four (in bold face) involve two. Overall, removing all trajectories for which an intermediate site is the P deep trap, there are 860 trajectories which comprise (6) for the 7 · 7 lattice, of which 80 are cycles and the rest are chains. That is, already for the 7 · 7 lattice, 9.3% of the set of six-step trajectories are cycles. So, returning to Table 3, Approximation I, which neglects all cycles,
Table 3 Comparison of exact versus approximate values of the ri(n) for the 7 · 7 square-planar Table 4
r1 r2 r2 (decimal) r3 r3 (decimal) r4 r4 (decimal) r5 r5 (decimal) r6 r6 (decimal) r7 r7 (decimal) r8 r8 (decimal) r9 r9 (decimal) Ænæ (fraction) Ænæ (decimal)
Exact
Approximation I
Approximation II
192
192
192
3144 13
4668072 19747
5240408 19747
241.8461538
236.3939839
265.3774244
3488 13
5513600 19747
4368928 19747
268.3076923
279.2120322
221.2451512
7376 13
10651184 19747
10861360 19747
567.3846154 304 304.0
539.3823872
550.0258267
403472 1519
426128 1519
265.6168532
280.5319289
3872 13
7540448 19747
6909920 19747
297.8461538
381.8528384
349.9225199
7920 13
12537360 19747
13397872 19747
609.2307692
634.8994784
678.4763255
8224 13
10013408 19747
10981344 19747
632.6153846
507.0850256
556.1018889
4216 13
5164680 19747
5648648 19747
324.3076923
261.5425128
286.0509444
931 13
83068 13
85127 1209
71.61538462
68.70802316
70.41108354
Lattice.
Table 4 One intersection and two intersection (bold) cycles for a n = 6 step walk passing through the nearest-neighbor site 18 to a deep trap positioned at site 25 on the 7 · 7 lattice diagrammed in Fig. 2 11 ! 10 ! 3 ! 4 ! 11 ! 18 ! 25 11 ! 12 ! 5 ! 4 ! 11 ! 18 ! 25 11 ! 4 ! 3 ! 10 ! 11 ! 18 ! 25 11 ! 18 ! 17 ! 10 ! 11 ! 18 ! 25 19 ! 18 ! 17 ! 10 ! 11 ! 18 ! 25 11 ! 4 ! 5 ! 12 ! 11 ! 18 ! 25 11 ! 18 ! 19 ! 12 ! 11 ! 18 ! 25 17 ! 18 ! 19 ! 12 ! 11 ! 18 ! 25 17 ! 16 ! 9 ! 10 ! 17 ! 18 ! 25 17 ! 18 ! 11 ! 10 ! 17 ! 18 ! 25 19 ! 18 ! 11 ! 10 ! 17 ! 18 ! 25 17 ! 10 ! 9 ! 16 ! 17 ! 18 ! 25 17 ! 24 ! 23 ! 16 ! 17 ! 18 ! 25 17 ! 16 ! 23 ! 24 ! 17 ! 18 ! 25 17 ! 18 ! 11 ! 12 ! 19 ! 18 ! 25 19 ! 18 ! 11 ! 12 ! 19 ! 18 ! 25 19 ! 20 ! 13 ! 12 ! 19 ! 18 ! 25 19 ! 12 ! 13 ! 20 ! 19 ! 18 ! 25 19 ! 26 ! 27 ! 20 ! 19 ! 18 ! 25 19 ! 20 ! 27 ! 26 ! 19 ! 18 ! 25
leads to an error of 4% when 9.3% of the total number of six-step trajectories is removed, as well as all cycles which arise in seven-step walks, eight-step walks, . . . , nstep walks. Although the Montroll–Weiss asymptotic estimate is numerically superior, the above trajectory analysis gives a deeper insight into the ÔmicroscopicÕ differences between the exact and approximate solutions to the random walk problem. The relevance of the above results on cycles to the problem of avoiding versus self-avoiding
R.A. Garza-Lo´pez, J.J. Kozak / Chemical Physics Letters 406 (2005) 38–43
walks and to problems in polymer physics will be explored in subsequent work. Acknowledgments R.A.G.L. gratefully acknowledges The American Chemical Society Petroleum Research Fund, PRF#39961-UFS for their support while on sabbatical at Caltech. This work has been supported by a grant from The James Irvine Foundation granted to Pomona College. The authors also thank Juan J. Jime´nez Go´mez for his help in entering some of the data for this paper.
43
References [1] R.A. Garza-Lo´pez, J.J. Kozak, Chem. Phys. Lett. 371 (2003) 365. [2] E.W. Montroll, Proc. Symp. Appl. Math. Am. Math. Soc. 16 (1964) 193. [3] E.W. Montroll, G.H. Weiss, J. Math. Phys. 6 (1965) 167. [4] E.W. Montroll, J. Math. Phys. 10 (1969) 753. [5] G.H. Weiss, Aspects and Applications of the Random Walk, North-Holland, New York/Amsterdam, 1994. [6] W.Th. den Hollander, P. Kasteleyn, Physica A 112 (1982) 523. [7] C.A Walsh, J.J. Kozak, Phys. Rev. Lett. 47 (1981) 1500. [8] C.A. Walsh, J.J. Kozak, Phys. Rev. B 26 (1982) 4166. [9] J.J. Kozak, Adv. Chem. Phys. 115 (2000) 245. [10] P.A. Politowicz, J.J. Kozak, Phys. Rev. B 28 (1983) 5549. [11] J.J. Kozak, V. Balakrishnan, Phys. Rev. E 65 (1–5) (2002) 021105. [12] J.J. Kozak, V. Balakrishnan, Int. J. Bifurc. Chaos 12 (2002) 2379.