However, the selection of states has to generate a final distribution which is the Boltzmann distribution. This is again the same we saw for a random walker, for the discrete case we had always a binomial distribution, whereas for the continuous case we had a normal distribution. The way we sample configurations should result, when equilibrium is established, in the Boltzmann distribution. Else, our algorithm for selecting microstates is wrong.

As stated above, we do in general not know the closed-form expression of the transition rate and we are free to model it as \( W(i\rightarrow j)=T(i\rightarrow j)A(i\rightarrow j) \). Our ratio between probabilities gives us $$ \begin{equation*} \frac{A_{j\rightarrow i}}{A_{i\rightarrow j}}= \frac{w_iT_{i\rightarrow j}}{w_jT_{j\rightarrow i}}. \end{equation*} $$ The simplest form of the Metropolis algorithm (sometimes called for brute force Metropolis) assumes that the transition probability \( T(i\rightarrow j) \) is symmetric, implying that \( T(i\rightarrow j)=T(j\rightarrow i) \).