by projecting the master operator to find an effective master operator given in terms of $\lambda_2$ and the maximum and minimum components of $\bra{l_2}$ $${\mathcal{L}}_{\text{eff}}=\frac{-\lambda_{2}}{\Delta{c}_{2}}
\begin{pmatrix}
-{c}_{2}^{\rm max} & -{c}_{2}^{\rm min}\\
{c}_{2}^{\rm max} & {c}_{2}^{\rm min}
\end{pmatrix}.
\label{EffectiveMMDynamics}$$

The meaning of the above is the following. Consider the system starting in an initial state $\ket{P_0}$ (either a specific configuration or a probability over configurations). At some time $t$, the state of the system will read in terms of the spectrum of ${\mathcal L}$, $$\ket{P_0} = \ket{P_\text{ss}} + e^{t \lambda_2} \langle l_2 | P_0 \rangle \ket{r_2} + \sum_{i \geq 3} 
e^{t \lambda_i} \langle l_i | P_0 \rangle \ket{r_i} .
\label{pot}$$ Due to the separation of timescales, for times $t$ such that $ 1/|\text{Re}(\lambda_3)|\ll t \ll 1/|\text{Re}(\lambda_2)|$, all but the first two terms in will be negligible, assuming that the overlap of the initial state with the modes $i\geq 3$ is small so that these terms are suppressed by the decaying modes for $i\geq 3$. Within these timescales, the initial state $\ket{P_0}$ evolves to a state in the one-dimensional MM, subsequently evolving within the MM and eventually reaching the unique stationary state for $t \gg 1/|\text{Re}(\lambda_2)|$, i.e., schematically $$\ket{P_0} \to p_1(t) \ket{\tilde{P}_1} 