Diffusive behaviour of self-attractive walks

Prasad, M. A.; Bhatia, D. P.; Arora, D.

doi:10.1088/0305-4470/29/12/011

Cited by 11 publications

(20 citation statements)

References 8 publications

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“…We also obtained an expression for the probability P u,s (S, t) of visiting S distinct sites in a time t, in terms of generating functions and found analytically that S t ∼ √ t in the absence of bias (recovering previous results [29]), and becomes ballistic with S t ∼ t in the presence of bias, as soon as s > 0. Thus, memory effects induce second order corrections in this case.…”

Section: Discussionsupporting

confidence: 71%

“…We preliminarily note that a positive field (s > 0) implies α > β > 1 − γ (see (6)) as well as t R t t L t ; therefore, we can write S t ∼ β t R t and x t ∼ t, from which we can infer t R t ∼ t, namely the propagation is ballistic, which is consistent with the above results. On the other hand, in the absence of bias (s = 0), we obtain 1 − γ = β and α = 1/2, from which S t ∼ t B t and x t t ∼ t B t , which is consistent with [29]. Now, we focus on the occupation of border sites, namely t B , and try to highlight its connection with S t .…”

Section: Border Timessupporting

confidence: 71%

“…Since t (0 → 1) does not depend on time, we have S t ≈ vt, hence recovering the same result as (25), obtained through the Tauberian theorem. We can improve the estimate of t (0 → 1) appearing in (29), taking into account also the presence of the left border, assumed as reflecting; this assumption allows us to obtain a lower bound for t (0 → 1). In this case, the time taken by the random walk to expand the visited region depends on the span length.…”

Section: The Linear Growth Rate Of S Tmentioning

confidence: 99%

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The true reinforced random walk with bias

2012

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Section: Discussionsupporting

confidence: 71%

Section: Border Timessupporting

confidence: 71%

Section: The Linear Growth Rate Of S Tmentioning

confidence: 99%

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The true reinforced random walk with bias

2012

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“…Prasad et.al. concluded that the SATW in one dimension is diffusive [14].In the present comment we report results of a Monte Carlo simulation for the self-attracting walk in one, two and three dimension. We find that SATW in one dimension is diffusive.…”

mentioning

confidence: 79%

“…This observation supports the recent calculation of Prasad et.al. [14]. However, in two and three dimensions the exponents ν decrease continuously when u decreases.…”

mentioning

confidence: 95%

Self-attracting walk on lattices

Lee

1998

J. Phys. A: Math. Gen.

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We have studied a model of self-attracting walk proposed by Sapozhnikov using Monte Carlo method. The mean square displacement R 2 (t) ∼ t 2ν and the mean number of visited sites S(t) ∼ t k are calculated for one-, two-and three-dimensional lattice. In one dimension, the walk shows diffusive behaviour with ν = k = 1/2. However, in two and three dimension, we observed a non-universal behaviour, i.e., the exponent ν varies continuously with the strength of the attracting interaction. 1There is great interest in random walks and interacting walks [1,2,3]. The mean square displacement of a random walk, R 2 (t) follows a power-law R 2 (t) ∼ t 2ν . The ordinary random walk(RW) is diffusive, with ν = 1/2, in all dimensions. For a walk with repulsion, such as self-avoiding walk(SAW), the exponent ν is greater than 1/2. Random walk on a fractal is anomalous with ν < 1/2 [2,3]. Various models of interacting walks have been studied, such as true self-avoiding walk [4,5], generalized true SAW[6], the Domb-Joyce model [7], and an interacting walk with a weight factor p for each new site that the random walker visits [8,9]. A comparative study of interacting random walk models was performed by Duxbury et.al. [10,11].Recently, Sapozhikov proposed a generalized walk in which the probability for the walker to jump to a given site is proportional to p = exp(−nu), where n = 1 for the sites visited by the walker at least once and n = 0 for other sites [12]. If u < 0, the walker is attracted to its own trajectory. This walk is called a self-attracting walk(SATW). Monte Carlo studies have suggested that ν < 1/2 for u = −1 and u = −2 on two dimension and 1/4 < ν < 1/3 on three dimension [12]. However, Aarão Reis obtained non-universal behaviour of the SATW in one to four dimension using exact enumeration method [13]. Prasad et.al. concluded that the SATW in one dimension is diffusive [14].In the present comment we report results of a Monte Carlo simulation for the self-attracting walk in one, two and three dimension. We find that SATW in one dimension is diffusive. However, SATW in two and three dimension shows non-universal behaviour. Monte Carlo simulations were performed on one dimensional lattice (10 6 -steps with 2000-configurational averages for each parameter u), square lattice (10 6 -steps with 2000-averages), and cubic lattice (10 5 -steps with 2000-averages). The lattice sizes used in this simulation were L = 10 6 (1D), 1024×1024 (2D) and 200×200×200(3D). We always used the periodic boundary conditions. We calculated the mean square displacement R 2 (t) and the mean number of visited sites S(t) . Fig. 1 (a) shows the log-log plot of the mean square displacement against the time. We obtained the exponent by a least-square fit: ν = 0.500 (7) for u = 0, ν = 0.500(9) for u = −0.5, ν = 0.499(8) for u = −1.0, and ν = 0.498(9) for u = −2.0. All the lines are parallel in the large-time limit. These results means that SATW in one dimension is diffusive and support the conclusions of Prasad et.al.[14]. This diffusive behaviour is furth...

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Reply to the comment by J W Lee. Self-attracting walk: are the exponents universal?

Sapozhnikov¹

1998

J. Phys. A: Math. Gen.

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Diffusive behaviour of self-attractive walks

Cited by 11 publications

References 8 publications

The true reinforced random walk with bias

The true reinforced random walk with bias

Self-attracting walk on lattices

Reply to the comment by J W Lee. Self-attracting walk: are the exponents universal?

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