Kinetic Path Summation, Multi--Sheeted Extension of Master Equation, and Evaluation of Ergodicity Coefficient

TL;DR

This paper develops a path summation formula for solving time-dependent master equations in Markov and chemical kinetics, introduces multi-sheeted extensions to estimate relaxation times and ergodicity, and explores the internal structure of kinetic systems.

Contribution

It introduces a novel multi-sheeted extension approach to analyze ergodicity and relaxation in kinetic equations, providing explicit solutions and estimates.

Findings

Path summation formula for kinetic equations

Explicit solutions for simple kinetic schemes

Estimates for relaxation time and ergodicity coefficient

Abstract

We study the Master equation with time--dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a path summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi--sheeted extensions of the initial kinetics. This approach allows us to exploit the internal ("micro")structure of the extended kinetics without perturbation of the base kinetics.