Critical dimensions for random walks on random-walk chains Savely Rabinovich, H. Eduardo Roman, Shlomo Havlin, and Armin Bunde Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat Gan, Israel arXiv:cond-mat/9604167v1 29 Apr 1996 and Institut für Theoretische Physik, Universität Gießen, Heinrich-Buff-Ring 16, D-35392 Gießen, Germany (received August 23, 2019) Abstract The probability distribution of random walks on linear structures generated by random walks in d-dimensional space, Pd (r, t), is analytically studied for the case ξ ≡ r/t1/4 ≪ 1. It is shown to obey the scaling form Pd (r, t) = ρ(r)t−1/2 ξ −2 fd (ξ), where ρ(r) ∼ r 2−d is the density of the chain. Expanding fd (ξ) in powers of ξ, we find that there exists an infinite hierarchy of critical dimensions, dc = 2, 6, 10, . . ., each one characterized by a logarithmic correction in fd (ξ). Namely, for d = 2, f2 (ξ) ≃ a2 ξ 2 ln ξ + b2 ξ 2 ; for 3 ≤ d ≤ 5, fd (ξ) ≃ ad ξ 2 + bd ξ d ; for d = 6, f6 (ξ) ≃ a6 ξ 2 + b6 ξ 6 ln ξ; for 7 ≤ d ≤ 9, fd (ξ) ≃ ad ξ 2 + bd ξ 6 + cd ξ d ; for d = 10, f10 (ξ) ≃ a10 ξ 2 + b10 ξ 6 + c10 ξ 10 ln ξ, etc. In particular, for d = 2, this implies that the temporal dependence of the probability density of being close to the origin Q2 (r, t) ≡ P2 (r, t)/ρ(r) ≃ t−1/2 ln t. Pacs: 5.40.+j, 05.60.+w, 66.30.-h 1 I. INTRODUCTION Random fractals represent useful models for a variety of disordered systems found in Nature. In addition to their structural properties, fractals have attracted much attention in recent years because of their interesting transport properties [1–4]. Of particular interest is the question of how the probability density of random walks, Pd (r, t), is changed on fractal structures with respect to its Gaussian form valid on regular d−dimensional systems, Pd (r, t) ∼ t−d/2 exp(−const × η 2 ), where η = r/t1/2 . The form of Pd (r, t) on fractals has been extensively studied in the asymptotic limit ξ = r/t1/dw ≫ 1 [2,5–12], where dw is the anomalous diffusion exponent characterizing the time behavior of the random walks, hr 2 (t)i ∼ t2/dw . As a result of these investigations, it is now generally accepted that Pd (r, t) displays a stretched Gaussian form Pd (r, t) ∼ ρ(r)t−ds /2 exp(−const × ξ u ), ξ ≫ 1, (1) where ρ(r) ∼ r df −d is the density of the fractal structure, df is the fractal dimension, ds = 2df /dw is the spectral dimension [1], u = dw /(dw − 1), and is normalized according to R dr r d−1 Pd (r, t) = 1. However, much less is known about the behavior of Pd (r, t) in the opposite limit when ξ approaches zero. In this paper we concentrate on diffusion in linear random fractal structures generated by random walks (random-walk chains, RWC) in d-dimensional systems, where Pd (r, t) can be obtained exactly. Recently, using numerical simulations, it has been suggested that for such linear fractals [13], Qd (r, t)/Qd (0, t) ∼ (1 − const × ξ d−2 ), ξ → 0, (2) for all dimensions d, where Qd (r, t) = ρ(r)−1 Pd (r, t) is normalized on the fractal chain, i.e., R dr r df −1 Qd (r, t) = 1, with df = 2, dw = 4 for RWC, ξ = r/t1/4 and Qd (0, t) is the probability density to return to the origin. In the following, we derive an exact expansion for Pd (r, t) in the limit of ξ → 0. Surprisingly, Pd (r, t) displays an extremely rich behavior as a function of both ξ and dimensionality 2 d. We show, among other results, that Eq. (2) can only be valid for 3 ≤ d ≤ 5, and Pd (r, t) ∼ ρ(r)t−1/2 (1 − const × ξ 4 ), for d ≥ 7. (3) Moreover, we find that the small-ξ expansion of Pd (r, t) is characterized by a hierarchy of critical dimensions, dc = 2, 6, 10, 14, . . ., where logarithmic corrections of the form ξ dc −2 log(1/ξ) occur. In particular, for d = 2 we obtain P2 (r, t) ≃ 2ρ(r)t−1/2 ln(t1/4 /r). II. RANDOM WALKS ON RANDOM-WALK CHAINS We consider linear structures generated by random walks in d-dimensional systems. Such structures are fractals with fractal dimension df = 2, independently of d. To study diffusion of particles along such linear chains, we assume that the diffusing particles (random walkers) can move only along the structure (path) which has been created sequentially by the generating walks. Thus, although the structure can intersect itself in space, the walkers see just a linear path. We denote such paths as random-walk chains (RWC). Along the linear path, the probability density of random walkers, at chemical distance ℓ along the RWC from their starting point after time t, p(ℓ, t), subject to the initial condition p(ℓ, 0) = δ(ℓ), approaches the well-known Gaussian distribution 2 ℓ2 p(ℓ, t) = , exp − (2πt)1/2 2t ! normalized according to R∞ 0 (4) dℓ p(ℓ, t) = 1. Thus, diffusion along the chain (i.e., ℓ-space) is normal and hℓ2 i = t. On the contrary, in Euclidean r-space diffusion is anomalous with dw = 2df = 4 (see, e.g., [2]). To obtain the behavior of the probability density in r-space, averaged over all RWC configurations, Pd (r, t), we note that it is related to p(ℓ, t) by Pd (r, t) = and is normalized according to the RWC fractal, i.e., R∞ 0 R Z 0 ∞ dℓ Φd (r, ℓ)p(ℓ, t) (5) dd r Pd (r, t) = 1. Another possibility is a normalization on dr r df −1 Qd (r, t) = 1. Both distributions are simply related to each other by Pd (r, t) = ρ(r)Qd (r, t). 3 In Eq. (5), Φd (r, ℓ) represents the probability for a site r to belong to a RWC at distance ℓ from the origin along the chain. The chemical distance ℓ plays the role of the time variable in Eq. (4), and one can immediately write Φd (r, ℓ) = Ad  1 2πℓ where Ad is a normalization factor such that 1 Pd (r, t) = 2π d/2 2Ad (2πt)1/2 Z ∞ 0 r2 exp − 2ℓ ! (6) dd r Φd (r, ℓ) = 1. Therefore, by inserting (4) R and (6) in (5) we infer [4,14]  d/2 ℓ2 r2 exp − . dℓ ℓ−d/2 exp − 2ℓ 2t ! ! (7) Now, the elementary transformation x = ℓ/r 2 brings (7) to the form Pd (r, t) = 2Ad (2π)−(d+1)/2 r −d fd (ξ), (8) where the scaling function fd (ξ) is defined by fd (ξ) = ξ 2 Z ∞ 0 −d/2 dx x 1 1 exp − ξ 4 x2 + 2 x    (9) for the scaling variable ξ ≡ r/t1/4 . If the RWC normalization is chosen, the distribution Qd (r, t) = ρ−1 (r)Pd (r, t) ∼ = t−1/2 f˜d (ξ), where f˜d (ξ) = ξ −2 fd (ξ). To deal now with the evaluation of fd (ξ) when ξ → 0, it is convenient to rewrite the integrand exponent as 1 1 exp − ξ 4 x2 + 2 x    1 1 = exp − ξ 4x2 + 2 2 x    1 1 1− exp − 2x x    and expand the second exponential factor in Taylor series. The remaining integrals can be solved exactly (see, e.g. [15]), and one arrives at the following expression for (9) ∞ X 1 1 − fd (ξ) = ξ 2 n=0 n! 2  n X n (−1) k=0 k ! n d/2−1+n+k ξ K 1 (d/2−1+n+k) (ξ 2 ), 2 k (10) where Kν is the modified Bessel function of order ν. Let us consider Eq. (10) in some particular cases of interest. The results for spatial dimensions d ≤ 7 are summarized in Table I. All the coefficients were calculated numerically 4 by computing the double sums explicitly. In some cases they are available in analytic form, but we include their numerical values to make the table uniform. However, besides the coefficients, the main properties of these expansions can be obtained readily as follows. The key parameter is s = 21 ( d2 − 1). The corresponding values of d = 2(2s + 1) for integer s should be referred to as critical dimensions, dc = 2, 6, 10, . . .. Each order in the expansion has its own critical dimension. The leading term has dc = 2, the first correction term has dc = 6, the second correction term has dc = 10, etc. This has to do with the functional form of fd (ξ) in the corresponding order which for d < dc depends on d, at d = dc it has a logarithmic correction and for d > dc becomes independent of d. In particular, the leading term of fd (ξ) behaves as ξ for d = 1 and ξ 2 ln(1/ξ) for d = dc = 2 and as ξ 2 for all d > dc = 2, the first correction term behaves as ξ d for 2 ≤ d < 6, and for d = dc = 6 as ξ 6 ln(1/ξ) and as ξ 6 , for all d > dc = 6, and so on. Mathematically, this behavior can be explained by the intrinsic properties of the Bessel function Ks (ξ 2). By its definition, Ks (ξ 2) = π2 cosec(πs)[Is (ξ 2 ) − I−s (ξ 2 )] for non-integer s and, in turn, Is (ξ 2 ) = ξ 2s P∞ k=0 bk (s)ξ 4k and ξ 2s I−s (ξ 2 ) = P∞ k=0 bk (−s)ξ 4k . As one can see this expansion has s-independent powers of ξ that form the invariant part of fd (ξ). The first terms of ξ 2s Ks (ξ 2 ) expansion are ξ 2s Ks (ξ 2 ) ≈ b0 (s)ξ 4s + b1 (s)ξ 4s+4 + . . . − b0 (−s) − b1 (−s)ξ 4 − . . . . Thus, for 0 < s < 1 (i.e., 2 < d < 6) fd (ξ) has the form fd (ξ) ≈ ξ 2 [a0 (−d) − a0 (d)ξ d−2 ]. We see that the first term of this expansion is the invariant part (up to numerical coefficients) of fd (ξ), which remains unchanged when varying d. The same argument shows that for 1 < s < 2 (6 < d < 10) fd (ξ) takes the form fd (ξ) ≈ ξ 2[a0 (−d) − a1 (−d)ξ 4 + a0 (d)ξ d−2 ] and now two first terms of this expansion are the invariant part of fd (ξ). A special case in our problem is d = 1, i.e., s = − 41 < 0. Then the leading term of fd (ξ) is ξ 2 ξ −4|s| = ξ, which is easily seen by noting that K−s (ξ 2) = Ks (ξ 2 ). For integer values of s, the small-ξ expansion of ξ 2s Ks (ξ 2 ) has a logarithmic term of form ξ 4s log(1/ξ) = ξ d−2 log(1/ξ) [15]. 5 In general, there are [s] + 1 ([s] is the integer part of s) terms in the invariant part of fd (ξ). III. CONCLUSIONS We have studied analytically the small-ξ expansion of the mean probability density, Pd (r, t), of random walks on random-walk chains in d-dimensional space. We have shown that the leading terms of the expansion of, Pd (r, t), behaves, in the limit ξ = r/t1/4 → 0, as Pd (r, t) ∝ ρ(r)t−1/2 (1 − ad ξ d−2 ), when 3 ≤ d ≤ 5, and as Pd (r, t) ∝ ρ(r)t−1/2 (1 − cd ξ 4 ), when d ≥ 7, where ρ(r) ∼ r df −d and df = 2. This implies that the probability density Qd (r, t) = Pd (r, t)/ρ(r) on the fractal chain behaves for d ≥ 7 as Qd (r, t) ∼ t−1/2 (1 − cd r 4 /t), consistent with the behavior of diffusion in ℓ-space, i.e., p(ℓ, t) ∼ t−1/2 (1 − ℓ2 /2t), for ℓ2 ≪ t, and the fact that Qd (r = 1, t) ∼ p(ℓ = 1, t) when d → ∞. We see that this already occurs when d ≥ 7. We have shown that logarithmic corrections occur at critical dimensions d = dc = 4n + 2, with n = 0, 1, 2, . . ., i.e., dc = 2, 6, 10, . . ., for the terms ξ dc −2 log(1/ξ). In particular for d = 2, Q2 (r, t) ≃ t−1/2 ln(t1/4 /r), for r ≪ t1/4 , and the probability density for the random walker to be close to the origin, Q2 (r, t) behaves as t−1/2 ln t. This logarithmic correction is due to the fact that in two dimensions the RWC returns to its starting point with probability one. In one dimension, Q1 (r, t) ≃ t−1/4 /r, and in d = 3, Q3 (0, t) ≃ t−1/2 . One can say that d = 2 plays the role of a marginal dimensionality for the probability density of being at the origin of random walks on RWC, while for larger r each order in the expansion has its own critical dimension. Acknowledgments: 6 We like to thank Julia Dräger, Markus Porto and Grisha Berkolaiko for valuable discussions. Financial support by the Alexander-von-Humboldt foundation, and the Deutsche Forschungsgemeinshaft is gratefully acknowledged. 7 REFERENCES [1] S. Alexander and R. Orbach, J. Phys. Lett. 43, L625 (1982) [2] S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987) [3] K. W. Kehr and R. Kutner, Physica A110, 535 (1982) [4] A. Bunde and S. Havlin, eds., Fractals and disordered systems (Springer Verlag, Heidelberg, 1991) [5] R. A. Guyer, Phys. Rev. A29, 2751 (1984) [6] B. O’Shaughnessy and I. Procaccia, Phys. Rev. Lett. 54, 455 (1985); J. P. Bouchaud and A. Georges, Phys. Rep. 195 (1990) [7] A. Bunde, S. Havlin, and H. E. Roman, Phys. Rev. A42, 6274 (1990) [8] J. Klafter, G. Zumofen and A. Blumen, J. Phys. A24, 4835 (1991) [9] H. E. Roman and M. Giona, J. Phys. A25, 2107 (1992); M. Giona and H. E. Roman, Physica A185, 87 (1992) [10] A. Aharony and A. B. Harris, Physica A25, (1992); G. H. Weiss, Aspects and Applications of the Random Walk (North Holland, Amsterdam, 1994) [11] H. E. Roman, Phys. Rev. E51, 5422 (1995) [12] S. B. Yuste, J. Phys. A28, 7027 (1995) [13] J. Dräger, S. Russ and A. Bunde, Europhys. Lett. 31, 425 (1995) [14] Actually, the function Φd (r, ℓ) = 0 when ℓ < ℓmin, and ℓmin = r when all RWC configurations are considered (see [16]). Thus, the lower integration limit in Eq. (7) corresponds to ℓmin = r. However, by performing the transformation x = ℓ/t1/2 , it can be shown to vanish, for fixed ξ = r/t1/4 , as ξ/t1/4 → 0 when t → ∞. Hence, we set the lower integration limit ℓmin = 0 in Eq. (7). 8 [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York 1965) [16] A. Bunde and J. Dräger, Phys. Rev. E52, 53 (1995) 9 TABLES TABLE I. The leading and correction terms for the series expansion of fd (ξ), with ξ ≡ r/t1/4 , when ξ → 0, Eq. (10), as a function of dimension d. d Leading term First correction Second correction 1 2.1558 ξ −2.5066 ξ 2 O(ξ 5 ) 2 2ξ 2 ln(1/ξ) 0.1738ξ 2 O(ξ 6 ) 3 2.5066 ξ 2 −0.0609 ξ 3 O(ξ 6 ) 4 2 ξ2 −1.2533 ξ 4 O(ξ 6 ) 5 2.5066 ξ 2 −1.4372 ξ 5 O(ξ 6 ) 6 4 ξ2 −ξ 6 ln(1/ξ) −0.3369ξ 6 7 7.5199 ξ 2 −1.2533 ξ 6 0.8244 ξ 7 10