Stochastic Polymer Modelling with Monte Carlo

Growth Sampling vs Metropolis Monte Carlo for Self-Avoiding Polymers

This repository presents a qualitative and statistically grounded comparison of two Monte Carlo approaches for generating three-dimensional self-avoiding polymer chains on a cubic lattice.

• Growth Sampling (GS)
  Polymer observables are estimated from independently generated chains:

  $$ \langle A \rangle \approx \frac{1}{M} \sum_{k=1}^{M} A!\left(\mathbf{R}^{(k)}\right) $$

• Metropolis Monte Carlo (TM)
  Polymer configurations are sampled via a Markov chain with equilibrium weighting:

$$ P_{\mathrm{acc}} = \min\left(1, \frac{\pi(X'),q(X \mid X')}{\pi(X),q(X' \mid X)}\right) $$

$$ \pi(X) \propto e^{-\beta E(X)} $$

where,

$\pi(X)$ is the equilibrium probability of configuration $X$,
$E(X)$ is the corresponding energy, and $\beta = 1/(k_B T)$.

The project is designed as a learning-focused, method-comparison study, with emphasis on understanding how different stochastic sampling strategies influence measured polymer statistics such as radius of gyration ($R_g$), end-to-end distance ($ee$), and scaling behaviour (eq shown below).

$$ \langle R_g \rangle \sim N^{\nu} $$

Project Objectives

The primary goals of this project are:

• To implement and compare two distinct Monte Carlo sampling strategies for self-avoiding walks.
• To evaluate how algorithmic choices affect equilibrium polymer statistics.
• To distinguish between visual realism and statistical correctness in Monte Carlo simulations.
• To develop a reproducible analysis workflow using replicate runs, paired comparisons, bootstrap confidence intervals, and scaling fits.

Methods Overview

Growth Sampling (GS)

Growth Sampling constructs polymer chains step-by-step from one end, selecting random local moves while enforcing self-avoidance. Each chain is generated independently from scratch.

This approach:

• Produces visually diverse and intuitive polymer conformations.
• Is computationally simple and efficient.
• Does not explicitly enforce equilibrium weighting over complete configurations.

Metropolis Monte Carlo (TM)

Metropolis Monte Carlo samples complete polymer configurations using a Markov chain approach. Trial moves are proposed and accepted or rejected according to self-avoidance constraints, and equilibrium statistics are obtained after burn-in and thinning.

This approach:

• Produces correlated but reproducible configurations.
• Enables systematic sampling of an equilibrium ensemble.
• Requires careful handling of burn-in, thinning, and replicate analysis.

Repository Structure

spmmc
├── Docs
│   ├── results_and_discussions.md
│   └── conclusion.md
|
├── Images
│   └── ExamplePolymerModel.png
│
├── MC
│   ├── GS_MC.md
│   ├── GS_monte_carlo.py
│   ├── TM_MC.md
│   └── TM_monte_carlo.py
│
├── Notebook
│   └── Statistical_comparison.ipynb
│
└── README.md

Analysis Workflow

All quantitative comparisons are performed in
Notebook/Statistical_comparison.ipynb.

The analysis includes:

• Ensemble averages of radius of gyration ($R_g$) and end-to-end distance ($R_{ee}$)
• Replicate-based comparisons across multiple random seeds
• Paired method differences per chain length
• Bootstrap confidence intervals for mean differences
• Log–log scaling fits of $\langle R_g \rangle \sim N^{\nu}$
• Histogram and kernel density analysis of fitted scaling exponents

Only results directly supported by these analyses are discussed.

Key Findings (Summary)

• Both GS and TM generate valid self-avoiding polymer configurations.
• Both methods show smooth, monotonic growth of $R_g$ and $R_{ee}$ with chain length.
• GS systematically produces smaller polymer size measures than TM at all $N$.
• The discrepancy between GS and TM increases with chain length.
• Bootstrap confidence intervals confirm that the differences are statistically decisive.
• Scaling fits show that GS yields consistently smaller values of the exponent $\nu$, indicating a bias toward more compact conformations.
• TM produces more stable and reproducible equilibrium statistics across replicates.

A detailed interpretation is provided in Docs/results_and_discussions.md, with a concise final assessment in Docs/conclusion.md.

Intended Use

This project is intended for:

• Learning and teaching Monte Carlo sampling concepts
• Understanding methodological bias in stochastic simulations
• Demonstrating statistical comparison techniques in computational physics or chemistry
• Building intuition about polymer conformational ensembles

It is not intended as a high-performance or production-grade polymer simulation package.

Reproducibility

• All simulations use explicit random seeds for reproducibility.
• Replicate analyses are performed to assess variability and robustness.
• Results can be regenerated by running the analysis notebook with the provided code.

License / Usage

This project is provided for educational and exploratory purposes ONLY.
Feel free to read, modify, and reuse the code with appropriate attribution.

Acknowledgement

This repository represents a structured exploration of Monte Carlo sampling strategies and their consequences, with emphasis on careful statistical reasoning rather than visual intuition alone.