Interacting Particle Systems in Time-Dependent Geometries

TL;DR

This paper develops an exact mapping technique for analyzing interacting particle systems in time-dependent geometries, enabling precise study of their asymptotic behavior and interactions in expanding or contracting domains.

Contribution

It introduces a non-linear time transformation mapping time-dependent geometries to fixed domains, applicable to self-affine models with or without memory, supported by exact computations.

Findings

Exact mapping preserves scale invariance in expanding/contracting domains.

The approach applies to models with memory and explicit time dependence.

Numerical examples include Lévy processes and fractional Brownian motion.

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 [A. Ali et al., Phys. Rev. E 87, 020102(R) (2013)] and here we provide a detailed discussion on its applicability for self-affince Markovian models, and how it can be adapted to self-affine models with memory or explicit time dependence. The mapping corresponds to a non-linear time transformation which convergences 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 L\'evy processes and fractional Brownian motion.