Quasi-stationary time asymmetry on topologically irreversible Markov chains

In this work, we investigate the property of time asymmetry in processes that are intrinsically time-irreversible. These are systems in which structural constraints prohibit certain time-reversed trajectories, rendering the entropy production rate undefined and, thus, unsuitable as a measure of irreversibility. To address this limitation, we focus on a specific class of systems, referred to as topologically irreversible Markov chains, in which irreversibility arises from topological constraints embedded in the transition structure. We introduce a method to assess time asymmetry in such systems by constructing a conditioned Markov chain, defined on the subset of trajectories whose time reversals occur with strictly positive probability. This subset, which we term the reversible interior, supports a restricted dynamics that enables a well-defined measure of irreversibility. We propose the quasi-stationary irreversibility index, defined as the Kullback-Leibler divergence between the forward and time-reversed path distributions of the conditioned process. This index serves as an analog to the entropy production rate for systems where the latter is undefined. We illustrate our approach with a simple Markov chain model for which all quantities can be computed analytically and then apply the method to biological data, specifically, mitochondrial DNA sequences from selected species of the Hominidae family, demonstrating its applicability to real-world intrinsically irreversible systems.