Kullback-Leibler Potential for Non-Ergodic Replication Dynamics:An Information-Theoretic Second Law

This study aims to quantify and visualize the degradation of fidelity (information degradation) that inevitably accompanies the replication of information within the framework of information thermodynamics and to propose an information-theoretic formulation of the second law based on this phenomenon. While previous research in information thermodynamics has focused on the thermodynamic costs associated with information "erasure'' or "measurement'' through concepts such as Landauer's principle and mutual information, little systematic discussion has addressed the inherently irreversible nature of "replication'' itself and the accompanying degradation of information structure. In this study, we construct a mathematical model of information replication using a discrete Markov model and Gaussian convolution, and quantify changes in information at each replication step: Shannon entropy, cross-entropy, and the Kullback--Leibler divergence (KLD). The monotonic decrease of KLD exhibits a Lyapunov-like property, which can be interpreted as a potential analogous to the free energy in the process by which a nonequilibrium system converges to a particular steady state. Furthermore, we extend this framework to the potential applicability to biological information processes such as DNA replication, showing that the free energy required for degradation and repair can be expressed in terms of KLD. This contributes to building a unified information-thermodynamic framework for operations such as replication, transmission, and repair of information.