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Finite-size scaling and edge effects in the Takayasu model of aggregation-diffusion with input
Rohan Banerjee Ravindran1, R Rajesh2,3
1Shiv Nadar Institution of Eminence, Department of Physics, Gautam Buddha Nagar, Uttar Pradesh 201314, India.
Physical Review. E
|June 19, 2026
Summary
Finite boundaries significantly alter particle diffusion and aggregation in the Takayasu model. Edge effects create distinct behaviors compared to bulk, especially in one-dimensional systems with steady monomer input.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Theoretical Physics
Background:
- The Takayasu model describes diffusing and aggregating particles with monomer input.
- Understanding boundary effects is crucial for realistic physical systems.
- One-dimensional systems often exhibit unique phenomena.
Purpose of the Study:
- To investigate the impact of finite spatial boundaries on the Takayasu model.
- To derive exact solutions for steady-state properties under different boundary conditions.
- To analyze the crossover behavior of mass distribution near boundaries.
Main Methods:
- Analytical calculations
- Numerical simulations
- Derivation of exact expressions for density profiles and correlation functions
- Scaling arguments for generalization
Main Results:
- Exact steady-state density profiles, correlation functions, and mean-squared density derived for open and periodic boundaries.
- Observed crossover in single-site mass distribution from bulk to edge power laws (P(m)∼m^{-4/3} to P(m)∼m^{-5/3}).
- Demonstrated breakdown of equivalence between boundary conditions for multipoint distributions near edges.
- Identified a boundary layer where mass currents dominate over constant flux.
Conclusions:
- Finite boundaries introduce significant deviations from bulk behavior in the Takayasu model.
- The edge anomaly is linked to the dominance of spatial mass currents.
- Results generalize to mass-dependent diffusion, highlighting the importance of boundary layer physics.
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