Subsections

• Ising Model and Lattice Gas
• Ising Model and Lattice Binary Mixture
• Ising-like Models

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 $-\epsilon$. We want to map this to Ising model. Introduce a spin

$$\sigma_i= \begin{cases} +1,&\text{cell $i$\space is occupied}\ -1,&\text{cell $i$\space is empty} \end{cases}$$

Number of particles in a cell:

$$(\sigma_i+1)/2 = \begin{cases} 1,&\text{cell $i$\space is occupied}\ 0,&\text{cell $i$\space is empty} \end{cases}$$

Total number of particles

$$N_p = \frac12\sum_i \left(\sigma_i+1\right) = \frac12 \sum_i \sigma_i + \frac{N}{2}$$

Interaction energy between cells 1 and 2

$$\epsilon_{12} = \begin{cases} -\epsilon,& \sigma_1=\sigma_2=1\ 0,& \text{otherwise} \end{cases}$$

This can be written as

$$\epsilon_{12} = -\frac14\epsilon\left(\sigma_1+1\right) \left(\sigma_2+1\right)$$

Total energy:

$$\begin{gathered} E_p = -\frac{\epsilon}{8}\sum_{(i,j)}(\sig... ...psilon}{4}\sum_i\sigma_i - \frac{zN\epsilon}{8} \end{gathered}$$

Grand partition function:  

$$\Xi = \sum_{\{\sigma\}} \exp\bigl(\beta\mu N_p-\beta E_p\bigr) = \sum_{\{\sigma\}} \exp\bigl(-\beta E_{\text{eff}}\bigr)$$ (2)

with effective energy:

$$E_{\text{eff}} = -\frac{\epsilon}{8}\sum_{(i,j)}\sigma_i\sig... ...right)\sum_i\sigma_i - \frac{zN\epsilon}{8} + \frac{\mu N}{2}$$

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

$$J = \frac{\epsilon}{4}, \quad H = \frac{\mu}{2}+\frac{z\epsilon}{4}$$

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

Ising Magnetic Lattice Gas

Canonical ensemble Grand canonical ensemble

$$\epsilon$$ Coupling constant J

$$\mu$$ External field H

$$\rho$$ Magnetization M

$$\Omega$$ Free energy A

$$\chi \alpha$$

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 $\mu=-z\epsilon/2\ne0$--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:

$$\begin{cases} 0,& \text{$AA$\space or $BB$\space contacts}\ \epsilon, & \text{$AB$\space contacts} \end{cases}$$

We introduce spins:

$$\sigma_i= \begin{cases} +1,&\text{cell $i$\space is occupi... ...$}\ -1,&\text{cell $i$\space is occupied by $B$} \end{cases}$$

Energy of interaction between 1 and 2

$$\epsilon_{12} = \frac12\epsilon(1-\sigma_1\sigma_2)$$

Total energy:

$$E_p= \frac{\epsilon}{4}\sum_{(i,j)}(1-\sigma_i\sigma_j) = \... ...zN\epsilon}{4} - \frac{\epsilon}{4}\sum_{(i,j)}\sigma_i\sigma_j$$

Difference between the numbers of particles:

$$N_A-N_B=\sum_i\sigma_i$$

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:

$$\Xi_{SG} = \sum_{\{\sigma\}} \exp\bigl(\beta\Delta\mu (N_A-N... ...r) = \sum_{\{\sigma\}} \exp\bigl(-\beta E_{\text{eff}}\bigr)$$

with effective energy:

$$E_{\text{eff}} = - \frac{\epsilon}{4}\sum_{(i,j)}\sigma_i\sigma_j - \Delta\mu\sum_i\sigma_i + \frac{zN\epsilon}{4}$$

Once again Ising model with

$$J = \frac{\epsilon}{2},\quad H = -\Delta\mu$$

Correspondence between Ising magnetic & binary mixture:

Ising Magnetic Binary Mixture

Canonical ensemble Semi-grand canonical ensemble

$$2\epsilon$$ Coupling constant J

$$\Delta\mu$$ External field H

$$\Delta\phi$$ Magnetization M

Isotropic phase Mixed phase

Ordered phase Separated phase

Curie point Critical mixing point

Ising-like Models

How can we extend Ising model?

• Let $\sigma_i$ have several values (Potts model)--useful for compressible multicomponent mixtures
• Introduce vector $\bm{\sigma}_i$ and write interaction as $\bm{\sigma}_i\bm{\sigma}_j$ (Heisenberg model)
• Introduce complex lattices (fcc, bcc, trigonal, hexagonal...)
• Introduce several sub lattices (decorated models)
• Introduce interaction of farther neighbors and $J(\mathbf{r}_1-\mathbf{r}_2)$
• Make J random--favorite toy in glass theory
• Introduce kinetics or re-orientation--make your model time-dependent!