Monte Carlo Basics

Introduction

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to obtain numerical results. They are particularly useful when dealing with problems that are difficult to solve analytically.

Key Concepts

Random Number Generation

Pseudorandom number generators
Quality of random numbers
Seeds and reproducibility

Basic Monte Carlo Integration

Estimating integrals using random sampling
Error analysis and convergence
Importance of sample size

Applications in Physics

Statistical mechanics
Particle physics simulations
Quantum Monte Carlo methods

Simple Example

Here’s a basic example of Monte Carlo integration to estimate π:

import random

def estimate_pi(n_samples):
    inside_circle = 0
    for _ in range(n_samples):
        x = random.uniform(-1, 1)
        y = random.uniform(-1, 1)
        if x*x + y*y <= 1:
            inside_circle += 1
    return 4 * inside_circle / n_samples

# Estimate π with 1,000,000 samples
pi_estimate = estimate_pi(1000000)
print(f"Estimated π: {pi_estimate}")

Siddharth Sule