---
title: "The Ising Model and the Mean-Field Approximation"
description: "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"
canonical: https://lightmysky.com/learn/science/the-ising-model-and-the-mean-field-approximation-mt_wPwyBTVI0C
source: https://lightmysky.com/learn/science/the-ising-model-and-the-mean-field-approximation-mt_wPwyBTVI0C.md
retrieved: 2026-09-12
---

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# 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.

Subject: Science · Area: Thermal & Statistical Physics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/the-ising-model-and-the-mean-field-approximation-mt_wPwyBTVI0C

## Ready when they can

- 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

## Lesson: A magnet made of averages

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Replace neighbours by their average, solve the loop for self-consistency, and remember the missing fluctuations near the critical point.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Phase Transitions, Order Parameters and Broken Symmetry](https://lightmysky.com/learn/science/phase-transitions-order-parameters-and-broken-symmetry-mt_3DQjfNJ-Ef)
- [Microstates, Multiplicity and Boltzmann's Entropy](https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6)

## Opens up

- [Critical Exponents, Scaling and Universality](https://lightmysky.com/learn/science/critical-exponents-scaling-and-universality-mt_EaJOjyvjjj)