eigenmodes defined by these gaps, with the late time relaxation given by a projection of the master operator onto the MM.

To illustrate this phenomenon in the simplest way possible, let us consider a Markov process with a unique stationary state and a large gap between the second and third eigenvalues, i.e., $|\text{Re}(\lambda_2)|\ll|\text{Re}(\lambda_3)|$. The MM of long-lived states corresponds in this case to a one-dimensional manifold of linear combinations of the stationary state and $\ket{r_2}$, with the coefficients of $\ket{r_2}$ bounded by the maximum and minimum values of $\bra{{l_2}}$ on the space of probability distributions, i.e., the maximum and minimum components of this vector in the configuration basis $c_2^{\text{max}}$ and $c_2^{\text{min}}$. These two values define the so-called *extreme metastable states* (eMSs) on the boundary of the manifold: $$\begin{aligned}
\ket{\tilde{P}_1}&=\ket{P_\text{ss}}+c_2^{\text{max}}\ket{r_2}\nonumber\\
\ket{\tilde{P}_2}&=\ket{P_\text{ss}}+c_2^{\text{min}}\ket{r_2},
\label{ems}\end{aligned}$$ in terms of which we can write the stationary state as $$\begin{aligned}
\ket{P_\text{ss}}&=\frac{1}{\Delta c_2}\left(-{c}_{2}^{\rm min}\ket{\tilde{P}_1}+{c}_{2}^{\rm max}\ket{\tilde{P}_2}\right)\label{SteadyStateMMExpansion}\\
    &=p_1^\text{ss}\ket{\tilde{P}_1}+p_2^\text{ss}\ket{\tilde{P}_2},\nonumber\end{aligned}$$ where $\Delta c_2={c}_{2}^{\rm max}-{c}_{2}^{\rm min}$. Note that $c_2^\text{min}\leq0$ as $\bra{l_2}$ is orthogonal to the stationary state which has purely positive components, so that the coefficients $p_1^\text{ss}$ and $p_2^\text{ss}$ in this expansion can be viewed as the probability weight of the two eMSs. Finally, we can construct an effective evolution on this subspace 