We obtain then (using the Boltzmann distribution) $$ \begin{equation*} \frac{A(j\rightarrow i)}{A(i\rightarrow j)}= \exp{(-\beta(E_i-E_j))} . \end{equation*} $$ We are in this case interested in a new state \( E_j \) whose energy is lower than \( E_i \), viz., \( \Delta E = E_j-E_i \le 0 \). A simple test would then be to accept only those microstates which lower the energy. Suppose we have ten microstates with energy \( E_0 \le E_1 \le E_2 \le E_3 \le \dots \le E_9 \). Our desired energy is \( E_0 \).