A Three-State Markov Process as a Pedagogical Example of the Continuous-Time Approximation

We present a pedagogical analysis of a fully connected three-state Markov process, focusing on the validity and limitations of the continuous-time approximation to discrete-time dynamics. While the correspondence between a discrete-time transition matrix and a continuous-time generator is often introduced formally, the conditions under which the approximation is accurate are rarely examined in detail. Using an exactly solvable three-state model, we derive both the discrete-time and continuous-time evolution operators and compare their transient behavior. We show explicitly that the continuous-time approximation is controlled by the relation between the time step and the characteristic relaxation time scales. The three-state model is the minimal example that can violate detailed balance, and thus provides a simple setting in which discrete and continuous dynamics may differ even though they share the same stationary distribution. The results provide a useful worked example for advanced undergraduate and graduate courses on stochastic processes and statistical physics.