Monte Carlo Methods in Statistical Mechanics and Quantum Field Theory

Overview

This summer, my work sat at the intersection of statistical mechanics, computational physics, and quantum field theory (QFT). I simulated various forms of random walks and lattice models, including the Ising model and percolation systems.
Please check the final write-up for detailed analysis. The contents of the Jupyter Notebooks are as follows:

Demonstration of Monte Carlo Methods

• Estimation of $\pi$ — Estimation_of_Pi.ipynb
• Test of Python's random number generator — RNG_test.ipynb

Random Walk

Simulated multiple variations of random walks:

• Basic Random Walk — Random_walk.ipynb
• Loops in Random Walk — Loops.ipynb
• Non-reversal Walk — Variations_of_random_walk.ipynb
• Self-avoiding Walk — Variations_of_random_walk.ipynb

Lattice Percolation

Simulated a 2D square lattice — Percolation.ipynb:

• Studied percolation probability and phase transition at the critical occupation probability
• Analyzed distribution of cluster size and number
• Investigated anomalous diffusion

Metropolis Algorithm

Implemented for the Ising model — Ising_model_metropolis.ipynb:

• Implemented and analyzed the Metropolis algorithm
• Computed:
  • Thermalization period
  • Auto-correlation period
  • Energy
  • Magnetization
  • Binder's Cumulant
  • Cluster Count
  • Susceptibility
  • Specific heat
  • 2-point correlation functions
• Performed finite-size scaling to extrapolate results to the infinite-lattice limit

Worm Algorithm

Developed and applied from theoretical principles — Worm_algorithm.ipynb:

• Applied it to the same systems as the Metropolis algorithm
• Verified matching results with the Metropolis approach
• Demonstrated accuracy and reliability through cross-validation

Connection to Lattice Quantum Field Theory

While the Ising model is a classical statistical physics system, the computational techniques and algorithms developed are closely related to those used in lattice QFT.
In lattice QFT:

• Many problems require evaluating infinite-dimensional integrals over all possible field configurations — the path integral
• These integrals are generally impossible to solve exactly, but lattice discretization of space-time makes them numerically tractable

By validating algorithms on simpler, well-understood systems like the Ising model, we are testing computational strategies that can later be applied to simulate the universe at its most fundamental level — including scenarios in quantum gravity.