Correlation functions, mean first passage times, and the Kemeny constant

Area of Science:

• Physical Chemistry
• Computational Chemistry
• Statistical Mechanics

Background:

• Markov processes are fundamental for modeling kinetic networks.
• Key quantities include mean first passage times (MFPTs), correlation functions, and the Kemeny constant.
• Existing frameworks lack explicit connections between these important metrics.

Purpose of the Study:

• To unify diverse results for kinetic network models within a single framework.
• To establish clear relationships between MFPTs, correlation functions, and the Kemeny constant.
• To introduce novel methods for network analysis and dimensionality reduction.

Main Methods:

• Collating and presenting existing results on kinetic networks.
• Developing theoretical insights into the Kemeny constant.
• Proposing and analyzing a protocol for dimensionality reduction of kinetic networks.
• Investigating a modified protocol that preserves the Kemeny constant.

Main Results:

• A simple physical interpretation of the Kemeny constant is provided.
• A method to infer equilibrium distributions and rate matrices from MFPTs is established.
• A dimensionality reduction protocol for kinetic networks is presented, proven to align with Hummer and Szabo's work, and shown to yield a variational principle for the Kemeny constant.
• A Kemeny constant-preserving modification of the reduction protocol is introduced.

Conclusions:

• The unified framework enhances understanding of kinetic network properties.
• The developed methods offer practical tools for analyzing complex systems.
• Applications include advancing algorithms in molecular simulations like milestoning and path sampling.