Adiabatic theorem for classical stochastic processes

TL;DR

This paper extends adiabatic theorems from quantum mechanics to classical stochastic processes, providing conditions on annealing time for the system to stay close to its instantaneous stationary state.

Contribution

It formulates a rigorous asymptotic expansion for classical stochastic processes and derives explicit conditions on annealing time based on the generator's properties.

Findings

Relaxation to the instantaneous stationary state under certain conditions

Derived bounds on annealing time T in terms of decay rate g

Established scaling T>const|ln g|/g^2 for process convergence

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 T that the state is close to the instantaneous stationary state. Depending on the time dependence of the generator, typical conditions are written as T> const/g^a with 1<a<2. We also find that a rigorous treatment gives the scaling T>const|ln g|/g^2.