Monte Carlo method in statistical physics

Remarks

For an irreducible chain on a finite set with stationary distribution $\pi$, the ergodic theorem for Markov chains gives $\bar A_n \to \langle A \rangle$ almost surely as $n \to \infty$, for every starting state. The Markov chain avoids computing $Z$, since only ratios $\pi(\tau)/\pi(\sigma) = e^{-\beta (H(\tau) - H(\sigma))}$ appear. Example: the Metropolis algorithm for the Ising model picks a uniform random site, proposes to flip its spin, and accepts with probability $\min(1, e^{-\beta \Delta H})$, where $\Delta H$ is the energy change. The proposal is symmetric, so $\pi(\sigma) P(\sigma, \tau) = \frac{1}{\abs{V}} \min(\pi(\sigma), \pi(\tau))$ for a single flip, where $V$ is the set of sites. This is symmetric in $\sigma, \tau$, which checks detailed balance. Non-example: averaging $A$ over uniform samples from $S$ estimates the average at infinite temperature, not $\langle A \rangle$. Its chain has the uniform stationary distribution, so clause 1 fails for $\beta > 0$ unless $H$ is constant. The error of $\bar A_n$ is proportional to $\sqrt{\tau / n}$, where $\tau$ is the integrated autocorrelation time of $A$; near a critical point $\tau$ becomes large, which motivates cluster algorithms.