Let's imagine that we have a Lindblad equation. Using that we can derive a Pauli master equation (rate equation) in form like this one (for the stationary case) \[\sum_\limits{k} (w_{nk}P_k - w_{kn}P_n)=0, \\\]

here \(w_{nk} \)is a probability of transition from state n to state k. Detailed balance condition looks like \[w_{nk}P_k = w_{kn}P_n, \,\, \forall n, k.\]Unfortunately, I am not good with this topic. So, I'd like to know if there are any typical ways how to proof that detailed balance condition is satisfied in the system according to the Lindblad equation.

P.S. In addition we know, that the density matrix is non-diagonal and our distribution is nonequilibrium.