The Metropolis Algorithm

The Metropolis algorithm provides an elegant solution to this sampling problems. Rather than trying to calculate absolute probabilities, it works with probability ratios:

Starting from a configuration \(i\):

1. Propose a random move to configuration \(j\).

2. Calculate the energy change \(\Delta E_{ij} = E_j - E_i\).

3. Accept the move with probability:

\[A(i \rightarrow j) = \min(1, \exp(-\Delta E_{ij}/kT))\]

This acceptance criterion ensures:

• Moves that lower energy (\(\Delta E < 0\)) are always accepted.

• Moves that raise energy are accepted with probability \(\exp(-\Delta E/kT)\)

• The partition function \(Z\) cancels out in the ratio