3D ising model simulation
- fitting data with 0th, 1st approximation

3D ising model is simulated with c++ code. As we move the temperature, we try Monte Carlo simulation and get the magnetization for each temperature.

Assumptions

• 3D lattice structure with size 1000 x 1000 x 1000.
• 1000 iterations are enough to get to the MDP equilibrium.
• Checker board scanning is applied.
• Periodic boundary condition is applied.

Generate data

Generate data with gen_magnetization.cpp
Monte carlo method(Metropolis) is used for getting to the equilibrium.
This file assumes that Curie temperature is at around 50 ~ 55K. 50 magnetization data are obtained for each temperature.

• If you want to see roughly where Curie temperature shows up (as above),
  

  run gen_rough.cpp
  

  It will produce 100 points which corresponds to magnetization at 0 ~ 100K.

• If you want to get the data to try fitting functions,
  

  run gen_magnetization.cpp
  

  It will produces 50 points for each temperature at 51K to 52.4K. Save the printed data to use for fitting.

Fitting data

After you load the data from the previous step, you can try 3 kinds of approximations, 0th, 1st(q move), 1st(q fixed). Each approximation is implemented in fit_data folder. Fitting is implemented with chi_square minimization & Jackknife resampling methods.
As for minimization, gradient descent method is implemented from scratch to find the optimum.

• 0th
  

• 1st(q move)
  

• 1st(q=6)
  

For analysis/discussion, please refer to the thesis in this repo.

References

[1] James P. Sethna, “Entropy, Order Parammeters, and complexity”, Oxford University Press.
[2] Pathria, R.K., “Statistical Mechanics”, Academic Press, 2011.
[3] Boram Yun, “Taste Non-Goldstone Pion Decay Constants and Beyond the Standard Model B-parameters in Lattice QCD with Staggered Fermions”, Seoul National University, 2013.
[4] E.Z. Meilikhov, “Curie Temperature for Small World Ising Systems of Different Dimensions”, Jornal of Magnetism and Magnetic Materials, 2006.