We consider continuous-time Markov chains (a.k.a. Markov jump processes) with pairwise rates of the form
L_{ij}=k_{ij}exp(-U_{ij}/ε). Such Markov chains arise in chemical physics applications such as modeling energy landscapes of Lennar-Jones clusters or biomolecules, or time-irreversible dynamics of molecular motors. As ε tends to zero, the spectral decomposition of the generator matrix can tell us a lot about the dynamics of the system on each timescale: eigenvectors approximate the indicator functions of quasi-invariant subsets of states while the corresponding eigenvalues give sharp estimates for the exit rates from these quasi-invariant states.

- An algorithm (Cameron, 2014) for computing asymptotic spectrum for time-reversible Markov chains. Application to LJ38.
- A different algorithm for computing asymptotic spectrum for Markov chains (Gan and Cameron, 2017). Time-reversibility is no longer required. Application to LJ75. Key word: optimal W-graphs.
- The algorithm (Gan and Cameron, 2017) has a nice probabilistic interpretation. It can be used not only for identifying quasi-invariant subsets of states and expected exit times from them but also for finding Freidlin's cycles and expected rotation times in them. Application to a time-irreversible network modeling a molecular motor. Key word: T-graph.Software is available upon request via email.