Monte Carlo
This course connects physical intuition, derivations, model boundaries and verifiable practice. Lessons state assumptions, work through examples and explain exercises; illustrations distinguish analytic models from simulation data.
1. Prerequisites and study method
Bring algebra, calculus and elementary probability. Quantum courses use complex numbers and linear algebra; sampling courses use partition functions. Read the overall roadmap, then follow the sequence below. For every formula, identify its variables, units, assumptions and failure conditions.
Recommended preparation: Molecular dynamics
2. Numbered lessons
- Probability, integration, and random estimators
- Canonical weights and thermodynamic estimators
- Importance sampling and reweighting
- Markov chains and Metropolis–Hastings balance
- Ergodicity, barriers, and convergence
- Ising updates and correlated error bars
- Particle, volume, and insertion moves
- Cluster updates, replica exchange, and Hamiltonian proposals
- Free energy, umbrella sampling, and rare events
- A reproducible Monte Carlo study
3. Core and advanced routes
Core route: complete the derivations and exercises in the first six lessons, then read the intuition and limitations of the remaining lessons. Advanced route: complete the full sequence, rederive the central equations independently, and connect them to numerical convergence or sampling design in the software tutorials. A shorter route does not waive the assumptions of advanced methods.
4. Completion task
Choose one target observable and separate its physical model, numerical approximation, statistical estimator and validation evidence. Supply one hand calculation, one convergence plan and one condition under which the interpretation fails. A proposed calculation must not be presented as an executed result.