Cluster Approximation for the Two-Dimensional Ising Model

In summary, the conversation is about a specific problem, 3.5, which involves calculating the dimensionless Hamiltonian in two cases. The question is how to explicitly calculate the value of "m" in both cases. The equation ##m = \frac{\sum_{\sigma_i}\sigma_i e^{-\beta H}}{\sum e^{-\beta H}}## is mentioned as a possible approach, but the solution is not clear. The discussion also involves sending attachments of pictures related to the problem.
  • #1
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Homework Statement


In the attachments there is the question and its solution, it's problem 3.5.

Homework Equations

The Attempt at a Solution


My question is how did they get the dimensionless Hamiltonian in both cases, and how did they explicitly calculated ##m## in both cases?

I assume it's with ##m = \frac{\sum_{\sigma_i}\sigma_i e^{-\beta H}}{\sum e^{-\beta H}}##, but I really don't see how to get to the same terms as in the solution.

Your help is appreciated!
https://www.physicsforums.com/attachments/214464 https://www.physicsforums.com/attachments/214465
 
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  • #2
I can see that the pictures didn't get uploaded, so I am sending it again.
Screenshot from 2017-11-06 16-26-56.png
Screenshot from 2017-11-06 16-26-38.png
 

Attachments

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  • Screenshot from 2017-11-06 16-26-38.png
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FAQ: Cluster Approximation for the Two-Dimensional Ising Model

1. What is the Two-Dimensional Ising Model?

The Two-Dimensional Ising Model is a mathematical model used to study the behavior of magnetic materials. It was first proposed by physicist Ernst Ising in 1925 and has since become a classic model in statistical mechanics.

2. How does the Cluster Approximation method work for the Two-Dimensional Ising Model?

The Cluster Approximation method is a mathematical technique used to approximate the behavior of complex systems, such as the Two-Dimensional Ising Model. It involves grouping the individual particles in the system together into clusters and then using statistical methods to analyze their behavior as a whole.

3. What are the advantages of using Cluster Approximation for the Two-Dimensional Ising Model?

One advantage of using Cluster Approximation for the Two-Dimensional Ising Model is that it allows for the study of larger systems, which would be computationally intractable using other methods. It also provides a better understanding of the critical behavior of the system, such as phase transitions.

4. What are some limitations of using Cluster Approximation for the Two-Dimensional Ising Model?

One limitation of using Cluster Approximation for the Two-Dimensional Ising Model is that it is an approximation and therefore may not accurately capture all the details of the system. It also assumes that the clusters are weakly interacting, which may not always be the case.

5. How is Cluster Approximation used in practical applications related to the Two-Dimensional Ising Model?

Cluster Approximation has been used in various practical applications related to the Two-Dimensional Ising Model, such as studying the behavior of magnetic materials and predicting phase transitions in materials. It is also used in computer simulations to model complex systems and in data analysis to identify patterns and trends in large datasets.

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