Computational Physics

Model and simulate physical systems using numerical methods, time integration, stability analysis, Monte Carlo and validation.

General idea of the topic

Computational physics bridges analytical theory and experimental science: it enables solving non-linear equations, many-body systems, and complex dynamics that lack closed-form analytical solutions on paper. Its core methodology relies on rigorous algorithmic modeling, numerical stability analysis, discretization error estimation, and deterministic and stochastic simulations. The central challenge is translating continuous physical laws (differential equations, field distributions, probability densities) into efficient, physically faithful discrete algorithms.

This module covers numerical fundamentals, field discretization methods, statistical simulations (Monte Carlo), molecular dynamics, chaotic dynamical systems, and reproducible scientific visualization. Each topic introduces key concepts: algorithmic convergence, preservation of physical invariants (energy, momentum), statistical fluctuations, and phase-space attractors.

What you will truly understand

By completing this module, you will design physical simulations from scratch, select the optimal numerical integrator for any dynamical regime, and distinguish genuine physical phenomena from numerical artifacts or algorithmic instability.

You will realize that computational physics is not merely writing code, but ensuring that computational algorithms strictly respect conservation laws and thermodynamic bounds.

Recommended learning route

1. Computational fundamentals: floating-point representation, round-off errors, numerical differentiation/integration, and root finding.
2. Field methods: spatial and temporal discretization, finite differences, relaxation methods for electrostatic potentials, and wave propagation.
3. Statistical simulation: probabilistic sampling, Markov chains, Metropolis algorithm, Ising model, and computational thermodynamics.
4. Applied physical simulation: symplectic integrators (Verlet, Leapfrog), molecular dynamics of gases and liquids, and particle fluid dynamics.
5. Dynamical systems and chaos: bifurcation diagrams, non-linear maps (logistic map), Lyapunov exponents, and Lorenz strange attractors.
6. Visualization and reproducibility: contour plots, phase portraits, vector field animations, and scientific validation workflows.

Physical criterion you must acquire

The primary physical criterion is invariant preservation: forward Euler methods artificially inject energy into conservative oscillators, whereas symplectic integrators conserve phase-space volume and Hamiltonian energy bounds indefinitely.

Connection with real problems

• Numerical astrophysics: N-body simulations of galactic evolution and chaotic orbital resonances.
• Materials science and biophysics: protein folding, cell membrane transport, and nanomaterial modeling via molecular dynamics.
• Meteorology and climate science: Navier-Stokes solutions and chaotic atmospheric modeling.
• Quantum and nuclear physics: path integral methods, lattice QCD, and magnetic confinement fusion design.

Final goal of the topic

The goal is to master the full lifecycle of scientific computation: formulating, implementing, benchmarking, and interpreting computational models in physics with methodological rigor.