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This article is cited in 1 scientific paper (total in 1 paper)
Canonical ensemble of particles in a self-avoiding random walk
V. I. Alkhimov Information Technology Faculty, Moscow State University of
Psychology and Education, Moscow, Russia
Abstract:
We consider an ensemble of particles not interacting with each other and randomly walking in the $d$-dimensional Euclidean space $\mathbb R^d$. The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is "marked" as a region in the form of a ball of diameter $r_0$, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in $\mathbb R^d$ of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than $r_0$. We describe a method for computing the asymptotic form of the probability density $W_n(\mathbf r)$ of the distance $r$ between the centers of the initial and final balls of a trajectory consisting of $n$ individual moves of a particle of the ensemble. The number $n$, the trajectory modulus, is a random variable in this model in addition to the distance $r$. This makes it necessary to determine the distribution of $n$, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density $W_n(\mathbf r)$ over the canonical distribution of the modulus $n$ allows finding the asymptotic behavior of the probability density of the distance $r$ between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in $\mathbb R^d$ for $2\le d<4$.
Keywords:
canonical ensemble, self-avoiding random walk, constitutive equation,
renormalization group, saddle-point method, asymptotic distribution.
Received: 25.01.2016 Revised: 14.02.2016
Citation:
V. I. Alkhimov, “Canonical ensemble of particles in a self-avoiding random walk”, TMF, 191:1 (2017), 100–115; Theoret. and Math. Phys., 191:1 (2017), 558–571
Linking options:
https://www.mathnet.ru/eng/tmf9154https://doi.org/10.4213/tmf9154 https://www.mathnet.ru/eng/tmf/v191/i1/p100
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Abstract page: | 464 | Full-text PDF : | 142 | References: | 76 | First page: | 14 |
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