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Subsections

Applications of Ising Model

Ising Model and Lattice Gas

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)
with effective energy:

This is (up to a constant) the Hamiltonian of an Ising system with

Correspondence between Ising magnetic & lattice gas (up to a constant or a factor):
 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

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

Ising Model and Lattice Binary Mixture

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 with

Correspondence 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

Ising-like Models

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!

Next: Conclusion Up: Ising Model and Its Previous: Phase Transitions of Ising

© 1997 Boris Veytsman and Michael Kotelyanskii
Mon Oct 13 22:07:20 EDT 1997