However, the condition that the rates should equal each other is in general not sufficient to guarantee that we, after many simulations, generate the correct distribution. We may risk to end up with so-called cyclic solutions. To avoid this we therefore introduce an additional condition, namely that of detailed balance $$ \begin{equation*} W(j\rightarrow i)w_j= W(i\rightarrow j)w_i. \end{equation*} $$ These equations were derived by Lars Onsager when studying irreversible processes. At equilibrium detailed balance gives thus $$ \begin{equation*} \frac{W(j\rightarrow i)}{W(i\rightarrow j)}=\frac{w_i}{w_j}. \end{equation*} $$ Rewriting the last equation in terms of our transition probabilities \( T \) and acceptance probobalities \( A \) we obtain $$ \begin{equation*} w_j(t)T_{j\rightarrow i}A_{j\rightarrow i}= w_i(t)T_{i\rightarrow j}A_{i\rightarrow j}. \end{equation*} $$