Entrainment in the Master Equation

TL;DR

This paper proves that the master equation with periodic transition rates naturally entrains to the same period, demonstrating convergence to periodic solutions in systems like biology and traffic models.

Contribution

It introduces a control-theoretic approach to show convergence of solutions to periodic states in master equations with periodic rates.

Findings

Solutions converge to periodic solutions matching the rate period

Theoretical proof of entrainment under mild conditions

Applications demonstrated in physics and epidemiology models

Abstract

The master equation plays an important role in many scientific fields including physics, chemistry, systems biology, physical finance, and sociodynamics. We consider the master equation with periodic transition rates. This may represent an external periodic excitation like the 24h solar day in biological systems or periodic traffic lights in a model of vehicular traffic. Using tools from systems and control theory, we prove that under mild technical conditions every solution of the master equation converges to a periodic solution with the same period as the rates. In other words, the master equation entrains (or phase locks) to periodic excitations. We describe two applications of our theoretical results to important models from statistical mechanics and epidemiology.