Adiabatic theorem for classical stochastic processes

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 <?CDATA τ> const.× g-α?> τ > const . × g − α with <?CDATA 1x2A7D α x2A7D 2?> 1 ⩽ α ⩽ 2 . We also find that a rigorous treatment gives the scaling <?CDATA τ.× g⁻²|ln g|?> τ > const . × g − 2 | ln ⁡ g | .