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

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:
\begin{figure}
 \includegraphics{lattice_gas}\end{figure}
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

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

Number of particles in a cell:

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

Total number of particles

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

Interaction energy between cells 1 and 2

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

This can be written as

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

Total energy:

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

Grand partition function:  
 \begin{displaymath}
 \Xi = \sum_{\{\sigma\}} \exp\bigl(\beta\mu N_p-\beta E_p\bigr) =
 \sum_{\{\sigma\}} \exp\bigl(-\beta E_{\text{eff}}\bigr) \end{displaymath} (2)
with effective energy:

\begin{displaymath}
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}\end{displaymath}

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

\begin{displaymath}
J = \frac{\epsilon}{4}, \quad H = \frac{\mu}{2}+\frac{z\epsilon}{4}\end{displaymath}

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 $\epsilon$
External field H Chemical potential $\mu$
Magnetization M Density $\rho$
Free energy A $\Omega$-potential; pressure P
Susceptibility $\chi$ Compressibility $\alpha$
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:
\begin{figure}
 \includegraphics{checkerboard}\end{figure}
Once again, only closest neighbor interact. Energy of interaction[*]:

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

We introduce spins:

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

Energy of interaction between 1 and 2

\begin{displaymath}
\epsilon_{12} = \frac12\epsilon(1-\sigma_1\sigma_2)\end{displaymath}

Total energy:

\begin{displaymath}
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\end{displaymath}

Difference between the numbers of particles:

\begin{displaymath}
N_A-N_B=\sum_i\sigma_i\end{displaymath}

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:

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

with effective energy:

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

Once again Ising model with

\begin{displaymath}
J = \frac{\epsilon}{2},\quad H = -\Delta\mu\end{displaymath}

Correspondence between Ising magnetic & binary mixture:

Ising Magnetic Binary Mixture
   
Canonical ensemble Semi-grand canonical ensemble
Coupling constant J Interaction energy $2\epsilon$
External field H Chemical potential diff. $\Delta\mu$
Magnetization M Composition $\Delta\phi$
Isotropic phase Mixed phase
Ordered phase Separated phase
Curie point Critical mixing point

Ising-like Models

How can we extend Ising model?


next up previous
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