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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 |
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?
© 1997 Boris Veytsman and Michael Kotelyanskii