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Ising model was invented to describe phase transition in magnetics. It also describes gas-liquid phase transition!

Consider a *lattice model:*

Each site can be either occupied or empty. Only nearest neighbors
interact. Energy of interaction . We want to map this to
Ising model. Introduce a spin

Number of particles in a cell:

Total number of particles Interaction energy between cells 1 and 2 This can be written as Total energy: Grand partition function:(2) |

Ising Magnetic | Lattice Gas |

Canonical ensemble | Grand canonical ensemble |

Coupling constant J |
Interaction energy |

External field H |
Chemical potential |

Magnetization M |
Density |

Free energy A |
-potential; pressure P |

Susceptibility | Compressibility |

Isotropic phase | Supercritical fluid |

Ordered phase | Liquid or gas |

Curie point | Critical point |

**Note about symmetry:**- In Ising model phase transition occur at
*H*=0. In lattice gas it corresponds to --there is no symmetry between particles & holes!

An *incompressible* mixture of *A* and *B*:

Once again, only closest neighbor interact. Energy of
interaction^{}:

We introduce spins:

Energy of interaction between 1 and 2 Total energy: Difference between the numbers of particles:
I cannot use grand canonical ensemble: the total # of particles is
constant (incompressibility). Semi-Grand Ensemble: the total number of
particle is constant, *but* I can switch *A* and *B*.

Partition function:

with effective energy: Once again Ising model withCorrespondence between Ising magnetic & binary mixture:

Ising Magnetic | Binary Mixture |

Canonical ensemble | Semi-grand canonical ensemble |

Coupling constant J |
Interaction energy |

External field H |
Chemical potential diff. |

Magnetization M |
Composition |

Isotropic phase | Mixed phase |

Ordered phase | Separated phase |

Curie point | Critical mixing point |

How can we extend Ising model?

- Let have several values
*(Potts model)*--useful for compressible multicomponent mixtures - Introduce vector and write interaction as
*(Heisenberg model)* - Introduce complex lattices (fcc, bcc, trigonal, hexagonal...)
- Introduce several sub lattices
*(decorated models)* - Introduce interaction of farther neighbors and
- Make
*J**random*--favorite toy in glass theory - Introduce
*kinetics or re-orientation*--make your model time-dependent!

© 1997

Mon Oct 13 22:07:20 EDT 1997