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Entrainment in the Master Equation
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