Simulation Parameters
Example input and sample output
| Parameter | Value |
|---|---|
| Lattice size | 100 |
| Initial coverage | 0.20 |
| k_ads | 0.10 |
| k_des | 0.05 |
| k_diff | 0.50 |
Sample (first rows) output when running with example parameters:
| time | coverage |
|---|---|
| 0.000000 | 0.200000 |
| 0.012345 | 0.210000 |
| 0.034567 | 0.205000 |
Formula / Algorithm used
This implementation uses the Gillespie (kinetic Monte Carlo) algorithm. At every step we enumerate all possible events and their rates
and compute the total rate R which equals the sum of individual rates. The next-event time increment \u0394t
is sampled from an exponential distribution: \u0394t = -ln(u)/R where u is a uniform random number in (0,1].
An event is chosen randomly with probability proportional to its rate (weighted selection). The lattice is updated accordingly.
How to use this calculator
- Set lattice size and initial coverage (fraction of occupied sites).
- Provide per-site rates for adsorption and desorption, plus diffusion hop rate.
- Choose stopping criteria: maximum time or maximum events.
- Set a record interval to control output density.
- Click Run simulation. Results will appear above the form.
- Download CSV or PDF for offline analysis.
Overview
Kinetic Monte Carlo (KMC) provides a statistically exact framework to simulate stochastic dynamics governed by discrete events and well‑defined rates. This simulator implements a one‑dimensional lattice model with adsorption, desorption, and nearest‑neighbour diffusion, enabling users to reproduce kinetic processes such as surface growth, catalysis, and thin film dynamics.
Physical model
The lattice contains N sites occupied (1) or empty (0). Adsorption fills empty sites at rate kads, desorption removes particles at rate kdes, and diffusion allows hops with rate kdiff. Periodic boundaries emulate an infinite ring; adjust for open or reflective boundaries if needed.
Lattice and boundary conditions
A 1D lattice is suitable for fast prototyping and analytic comparison. Use periodic boundary conditions to remove edge effects. For spatially resolved phenomena on surfaces, map the 1D results qualitatively to 2D/3D systems, or extend the event generation to higher dimensions for quantitative fidelity.
Event types and rates
Events are enumerated per site: adsorption on empty sites (rate kads), desorption from occupied sites (rate kdes), and diffusion hops into adjacent empty sites (rate kdiff). The total rate R is the sum of all individual event rates; event selection is proportional to rate contribution.
Gillespie / KMC algorithm
The implementation follows Gillespie’s first reaction approach: sample the inter‑event time Δt from an exponential distribution Δt = −ln(u)/R and select an event by weighted sampling. This yields exact stochastic trajectories consistent with the master equation for Markov jump processes.
Numerical considerations
Choose lattice size, tmax, and maximum events to balance statistical convergence and compute cost. Use multiple independent trajectories for ensemble averages and reduce recorded event density via the record interval to control output size. Watch for long tails in rare‑event regimes.
Output and analysis
The calculator produces time‑series coverage data (fraction occupied) and summary statistics. Export CSV or PDF for post‑processing: compute ensemble means, variances, correlation functions, or fit to rate equations to extract effective kinetic parameters.
Extensions and best practices
Extend the model to site heterogeneity, multi‑species reactions, or 2D lattices for surface diffusivity studies. Validate KMC results against mean‑field rate equations and, where possible, experimental kinetics or kinetic Monte Carlo benchmarks to ensure model fidelity.
Frequently Asked Questions
Q1: What is the difference between KMC and molecular dynamics?
A1: KMC simulates discrete stochastic events using rates and is efficient for rare‑event kinetics. Molecular dynamics resolves deterministic particle trajectories and is costlier for long‑time kinetics.
Q2: How many trajectories are needed for reliable averages?
A2: Typically tens to hundreds of independent runs give stable ensemble averages; the required number depends on noise amplitude and desired statistical confidence.
Q3: Can I model multi‑species reactions?
A3: Yes. Add species labels per site and extend event generation to include reaction rules and corresponding rates.
Q4: Why use periodic boundaries?
A4: Periodic boundaries remove finite‑size edge effects and approximate an infinite system, simplifying comparisons. Use open or reflective boundaries if edges are physically relevant.
Q5: How to convert 1D results to 2D intuition?
A5: 1D simulations provide qualitative trends. For quantitative predictions, implement a 2D lattice and adjust diffusion connectivity and rates accordingly.
Q6: What should I watch for in rare‑event regimes?
A6: Rare events produce long waiting times and heavy tails. Increase maximum simulation time, use importance sampling techniques, or accelerate dynamics with adaptive algorithms.
Q7: How to validate my KMC model?
A7: Validate against analytic rate equations, experimental kinetics, or established KMC benchmarks; compare ensemble means and transient behaviours for consistency.