Theoretical Principles & Computational Physics Guide
Monte Carlo Method: Statistical Physics & Stochastic Modeling
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Stochastic Processes: Simulates random physical phenomena including Brownian diffusion, radioactive decay, and thermal fluctuations.
Ensemble Averages: Computes macroscopic thermodynamic observables as statistical ensemble averages over microscopic configurations.
Monte Carlo Convergence: Statistical error scales inversely with the square root of trial count: $\sigma \propto 1/\sqrt{N}$.
2. Mathematical Formulation & Algorithmic Pipeline
Random walk diffusive spreading and Monte Carlo error scaling:
$$ \langle x(t) \rangle = 0, \quad \langle x^2(t) \rangle = 2 D t, \quad \text{Statistical Uncertainty: } \sigma_N = \frac{\sigma}{\sqrt{N}} $$
3. Numerical Mechanics & Implementation Logic
- Random Number Seeding: Initialize pseudo-random generators with fixed seeds to ensure reproducible simulation results.
- Ensemble Sampling: Execute $M$ independent trials of $N$ steps each to build accurate probability distributions.
- Histogramming: Bin particle final positions to verify convergence toward Gaussian/Poisson distributions.
4. Relevant Computational Practice Exercises
Exercise 1
Simulate $M = 1000$ independent random walk trajectories and confirm that root-mean-square displacement scales as $\sqrt{t}$.
Exercise 2
Apply Monte Carlo integration to estimate $\pi$ using hit-or-miss circle sampling and plot error vs trial count $N$.
Exercise 3
Model radioactive nuclear decay for $N_0 = 10^4$ nuclei and compare against the continuous decay law $N(t) = N_0 e^{-\lambda t}$.
- F. Reif, "Fundamentals of Statistical and Thermal Physics", Waveland Press (2009).
- Mark Newman, "Computational Physics", University of Michigan (2013).