2D Ising Model (analytical expressions)

In summary, the conversation discusses a program to simulate the 2D Ising Model using the Metropolis algorithm. The individual is looking for help in finding analytical expressions for the mean energy, magnetization, specific heat, and magnetic susceptibility. They have found the expressions for the mean energy, magnetization, and specific heat, but are having trouble with the results for the mean energy. They are also unable to find an analytical expression for the magnetic susceptibility. Another individual suggests using the expression for magnetization to find the susceptibility, but the original individual is only interested in the case with B=0 and does not have an analytical expression for the susceptibility. They are seeking assistance in finding this expression.
  • #1
Orion_PKFD
9
0
Hi all,

I am doing a program to simulate the 2D Ising Model under the metropolis algorithm. In order to check my results I would like to compare them with the analytical expressions for the mean energy, magnetization, specific heat and magnetic susceptibility.

I already found the expressions for the mean energy, magnetization and specific heat. However, when I plot the expression for the mean energy the result does not look right. I used the one in Huang's book. Concerning the magnetic susceptibility, I couldn't find any analytical expression...

Anyone could give me some help? Thanks.

Best regards!
 
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  • #2
If you have the expression for the magnetisation for a given applied field you should be able to go from there to the susceptibility...
 
  • #3
Hi,

Thank you for your reply. I am only interested in the case with B=0.

The expression for the magnetization is [itex]M=[1-\sinh^{-4}(2/k_B T)]^{1/8}[/itex].

The susceptibility is obtained from [itex]\chi = \frac{<M^2>- <M>^2}{k_B T}[/itex].

But I don't know the analytical expression for [itex]\chi[/itex]...
 

1. What is the 2D Ising Model?

The 2D Ising Model is a mathematical model used in statistical mechanics to study the properties of magnetic materials. It describes the behavior of a lattice of interacting magnetic spins in two dimensions.

2. What are the main assumptions of the 2D Ising Model?

The main assumptions of the 2D Ising Model include a regular lattice structure, nearest-neighbor interactions between spins, and a simplified energy function that only considers the alignment of neighboring spins.

3. What are the analytical expressions for the energy and magnetization in the 2D Ising Model?

The energy of the 2D Ising Model can be expressed as E = -J∑s_is_j, where J is the coupling constant and s_i and s_j are the neighboring spin values. The magnetization can be calculated as M = ∑s_i, where s_i is the spin value at each lattice site.

4. How is the critical temperature derived in the 2D Ising Model?

The critical temperature, T_c, is derived by solving the self-consistent equation for the average magnetization, which is given by M = (1/N)∑_is_i = tanh(JM/T). The critical temperature is the temperature at which this equation has two solutions, representing the phase transition from a magnetized to a non-magnetized state.

5. What are some applications of the 2D Ising Model?

The 2D Ising Model has been used in various fields, such as statistical physics, materials science, and computer science. It has been applied to study phase transitions, magnetic materials, and optimization problems, and has also been used as a simple model for neural networks and social dynamics.

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