The mean field approximation
The mean field approximation
The mean field approximation is a technique that can be used to calculate approximate partition functions for systems composed of interacting particles. The problem with calculating such partition functions exactly comes when one attempts to enumerate all the possible microstates and calculate their energy. This combinatorial problem is too expensive to solve by brute force and is impossible to solve for complex Hamiltonians. In mean field theory the $N$-body system is thus replaced by a 1-body system that sits in a suitably chosen external field. This external field is set equal to the average field due to the remaining particles. In essence mean field theory is an approximation technique that allows us to map a multi-body problem onto a one-body problem. Similar techiniques appear in many different fields within physics and are often given different names.
Syllabus Aims
- You should be able to write down the mean field Hamiltonian for the 1D and 2D closed Ising models.
- You should be able to derive expressions for the mean field canonical partition function for 1D and 2D Ising models.
- You should be able to find average energies and average spins for 1D and 2D, mean-field Ising models by taking appropriate derivatives of the partition function.
- You should be able to find the temperature at which the order disorder transition takes place for an Ising model simulated using mean field theory.
- You should be able to perform derivations with more refined mean field theories in which the interaction between a subset of the spins are modelled explicitly.
Description and link | Module | Author | ||
| An video describing how we can use the mean field approximation to calculate an approximation to the partition function for the Ising model. | AMA4004 | G. Tribello |
Description and link | Module | Author | ||
| Problems in which the mean field approximation is used to examine Ising models in one and two dimensions. | AMA4004 | G. Tribello |
Contact Details
School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN
Email: g.tribello@qub.ac.uk
Website: mywebsite