For the Boltzmann distribution, detailed balance results in $$ \begin{equation*} \frac{w_i}{w_j}= \exp{(-\beta(E_i-E_j))}. \end{equation*} $$

Let us now specialize to a system whose energy is defined by the orientation of single spins. Consider the state \( i \), with given energy \( E_i \) represented by the following \( N \) spins $$ \begin{equation*} \begin{array}{cccccccccc} \uparrow&\uparrow&\uparrow&\dots&\uparrow&\downarrow&\uparrow&\dots&\uparrow&\downarrow\\ 1&2&3&\dots& k-1&k&k+1&\dots&N-1&N\end{array} \end{equation*} $$