README.md

pottsmodel-python

Performing the Wang Landau Algorithm for the Q-State Potts Model on a two-dimensional square lattice.

Theoretical Background

Note

The Wang Landau algorithm ^{1 2} estimates the density of states (DOS) by performing random moves, updating probabilities based on energy changes, and iteratively flattening a histogram. It allows uniform energy space sampling, facilitating accurate thermodynamic property calculations over various temperatures, overcoming limitations of traditional Monte Carlo methods dependent on specific temperatures. The Potts model ³ is a generalization of the Ising model in statistical mechanics. It describes interacting spins on a lattice, where each spin can be in one of [0,Q) states. The model is used to study phase transitions, critical phenomena, and various problems in condensed matter physics and materials science.

How to use

python main.py -g 10 -f example -z 0.8 -m 0.001 -n 100 -q 2

Parameter	Default	Description
-g	10	gridsize
-f	WLA-RUN	directory name
-z	0.8	WLA histogram flatness
-m	0.000001	Final ln(f) value
-n	100	number of bins
-q	2	number of possible q states

Thermodynamic Results

Ising Model (Q=2)

For the $Q=2$ case a second order phase tranisition can be observed. The vertical line indicates the analytical Onsager solution.⁴ The label HI indicated that only the energy interval [-2;0] was sampled and then mirrored. The results match thermodynamic calculations reported in the literature.²

Higher Order (Q=8)

For the $Q=8$ case a first order phase tranisition can be observed. The results match thermodynamic calculations reported in the literature.²

Known bugs and To-Do's

Warning

• Implement proper energy boundaries (upper and lower energy limits for proper sampling). This currently leads to a small inconsistency at $E=-1.0$ for the $Q=8$ case.

• Parallelization

• Include calculations for order parameter depending on the temperature

Footnotes

1. Phys. Rev. Lett. 86, 2050 ↩

2. Am. J. Phys. 72, 1294–1302 (2004) ↩ ↩² ↩³

3. Rev. Mod. Phys. 54, 235 ↩

4. Phys. Rev. 65, 117 ↩