Effect of sample size when using periodic boundary conditions in 2D Ising model

In summary, the conversation discusses the use of the Monte Carlo Metropolis algorithm to study the 2D Ising model and the effect of lattice size on the sharpness of the phase transition. The speaker is experiencing critical opalescence and is questioning why the size of the unit cell affects their results when using nearest neighbor interactions and periodic boundary conditions. The suggested solution is to calculate the correlation length and compare the temperature dependence for different grid sizes to that of an infinite lattice.
  • #1
DavidwN
1
0
Hi,

I'm currently using the Monte Carlo Metropolis algorithm to investigate the 2D Ising model.

I have an NxN lattice of points with periodic boundary conditions imposed. I was wondering if anyone could explain why the sharpness of the phase transition is affected by the size of N?
I.e. if N is small I get a slow transition and as N is increased, the transition approaches a step function.

I don't understand why this is as I am only considering nearest neighbour interactions and by using periodic boundary conditions surely I am effectively modelling an infinite lattice? So why does the size of the unit cell affect my results?

Thanks!
 
Physics news on Phys.org
  • #2
You are experiencing critical opalescence. As you approach the phase transition, the correlation length increases exponentially. When this domain size reaches the size of your simulation, then the simulation breaks down, i.e. does not describe the physics correctly anymore.

http://en.wikipedia.org/wiki/Ising_critical_exponents

Try and find a definition of the correlation length and calculated that on your grid. Then compare the temperature dependence for different grid sizes to what one would expect for an infinite lattice.
 

1. How does sample size affect the results of the 2D Ising model with periodic boundary conditions?

Sample size is an important factor in the 2D Ising model with periodic boundary conditions as it directly affects the statistical significance of the results. A larger sample size tends to produce more accurate and reliable results compared to a smaller sample size. This is because a larger sample size reduces the impact of random variations and increases the precision of the results.

2. What is the optimal sample size for studying the 2D Ising model with periodic boundary conditions?

The optimal sample size for studying the 2D Ising model with periodic boundary conditions depends on the specific research question and the desired level of accuracy. In general, a larger sample size is recommended to ensure more precise results. However, it is important to balance the cost and time of data collection with the potential benefits of a larger sample size.

3. Does the sample size affect the critical temperature in the 2D Ising model with periodic boundary conditions?

Yes, the sample size can affect the critical temperature in the 2D Ising model with periodic boundary conditions. The critical temperature is a thermodynamic property that is influenced by the size of the system. As the sample size increases, the critical temperature tends to approach the theoretical value for an infinite system.

4. Are there any limitations to using a small sample size in the 2D Ising model with periodic boundary conditions?

Yes, there are limitations to using a small sample size in the 2D Ising model with periodic boundary conditions. A small sample size can lead to less accurate and less reliable results, making it difficult to draw meaningful conclusions. Additionally, a small sample size may not adequately represent the entire system, leading to biased results.

5. How can the effects of sample size be mitigated in the 2D Ising model with periodic boundary conditions?

The effects of sample size can be mitigated in the 2D Ising model with periodic boundary conditions by using statistical techniques such as bootstrapping or Monte Carlo simulations. These methods allow for the generation of multiple samples from a small dataset, providing a more comprehensive understanding of the system. Additionally, conducting sensitivity analyses can help identify the impact of sample size on the results and provide insights on how to improve the study design.

Similar threads

  • Atomic and Condensed Matter
Replies
5
Views
2K
Replies
14
Views
3K
  • Atomic and Condensed Matter
Replies
2
Views
4K
Replies
1
Views
1K
Replies
1
Views
2K
  • Atomic and Condensed Matter
Replies
2
Views
3K
  • Atomic and Condensed Matter
Replies
1
Views
2K
Replies
2
Views
600
  • Programming and Computer Science
Replies
6
Views
4K
Replies
4
Views
7K
Back
Top