Grand Canonical Ensemble: Particle Number Fluctuations

Most introductions to statistical mechanics stop at the canonical ensemble, where particle number N is fixed and only energy is exchanged with a heat bath. This simulator instead builds the grand canonical (μVT) ensemble directly: a 3D lattice gas exchanges both energy and particles with an implicit reservoir held at fixed temperature and chemical potential, using a real Metropolis grand-canonical Monte Carlo move (ΔΩ = ΔE − μΔN) on every proposed insertion or removal. Watch the equilibrium density respond live as you sweep the chemical potential, see attractive interactions drive clustering as temperature drops, and read off the particle-number variance ⟨(ΔN)²⟩ that — via the fluctuation-compressibility relation — directly measures how compressible the simulated system is.