Finite size scaling and edge effects in the Takayasu model of aggregation diffusion with input

TL;DR

This paper analytically and numerically investigates how finite boundaries influence the steady-state properties of the Takayasu aggregation-diffusion model, revealing boundary-induced effects and a crossover in mass distribution.

Contribution

The study provides exact solutions for boundary effects in the Takayasu model, including density profiles and mass distributions, highlighting the impact of finite size and boundary conditions.

Findings

Identification of a crossover in mass distribution near boundaries

Exact expressions for steady-state density and correlations

Boundary effects cause a breakdown of equivalence between boundary conditions

Abstract

We analytically and numerically study the effect of finite spatial boundaries on the Takayasu model of diffusing and aggregating particles with steady monomer input in one dimension. Exact expressions are derived for the steady-state density profile, two-point correlation functions, and mean-squared density under both open and periodic boundary conditions. The single-site mass distribution exhibits a crossover from a bulk power law $P(m)\sim m^{-4/3}$ to an edge power law $P(m)\sim m^{-5/3}$, occurring near the boundaries or the condensate that forms in periodic systems. The equivalence between the two boundary conditions is shown to break down in the case of multipoint probability distributions near the edge. The exact solution identifies a distinct boundary layer and shows that the edge anomaly arises when spatial mass currents, which scale as $\mathcal{O}(L)$, dominate over the $\mathcal{O}(1)$ constant flux in mass space. We further generalize these results to mass-dependent diffusion.