The hard-sphere Monte Carlo algorithm is a computational method used to simulate systems of hard spheres, which are particles that interact only through excluded volume effects—meaning they cannot overlap but otherwise do not interact.
The Monte Carlo (MC) method for hard spheres typically follows the Metropolis algorithm, where random trial moves are generated, and only those that do not violate the hard-sphere constraint (no overlap) are accepted.
Basic Steps
- Initialize the System:
- Place $N$ hard spheres randomly in a simulation box.
- Ensure there are no overlapping spheres.
- Perform Monte Carlo Moves:
- Randomly select a sphere.
- Propose a random displacement within a chosen step size $\delta r$.
- Check for overlap with other spheres.
- If no overlap, accept the move.
- If overlap, reject the move and keep the previous position.
- Repeat Steps (2) for Many Iterations:
- The number of moves should be large enough to ensure equilibrium sampling.
- Measure properties such as radial distribution function, equation of state, etc.
Key Considerations
- Packing Fraction ($\phi$): The density of the system affects the acceptance rate. As ϕ increases, the probability of rejecting moves increases.
- Step Size Optimization: The displacement step $\delta r$ should be tuned to maintain an acceptance ratio around 30−50% for efficiency.
- Boundary Conditions: Usually, periodic boundary conditions (PBCs) are used to simulate an infinite system.
- Equilibration & Sampling: The system must be equilibrated before collecting data.
Applications
- Modeling fluids and dense packings.
- Studying phase transitions (fluid-solid transition).
- Understanding crystallization and jamming.
🧠Simulates a system of hard spheres in a 2D box with periodic boundary conditions
https://gist.github.com/viadean/a7ac0057f133b19f40c8cc940aed2110
How It Works:
- Initializes spheres randomly in a 2D box.
- Uses periodic boundary conditions (PBCs) to maintain an infinite system.