Interacting Systems and Mean-Field Theory

Define the Ising model Hamiltonian as a paradigm for interacting systems and cooperative phenomena.

Derive the mean-field (Bragg-Williams) approximation for the Ising model by neglecting fluctuations.

Solve the mean-field self-consistency equation for the magnetization and find the critical temperature T_c.

Expand the mean-field free energy near T_c to calculate the critical exponents β, γ, and δ.

Derive the van der Waals equation of state from physical arguments and show that it is a mean-field theory of the liquid-gas transition.

Identify the limitations of mean-field theory by comparing its predictions with general principles and exact results.