Diffusion-Driven Surface Autocatalysis in Exterior Domains
Abstract
We study the collective dynamics of independent particles that diffuse outside a spherical surface, on which they are replicated with a prescribed catalytic rate. In spatial dimensions three and higher, the transient nature of diffusion creates a competition between autocatalytic branching and escape to infinity, leading to three distinct regimes: subcritical (finite limiting population), critical (algebraic growth), and supercritical (exponential growth). The rotational symmetry of the domain and an explicit form of the single-particle diffusion propagator allow us to fully investigate the time evolution of the population size, including the mean population size, variance, higher-order moments, as well as the whole distribution. In particular, we obtain an explicit form of the distribution at long times and describe a slow, power-law approach to this steady-state limit. We further extend the steady-state analysis to general exterior domains using the Dirichlet-to-Neumann operator, which determines the critical catalytic rate and reveals how boundary geometry affects the population-size distribution.
Related
- Diffusion-controlled reactions with time-dependent surface reactivity
- Diffusion under competing bulk and surface stopping mechanisms
- Molecular Autocatalysis through Interfacial Recruitment of Catalytic Functional Groups
- Molecular Diffusion-Driven Graded Polymer Hole-Transport Layers Enabling Efficient and Durable Organic Photovoltaics
- Multimodal Spectroscopy–Driven Reconstruction of Three-Dimensional Molecular Structures via Conditional Diffusion Model
BibTeX
@misc{grebenkov2026diffusiondriven,
title = {{Diffusion-Driven Surface Autocatalysis in Exterior Domains}},
author = {Denis S. Grebenkov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.14133},
eprint = {2606.14133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14133},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diffusion-Driven Surface Autocatalysis in Exterior Domains
AU - Denis S. Grebenkov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2606.14133
UR - https://arxiv.org/abs/2606.14133
AB - We study the collective dynamics of independent particles that diffuse outside a spherical surface, on which they are replicated with a prescribed catalytic rate. In spatial dimensions three and higher, the transient nature of diffusion creates a competition between autocatalytic branching and escape to infinity, leading to three distinct regimes: subcritical (finite limiting population), critical (algebraic growth), and supercritical (exponential growth). The rotational symmetry of the domain and an explicit form of the single-particle diffusion propagator allow us to fully investigate the time evolution of the population size, including the mean population size, variance, higher-order moments, as well as the whole distribution. In particular, we obtain an explicit form of the distribution at long times and describe a slow, power-law approach to this steady-state limit. We further extend the steady-state analysis to general exterior domains using the Dirichlet-to-Neumann operator, which determines the critical catalytic rate and reveals how boundary geometry affects the population-size distribution.
ER -