Grand Canonical and Isothermal Ensembles

This lecture covers the derivation of the grand canonical ensemble, which describes systems with a fluctuating number of particles and a fixed chemical potential, and the isothermal isobaric ensemble, where temperature and pressure are held constant. The instructor explains the extended Lagrangian used to define these ensembles, the role of Lagrange multipliers in enforcing thermodynamic conditions, and the link between thermodynamic potentials and partition functions. Through detailed derivations, the lecture illustrates how to obtain the probability distributions for these ensembles and how they are related to the Gibbs free energy. The audience is encouraged to engage with the material by attempting derivations independently.