Grand Canonical Ensemble 2D: Lattice-Gas 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 on a 2D square lattice gas: sites exchange 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 into amber domains 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.