An exact and efficient first passage time algorithm for reaction-diffusion processes on a 2D-lattice

Andri Bezzola, Benjamin B. Bales, Richard C. Alkire, Linda R. Petzold

Research output: Contribution to journalArticlepeer-review


We present an exact and efficient algorithm for reaction-diffusion-nucleation processes on a 2D-lattice. The algorithm makes use of first passage time (FPT) to replace the computationally intensive simulation of diffusion hops in KMC by larger jumps when particles are far away from step-edges or other particles. Our approach computes exact probability distributions of jump times and target locations in a closed-form formula, based on the eigenvectors and eigenvalues of the corresponding 1D transition matrix, maintaining atomic-scale resolution of resulting shapes of deposit islands. We have applied our method to three different test cases of electrodeposition: pure diffusional aggregation for large ranges of diffusivity rates and for simulation domain sizes of up to 4096 × 4096 sites, the effect of diffusivity on island shapes and sizes in combination with a KMC edge diffusion, and the calculation of an exclusion zone in front of a step-edge, confirming statistical equivalence to standard KMC simulations. The algorithm achieves significant speedup compared to standard KMC for cases where particles diffuse over long distances before nucleating with other particles or being captured by larger islands.

Original languageEnglish (US)
Pages (from-to)183-197
Number of pages15
JournalJournal of Computational Physics
StatePublished - Jan 1 2014


  • Discrete Laplacian
  • Electrodeposition
  • First passage time
  • Nucleation and growth
  • Reaction-diffusion
  • Stochastic algorithm

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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