Numba-Accelerated 2D Diffusion-Limited Aggregation: Implementation and Fractal Characterization

TL;DR

This work develops dla-ideal-solver, a Numba-accelerated Python framework for 2D diffusion-limited aggregation on a lattice, enabling large ensembles necessary to resolve fractal scaling. It implements a 2D DLA model with random and radial injection geometries and a re-injection mechanism to focus computation near the cluster. Key findings validate the canonical fractal dimension $D_f ≈ 1.71$ in dilute regimes and reveal a finite-density crossover to Eden-like $D_f ≈ 1.87$ at high walker density, attributed to screening-length saturation; Renyi dimensions and lacunarity quantify monofractal structure and spatial heterogeneity. The open-source framework provides a reproducible platform for exploring non-equilibrium phase transitions in Laplacian growth and related pattern formation problems.

Abstract

We present dla-ideal-solver, a high-performance framework for simulating two-dimensional Diffusion-Limited Aggregation (DLA) using Numba-accelerated Python. By leveraging just-in-time (JIT) compilation, we achieve computational throughput comparable to legacy static implementations while retaining high-level flexibility. We investigate the Laplacian growth instability across varying injection geometries and walker concentrations. Our analysis confirms the robustness of the standard fractal dimension $D_f \approx 1.71$ for dilute regimes, consistent with the Witten-Sander universality class. However, we report a distinct crossover to Eden-like compact growth ($D_f \approx 1.87$) in high-density environments, attributed to the saturation of the screening length. Beyond standard mass-radius scaling, we employ generalized Rényi dimensions and lacunarity metrics to quantify the monofractal character and spatial heterogeneity of the aggregates. This work establishes a reproducible, open-source testbed for exploring phase transitions in non-equilibrium statistical mechanics.

Figures (4)

• Figure 1: Spatiotemporal evolution of DLA aggregates across four test configurations. Rows correspond to: (a--c) classic DLA with single central seed; (d--f) multiple seeds with 12 randomly distributed nucleation sites; (g--i) radial injection from fixed radius $R_{\mathrm{inj}} = 180$; (j--l) high density with $N_w = 25{,}000$ walkers. Columns represent early ($\sim$10%), middle ($\sim$50%), and final (100%) growth stages. Particle counts $N$ and completion percentages are indicated in each panel. Color encodes radial distance from aggregate centroid (dark red: center; bright orange: tips). Lattice size $512 \times 512$ for all cases.
• Figure 2: Mass-radius scaling analysis for fractal dimension extraction. (a) Classic DLA: $D_f = 1.711 \pm 0.080$, $R^2 = 0.9875$. (b) Radial injection: $D_f = 1.713 \pm 0.077$, $R^2 = 0.9854$. (c) High density: $D_f = 1.870 \pm 0.055$, $R^2 = 0.9950$. Circles denote simulation data; solid red lines show linear regression fits; shaded bands indicate 95% confidence intervals; dashed teal lines represent the theoretical prediction $D_f = 1.71$. The multiple seeds case is excluded as mass-radius analysis requires a single connected aggregate. Data filtered to $M(R) > 10$ and $R < 0.8 R_{\mathrm{max}}$ to exclude boundary effects and small-number fluctuations.
• Figure 3: Growth dynamics analysis for all four DLA configurations. (a) Cumulative particle count $N(t)$ versus snapshot index, showing characteristic decelerating growth as aggregates enlarge. (b) Instantaneous growth rate $\mathrm{d}N/\mathrm{d}t$ versus particle count $N$, demonstrating monotonic decrease due to diffusive screening. (c) Normalized growth curves $N(t)/N_{\mathrm{max}}$ versus normalized time $t/T$, enabling direct comparison of growth kinetics. (d) Growth efficiency $N/R_{\mathrm{eff}}$ (normalized) versus $t/T$, where $R_{\mathrm{eff}} = N^{1/D_f}$ estimates the effective aggregate radius. Legend applies to all panels.
• Figure 4: Information-theoretic characterization of DLA aggregate structure. (a) Shannon entropy $H(\varepsilon)$ versus box size $\varepsilon$ on semi-logarithmic axes, quantifying spatial information content at each scale. (b) Lacunarity $\Lambda(\varepsilon)$ versus $\varepsilon$ on log-log axes, measuring scale-dependent heterogeneity ("gappiness"). (c) Box-counting dimension $D_0$ extraction via linear regression of $\log N(\varepsilon)$ versus $\log \varepsilon$; fitted values indicated for each case. (d) Information dimension $D_1$ extraction from $H(\varepsilon)$ versus $\log_2(1/\varepsilon)$; note consistent use of $\log_2$ matching Shannon entropy definition. For monofractal structures, $D_0 \approx D_1 \approx D_2 \approx D_f \approx 1.71$.