We consider continuous-time Markov chains (a.k.a. Markov jump processes) with pairwise rates of the form Lij=kijexp(-Uij/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 illustration: asymptotic dymanics of LJ75.
Largest quasi-invariant sets in LJ75 are represented by circles with radii proportional to the number of states in them. the arcs represent the maximum likelihood transitions from them. For each quasi-invariant set: the main state is the bold number (the first number), the number of states is the second number, and the size of the corresponding Freidlin's cycle is the third number. The numbers next to the edges are the exponential factors Uij in the rate formulas. Input data for the LJ75 network: courtesy of Prof. D. Wales. |
My projects
The asymptotic spectral decomposition can be computed by greedy/dynamical programming algorithms:- 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.