It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time. | It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time. |