The Ising Model and the Mean-Field Approximation
Replacing the neighbours of a spin by their average turns an intractable interacting model into one self-consistent equation with a spontaneous solution below a critical temperature. The approximation gets the transition right and the exponents wrong, and knowing why is the point.
What a learner can do afterwards
- Writes the self-consistency equation and solves it graphically
- Finds the critical temperature the approximation predicts
- Explains which fluctuations the approximation discards and when that matters
1 · Read
The Ising model lines up tiny spins that each feel their neighbours, which is intractable exactly. The mean-field trick replaces every neighbour by the average spin, turning the tangle into one spin sitting in an average field.
That gives one self-consistent equation: assume an average, compute what each spin does in it, and demand the result reproduces your average. Solve it graphically, line against curve, and below a critical temperature a nonzero solution appears on its own: spontaneous order.
High above the critical point only zero solves the equation, so there is no lasting magnetisation. Cool below it and new nonzero solutions grow: the system picks one and stays magnetised with no outside push.
The trick gets the transition right and the exponents wrong, because it discards fluctuations: real neighbours wobble together in patches, and the average cannot see patches. Trust it far from the critical point, not inside the wobble zone.
Replace neighbours by their average, solve the loop for self-consistency, and remember the missing fluctuations near the critical point.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.