Grand Canonical Monte Carlo

Goal

To perform Grand Canonical Monte Carlo (GCMC) simulations using cluster expansion models to study composition-dependent thermodynamics and generate temperature-composition (T-x) phase diagrams. GCMC allows the system composition to vary by controlling the chemical potential ($\mu$) instead of fixing composition directly.

Background

In the canonical ensemble (fixed N, V, T), Monte Carlo simulations explore configurational space at a fixed composition. In contrast, the grand canonical (or semigrand canonical) ensemble allows composition to fluctuate in response to specified chemical potentials. This is particularly useful for:

• Mapping phase diagrams (composition vs. temperature)
• Identifying miscibility gaps and phase transitions
• Studying composition-dependent thermodynamics
• Exploring equilibrium compositions at different chemical potentials

For binary alloys (e.g., Cu-Ag), we typically control the chemical potential difference Δμ = μ_A - μ_B by setting one species to μ = 0 and varying the other.

Workflow