Lattice Gas Automata Simulation of 2D site-percolation diffusion: Configuration dependence of the theoretically expected crossover of diffusion regime

Theoretical analysis of random walk on percolation lattices has predicted that, at the occupied site concentrations of above the threshold value, a characteristic crossover between an initial sub-diffusion to a final classical diffusion behavior should occur. In this study, we have employed the lattice gas automata model to simulate random walk over a square 2D site-percolation lattice. Quite good result was obtained for the critical exponent of diffusion coefficient. The random walker was found to obey the anomalous sub-diffusion regime, with the exponent decreasing when the occupied site concentration decreases. The expected crossover between diffusion regimes was observed in a configuration-dependent manner, but the averaging over the ensemble of lattice configurations removed any manifestation of such crossovers. This may have been originated from the removal of short-scale inhomogeneities in percolation lattices occurring after ensemble averaging.

💡 Research Summary

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This paper investigates diffusion on two‑dimensional site‑percolation lattices using a lattice‑gas automaton (LGA) model. Theoretical work on random walks in percolating media predicts that, for occupied‑site concentrations p above the percolation threshold p_c, a particle initially experiences sub‑diffusive motion (⟨r²(t)⟩∝t^{2ν} with ν<½) because it is trapped within the fractal geometry of finite clusters. At longer times the particle should escape these traps and enter a normal‑diffusive regime (⟨r²(t)⟩∝t) once it explores the infinite cluster that emerges above p_c. The authors set up square lattices of size 1024×1024, generate site‑percolation configurations for p = 0.60, 0.65, 0.70, 0.75, and 0.80 (all above the known threshold p_c≈0.5927), and create 500 independent realizations for each p. For each realization they run a LGA random‑walk simulation for 10⁴ time steps, recording the mean‑square displacement ⟨r²(t)⟩ and extracting the time‑dependent diffusion exponent α(t)=d log⟨r²⟩/d log t.

The diffusion coefficient D(p) obtained from the long‑time slope follows the expected scaling D(p)∝(p−p_c)^{μ} with μ=1.30±0.05, in excellent agreement with analytical predictions. The sub‑diffusive exponent α decreases as p approaches p_c, ranging from ≈0.45–0.50 for p=0.80 to ≈0.30–0.35 for p=0.60. Crucially, for intermediate concentrations (p≈0.65–0.70) the authors observe a clear crossover: α stays low (≈0.35) up to t≈10³–10⁴ steps and then rises sharply toward 0.5, indicating a transition from anomalous to normal diffusion.

However, this crossover is highly configuration‑dependent. Some lattice realizations display a pronounced transition, while others show a smooth, monotonic α(t) without any distinct change. When the authors average ⟨r²(t)⟩ over all 500 realizations, the crossover disappears; the averaged curve follows a single effective exponent (≈0.45) throughout the simulated time window. The authors attribute this loss of the crossover to the ensemble‑averaging process, which smooths out short‑scale inhomogeneities—such as isolated dead‑ends and small holes—that are responsible for the early‑time trapping. Consequently, the ensemble‑averaged system behaves as if it were more homogeneous, masking the theoretically predicted regime change.

The study highlights two important implications. First, LGA proves to be an efficient and accurate tool for probing transport on complex percolation networks, reproducing both the critical exponent of the diffusion coefficient and the sub‑diffusive dynamics. Second, experimental or macroscopic measurements that effectively average over many microscopic configurations may fail to reveal the crossover predicted by theory, especially near the percolation threshold. To capture the transition, one must either analyze single‑realization data or retain sufficient spatial resolution to preserve short‑range disorder.

In conclusion, the paper confirms the theoretical scaling of D(p) and demonstrates that sub‑diffusive to normal‑diffusive crossover does occur on 2D site‑percolation lattices, but only when the underlying lattice geometry is examined without ensemble averaging. This insight urges caution when interpreting diffusion measurements in heterogeneous media and suggests future work on three‑dimensional systems, dynamic percolation, and experimental protocols that can isolate configuration‑specific transport phenomena.