3.3.5 Monte Carlo Method

Central theorem:

If the stationary Markov chain characterized by $\mbox{$\bf p$} \equiv \{ p_{\alpha} \}$ and $\mbox{${\bf P}$} \equiv \{ p_{\alpha \beta}\}$ is reversible, then each state $\mbox{$\bf x$}_{\alpha}$ will be visited, in the course of a sufficiently long chain, with the relative frequency $p_{\alpha}$.

$\Longrightarrow$Here is yet another recipe for generating random numbers with a given probability density $\mbox{$\bf p$}$:





EXERCISE: Let $p(x)=A\, exp[-x^{2}]$ be the desired probability density. Apply the Metropolis' prescription to generate random numbers with this density. Confirm that $\langle x(n)\,x(n+k) \rangle \neq 0$.




Advantage of Metropolis' method: only $p_{\beta}/p_{\alpha}$ is needed, not $p_{\alpha}$.

$\Longrightarrow$ Statistical-mechanical Monte Carlo simulation: only relative thermodynamic probabilities needed!