An Introduction to Thermal Physics · Problem 8.20

Mean field theory replaces the complex interactions in an Ising model with an effective average field, letting you solve for magnetization $\bar{s}$ self-consistently. The key insight is that the transition from ordered (magnetized) to disordered (paramagnetic) phases happens at a critical temperature $kT_c/\epsilon$ that depends on lattice dimension and coordination number—for a 2D square lattice, this occurs around $kT_c/\epsilon \approx 2$. Plotting $\bar{s}$ versus $kT/\epsilon$ reveals the characteristic S-shaped curve with a sharp drop at criticality, showing how thermal fluctuations destroy long-range order.

Mean field theory replaces the complex interactions in an Ising model with an effective average field, letting you solve for magnetization $\bar{s}$ self-consistently. The key insight is that the transition from ordered (magnetized) to disordered (paramagnetic) phases happens at a critical temperature $kT_c/\epsilon$ that depends on lattice dimension and coordination number—for a 2D square lattice, this occurs around $kT_c/\epsilon \approx 2$. Plotting $\bar{s}$ versus $kT/\epsilon$ reveals the characteristic S-shaped curve with a sharp drop at criticality, showing how thermal fluctuations destroy long-range order.