Monte Carlo
in statistical physics

• Simple Monte Carlo integration
• Importance sampling
• Metropolis–Hastings algorithm

Monte Carlo integration

Importance sampling

Sharply peaked distributions

Metropolis

When direct sampling is impossible/non-practical ...

eg. ρ~exp[-βH]

1. Attempt a random
   trial displacement.
2. Accept or reject
   according to ρnew/ρold.
3. Accumulate local observables.

Variational Monte Carlo

• H=∑(ℏ²∇²/2m) + ∑∑vij
• Parametrize Ψ({r}) → Ψ({r};α)
• Use |Ψ({r};α)|² as probability density
• Optimize according to lowest E
  or lowest σE²

Diffusion Monte Carlo

• Bullet One
• Bullet Two
• Bullet Three

More advanced qchem methods

• Bullet One
• Bullet Two
• Bullet Three

Inadequacy of MO/VB

• Bullet One
• Bullet Two
• Bullet Three

Quantum entanglement

• Bullet One
• Bullet Two
• Bullet Three

Partial traces and Rényi entropies

• Bullet One
• Bullet Two
• Bullet Three

Quantum Monte Carlo

• Bullet One
• Bullet Two
• Bullet Three

Variational Monte Carlo

• Bullet One
• Bullet Two
• Bullet Three

Some triumphs of QMC

• Bullet One
• Bullet Two
• Bullet Three