Derivation of a nonequilibrium Markov six-state model in the presence of a time-reversal antisymmetric variable

A Markov model that does not satisfy the detailed balance condition is often used for modeling a system continuously driven out of equilibrium by an external agent. I demonstrate how such a model is obtained as an early-time approximation after coarse-graining a Markov model with the detailed balance, by studying a simple six-state model, the current result extends a previous study by considering a more general case that includes a time-reversal antisymmetric variable as well as symmetric one. I explicitly show that the nonequilibrium steady state in the six-state model is actually a quasi-stationary state in the refined model.