Lattice random walks for sets of random walkers. First passage times

Lindenberg, Katja; Seshadri, V.; Shuler, Kurt E.; Weiss, George H.

doi:10.1007/bf01014427

Cited by 48 publications

(36 citation statements)

References 7 publications

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“…Or, in other words, we give an asymptotic description of the order statistics [9,10] of the diffusion process. Some results concerning the order statistics of a set of random walkers on Euclidean lattices are known [11]. However, for the diffusion limit there are rigorous asymptotic (N ¿ 1) results only for one-dimensional processes (with only certain conjectures for higher dimensions [9,12]).…”

Section: S Bravo Yustementioning

confidence: 99%

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Escape Times ofjRandom Walkers from a Fractal Labyrinth

Yuste

1997

Phys. Rev. Lett.

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The problem of the statistical description of the first passage time t j,N to a given distance r of the first j of a set of N noninteracting diffusing particles, all starting from the same origin on fractal substrates, is addressed. Asymptotic expressions (the main and two corrective terms) for large N of the (arbitrary) moments of t j,N are given. It is shown that, to first order and for 1 # j ø N, the mth moment of t j,N goes as ͑ln N͒ m͑12dw ͒ , and its variance as ͑ln N͒ 22dw , d w being the anomalous diffusion exponent of the fractal medium. [S0031-9007 (97)04503-1] PACS numbers: 05.40. + j, 05.60. + w, 66.30.DnThe random walk formalism has proven to be extremely fruitful in science [1,2]. Usually it is the random walk of only one particle which is studied, but the behavior of many random walkers is also presently an area of active research [3]. Thus, for example, Larralde et al. have recently found very nice results regarding the number of distinct sites, S N ͑t͒, visited by a set of N ¿ 1 independent random walkers on Euclidean lattices of one, two, and three dimensions [4]. Shortly thereafter, these results were extended to fractal substrates [5]. Diffusion in these fractal media has attracted much attention because it exhibits new, qualitatively different properties (anomalous diffusion) also present in geometrically disordered media (indeed, fractals are considered good models for disordered media) which are unexplained by the classical theories of diffusion [6,7]. For example, the mean-square displacement of a random walker is given by ͗r 2 ͘ ഠ 2Dt 2͞d w , d w fi 2 being the anomalous diffusion exponent (or fractal dimension of a random walk) and D the diffusion coefficient.In this Letter we give an answer to another basic question about the diffusion of a group of particles in a fractal medium that, as we will see, is closely related to the problem of calculating S N ͑t͒. The question is: If a set of N independent random walkers (ants, in the language coined by de Gennes [1,8]) are initially placed (parachuted ) onto a site of a fractal structure (the labyrinth), how long will it take the first j random walkers to reach a given distance r from the origin? In other words, if the exits of the maze are placed at a distance r, what are the escape times of the first j ants of this battalion of N members? (Notice that not only the first passage time of the first particle is important if more than one particle must arrive at a certain place in order to trigger some effect there.) Explicitly, in this Letter we give asymptotic expressions (N ¿ 1) for the moments of the jth passage time, t j,N ͑r͒, i.e., of the time to first reach a given distance r of the jth random walker of a set of N independent diffusing random walkers all starting from the same origin on a fractal substrate. Or, in other words, we give an asymptotic description of the order statistics [9,10] of the diffusion process. Some results concerning the order statistics of a set of random walkers on Euclidean lattices are known [11]. However, for ...

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Section: S Bravo Yustementioning

confidence: 99%

“…It is clear that the numerically evaluated ratio t 1,N ͞s 1,N closely follows the theoretical prediction given by Eq. (11). It should be kept in mind that we have chosen the mean time for a single particle to reach the distance r as a time unit.…”

Section: S Bravo Yustementioning

confidence: 99%

Escape Times ofjRandom Walkers from a Fractal Labyrinth

Yuste

1997

Phys. Rev. Lett.

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“…Instead of considering a single random walk, one may generalize to the case of ν statistically independent random walks. Such studies were first concerned with the properties of first passage times [27][28][29][30]. The study of the mean number of distinct sites visited by ν random walkers, s ∞,ν (t), was initiated in [31] where asymptotic expressions for ν large were obtained.…”

Section: Introductionmentioning

confidence: 99%

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Turban¹

2014

J. Phys. A: Math. Theor.

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Abstract. The probability distribution of the number s of distinct sites visited up to time t by a random walk on the fully-connected lattice with N sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of s and r, where r is the number of sites visited only once up to time t. Mean values, variances and covariance are deduced from the generating functions and their finitesize-scaling behaviour is studied. Introducing properly centered and scaled variables u and v for r and s and working in the scaling limit (t → ∞, N → ∞ with w = t/N fixed) the joint probability density of u and v is shown to be a bivariate Gaussian density. It follows that the fluctuations of r and s around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension d c = 2.

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“…The statistical problem of calculating the mean first passage time (FPT) for random walkers has a distinguished history (18)(19)(20)(21)(22). Here we will find the first passage time and the survival probability in terms of a dynamic probability distribution.…”

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Statistics of cellular signal transduction as a race to the nucleus by multiple random walkers in compartment/phosphorylation space

Shen

Zong

et al. 2006

Proc. Natl. Acad. Sci. U.S.A.

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Cellular signal transduction often involves a reaction network of phosphorylation and transport events arranged with a ladder topology. If we keep track of the location of the phosphate groups describing an abstract state space, a simple model of signal transduction involving enzymes can be mapped on to a problem of how multiple biased random walkers compete to reach their target in the nucleus yielding a signal. Here, the first passage time probability and the survival probability for multiple walkers can be used to characterize the response of the network. The statistics of the first passage through the network has an asymmetric distribution with a long tail arising from the hierarchical structure of the network. This distribution implies a significant difference between the mean and the most probable signal transduction time. The response patterns for various external inputs generated by our model agree with recent experiments. In addition, the model predicts that there is an optimal phosphorylation enzyme concentration for rapid signal transduction.stochastic biology ͉ first passage time ͉ NFAT A chieving a quantitative understanding of the reaction networks that transduce cellular signals is one of the major challenges in biology. Signaling networks are found in a diverse set of organisms, ranging from prokaryotes to eukaryotes, and provide mechanisms for fundamental processes such as generegulatory control and cellular communication. Qualitative descriptions of the biomolecular components and mechanisms of cellular signaling have greatly improved our understanding of how cells function and have given insights into how to intervene therapeutically when such signals are miscommunicated. Experimental advances now allow quantitative studies of signal transduction and thereby inspire theoretical treatments. Many networks of nonlinear reactions exhibit interesting behavioral features as ultrasensitivity, adaption, robustness, and discrete ''all-or-none'' response, which have been quantitatively explored (1-8).A commonly occurring network topology is the reaction ladder network. This network may be viewed as a generalization of multiple-site phosphorylation͞dephosphorylation cascades, such as the pathway governing nuclear factor activation of T cells (NFAT), which regulates the response of T cells to antigen signaling (9-13). To stimulate T cells, NFAT must be transported to the nucleus. This transition occurs in response to a conformational change that exposes a nuclear localization sequence (NLS), which is normally buried in the protein interior in the inactive conformation and thus makes the NFAT inaccessible to transport by importin. The NLS becomes exposed in response to the progressive dephosphorylation of specific serine residues in its regulatory domain. This dephosphorylation occurs in response to an increase of intracellular calcium ions that activate calcineurin, which then dephosphorylates the masking residues. Once a sufficient number of sites have been dephosphorylated, conformational changes expose...

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Lattice random walks for sets of random walkers. First passage times

Cited by 48 publications

References 7 publications

Escape Times ofjRandom Walkers from a Fractal Labyrinth

Escape Times ofjRandom Walkers from a Fractal Labyrinth

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Statistics of cellular signal transduction as a race to the nucleus by multiple random walkers in compartment/phosphorylation space

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