Statistical Simulation

Monte Carlo, Markov chains, spin models, sampling, and many-particle systems.

General idea of the topic

Computational statistical physics connects microscopic particle states with macroscopic thermodynamics through probabilistic sampling. When a system spans $10^{23}$ microscopic configurations, exact analytical summation is impossible; Monte Carlo methods overcome this hurdle by sampling statistically dominant microstates weighted by the Boltzmann distribution.

This node covers pseudorandom number generators, multidimensional Monte Carlo integration, importance sampling via Metropolis-Hastings, and phase transitions in the Ising model.

What you will truly understand

You will grasp computational thermal equilibrium: how a Markov chain satisfying detailed balance asymptotically converges to the canonical ensemble regardless of the initial configuration.

You will compute internal energy, mean magnetization, magnetic susceptibility, and heat capacity $C_v$ directly from microscopic fluctuation theorems.

Recommended learning route

1. Monte Carlo integration: hit-or-miss method, uniform sampling, and variance reduction in high-dimensional phase space.
2. Markov chains and Metropolis: detailed balance, acceptance probability $P = \min(1, e^{-\Delta E / k_B T})$, and autocorrelation times.
3. 2D Ising model: ferromagnetic spin lattice, spontaneous magnetization, and symmetry breaking.
4. Critical phenomena and scaling: locating Curie critical temperature $T_c$, thermodynamic fluctuations, and finite-size scaling.

Physical criterion you must acquire

Properly discarding the thermalization transient before recording equilibrium data, and computing statistical error bars accounting for auto-correlated time series.

Connection with real problems

• Condensed matter physics: ferromagnetism, spin glasses, and disorder in magnetic alloys.
• Computational chemistry: polymer conformation sampling and molecular adsorption equilibria.
• Radiation transport and reactor design: stochastic neutron transport and dosimetry modeling.

Final goal of the topic

Implement complete Monte Carlo Metropolis simulations to compute thermodynamic observables and characterize critical phase transitions.