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

TL;DR

The study provides exact analytical and numerical insights into finite-size and edge effects in the Takayasu aggregation-diffusion model with input, revealing a robust bulk exponent $\tau_1=\frac{4}{3}$ and a distinct edge exponent $\tau_1=\frac{5}{3}$, with a boundary layer that governs edge behavior. By solving for density profiles and two-point correlations under open and periodic boundaries, the authors show that edge physics and multipoint distributions depend sensitively on boundary conditions, and they establish a comprehensive scaling framework including mass-dependent diffusion. The work combines exact continuum-limit solutions, Fourier-Bessel-type expansions, and Monte Carlo simulations to verify bulk and edge scalings, including explicit expressions for $R(x)$, $R_P(x)$, $C(x,y)$, and $E(x)$, and extends to general $\alpha$-dependent diffusion. These results clarify finite-size effects in driven-dissipative mass-aggregation systems and provide precise predictions for edge-dominated statistics that may be observed in constrained physical and biological systems.

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.

Figures (14)

• Figure 1: The scaled density, $R(x)$, obtained from Monte Carlo simulations is compared with the exact result, Eq. (\ref{['eq12']}). The data are for open boundary conditions.
• Figure 2: The scaled second moment of density, $E(x)$, obtained from Monte Carlo simulations is compared with the exact result, Eq. (\ref{['eq31']}). The solid line has slope $g$ as given in Eq. (\ref{['eq36']}). The data are for open boundary conditions.
• Figure 3: The CCDF for the single site mass distribution, $F_1^B$, at the middle site $L/2$ for different system sizes. The data are consistent with the power law $m^{-1/3}$. Inset: The data for different $L$ collapse onto one curve when scaled with the exponents in Eq. (\ref{['eq33']}). The data are for open boundary conditions.
• Figure 4: The CCDF for multipoint probabilities, $F_n^B$ at the bulk middle site $L/2$ for different system sizes collapse onto one curve when scaled with the exponents in Eq. (\ref{['eq34']}) for $n=2, 3, 4$. The solid lines correspond to power-laws with exponents $\zeta_n^B$. The data are for open boundary conditions.
• Figure 5: The CCDF for the single site mass distribution, $F_1^E$ at the edge site $1$ for different system sizes. The data are consistent with the power law $m^{-2/3}$. Inset: The data for different $L$ collapse onto one curve when scaled with the exponents in Eq. (\ref{['eq37']}). The data are for open boundary conditions.