Adiabatic theorem for classical stochastic processes

Abstract

We apply adiabatic theorems developed for quantum mechanics to stochastic annealing processes described by the classical master equation with a time-dependent generator. When the instantaneous stationary state is unique and the minimum decay rate g is nonzero, the time-evolved state is basically relaxed to the instantaneous stationary state. By formulating an asymptotic expansion rigorously, we derive conditions for the annealing time τ that the state is close to the instantaneous stationary state. Depending on the time dependence of the generator, typical conditions are written as τ>const.×g−α with 1⩽α⩽2 . We also find that a rigorous treatment gives the scaling τ>const.×g−2|ln⁡g| .