Optimization of Markov process violates detailed balance condition

TL;DR

This paper applies the brachistochrone method to optimize Markov process dynamics, revealing solutions that violate detailed balance and enhance convergence speed, with implications for counterdiabatic driving.

Contribution

It introduces a brachistochrone approach to optimize Markovian dynamics, explicitly solving for systems where detailed balance is violated to achieve faster convergence.

Findings

Optimal transition-rate matrices violate detailed balance.

Explicit solutions for three-state systems demonstrate improved convergence.

Counterdiabatic terms can be incorporated into Markov dynamics.

Abstract

We consider the optimization of Markovian dynamics to pursue the fastest convergence to the stationary state. The brachistochrone method is applied to the continuous-time master equation for finite-size systems. The principle of least action leads to a brachistochrone equation for the transition-rate matrix. Three-state systems are explicitly analyzed, and we find that the solution violates the detailed balance condition. The properties of the solution are studied in detail to observe the optimality of the solution. We also discuss the counterdiabatic driving for the Markovian dynamics. The transition-rate matrix is then divided into two parts, and the state is given by an eigenstate of the first part. The second part violates the detailed balance condition and plays the role of a counterdiabatic term.