Ali, Adnan, Ball, Robin, Grosskinsky, Stefan and Somfai, Ellák (2013) Interacting particle systems in time-dependent geometries. Journal of Statistical Mechanics: Theory and Experiment, Volume 2013 (Number 09). Article number P09006. doi:10.1088/1742-5468/2013/09/P09006 ISSN 1742-5468.

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Abstract

Many complex structures and stochastic patterns emerge from simple kinetic rules and local interactions, and are governed by scale invariance properties in combination with effects of the global geometry. We consider systems that can be described effectively by space–time trajectories of interacting particles, such as domain boundaries in two-dimensional growth or river networks. We study trajectories embedded in time-dependent geometries, and the main focus is on uniformly expanding or decreasing domains for which we obtain an exact mapping to simple fixed domain systems while preserving the local scale invariance properties. This approach was recently introduced in Ali et al (2013 Phys. Rev. E 87 020102(R)) and here we provide a detailed discussion on its applicability for self-affine Markovian models, and how it can be adapted to self-affine models with memory or explicit time dependence. The mapping corresponds to a nonlinear time transformation which converges to a finite value for a large class of trajectories, enabling an exact analysis of asymptotic properties in expanding domains. We further provide a detailed discussion of different particle interactions and generalized geometries. All our findings are based on exact computations and are illustrated numerically for various examples, including Lévy processes and fractional Brownian motion.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Research Centres > Centre for Complexity Science Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Physics
Journal or Publication Title:	Journal of Statistical Mechanics: Theory and Experiment
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	1742-5468
Official Date:	13 September 2013
Dates:	Date Event 13 September 2013 Published
Volume:	Volume 2013
Number:	Number 09
Page Range:	Article number P09006
DOI:	10.1088/1742-5468/2013/09/P09006
Status:	Peer Reviewed
Publication Status:	Published
Open Access Version:	ArXiv

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