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 $\mathrm{\Delta }{E}_{ij}=E_j-E_i$.

3. Accept the move with probability:

This acceptance criterion ensures:

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

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

• The partition function $Z$ cancels out in the ratio

Accepting Moves with Probability p#

The Metropolis criterion tells us to accept or reject each proposed move according to the following rules:

• If $\mathrm{\Delta }E\le 0$, we always accept the move.

• If $\mathrm{\Delta }E>0$, accept the move with a probability $p={\mathrm{e}}^{-\mathrm{\Delta }E/kT}$.

Accepting Moves with Probability p#

What exactly does it mean to accept a move “with probability $p$”? A probability is a number between 0 and 1 that represents the chance of an event occurring. A probability of 0 means the event cannot occur, while 1 means it must occur. Take flipping a fair coin — if we have no information about how the coin is flipped, we would reasonably assign each outcome a probability 0.5: $p(\text{heads})=p(\text{tails})=0.5$.

In Metropolis Monte Carlo, the probability comes from the Boltzmann factor $p=\exp (-\mathrm{\Delta }E/kT)$. For moves that increase the system’s energy ($\mathrm{\Delta }E>0$), the Boltzmann factor is a number between 0 and 1. For moves that decrease energy ($\mathrm{\Delta }E<0$), the expression would exceed 1, but we have already decided to accept these moves automatically.

To turn these probabilities into decisions, we use a random number generator. The strategy is straightforward: generate a random number $r$ between 0 and 1, then accept the move if $r\le p$. This ensures our decisions match the assigned probabilities.

Take our coin flip example where $p(\text{heads})=0.5$. We can code this decision as:

is_heads = np.random.rand() <= 0.5

The Metropolis algorithm uses the same logic. For a move that increases energy by $\mathrm{\Delta }E$:

1. Calculate $p=\exp (-\mathrm{\Delta }E/kT)$

2. Generate a random number $r$ between 0 and 1

3. Accept the move if $r\le p$

Here’s what this looks like for a move where $\mathrm{\Delta }E=2kT$:

p = np.exp(-delta_E/kT)  # ≈ 0.135
accept = np.random.rand() <= p

In this case, there is about a 13.5% chance of accepting the move. Larger energy increases mean lower acceptance probabilities — the move becomes less likely but remains possible. This mechanism lets our system occasionally climb uphill in energy while generally favoring lower energy states.

This simple accept/reject rule, guided by physically motivated probabilities, lets us efficiently explore possible states while maintaining the correct Boltzmann distribution.