Report
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Abstract
We perform Monte Carlo simulations on the 2-dimensional Ising model in a zero magnetic field, using the Metropolis and Wolff Cluster algorithms to obtain computational results for thermodynamic quantities and the dynamic critical exponents of the algorithms. The Ising model in 2-dimensions and zero magnetic field undergoes a second order phase transition from a phase with no magnetisation to one with spontaneous magnetisation. This phase transition is modelled by our simulations with the behaviour of magnetisation, energy, magnetic susceptibility and heat capacity at phase transition all being accurately valued. We used finite size scaling of the lattice at critical temperature to derive the value of the dynamic critical exponent for both algorithms of z = 2:11 +- 0:073 for Metropolis and 0:47 +- 0:051 for Wolff Cluster which compares well with accepted quantities of zmetropolis = 2:17 and zcluster = 0:52. Critical exponents of derived observables magnetic susceptibility (gamma) and heat capacity (alpha) were found to be gamma = 1:76 +- 0:023 and alpha = 0:29 +- 1:5. compared well with the accepted value of gamma 7/4. The discrepancy of alpha from the accepted value 0 was discussed.
Source code and other Documentation
Metropolis Algorithm source code: View code files
Cluster Algorithm source code: View code file
I've compiled a sort Readme text document with some helpful information on using the above simulation codes: Readme.pdf
Here are codes written in python to calculate the exact solution for 3x3 and 4x4 lattices. View code files.
Finally, data analysis was done in python, the main scripts used can be found here.
Due to time restraints, no Readme file has been made for these and they are poorly documented. View code files.