Shannon entropy and quantitative time irreversibility for different and even contradictory aspects of complex systems

The Shannon entropy and quantitative time irreversibility (qTIR) are statistical quantifiers that are widely used for characterizing complex processes. However, the differences and associations between them have not been subjected to detailed investigation. In this Letter, we report a comparative analysis of the Shannon entropy and qTIR using model series and real-world heartbeats. We find that the permutation-based Shannon entropy (PEn) and time irreversibility (PYs) detect nonlinearities in the model series differently according to the surrogate theory. Moreover, PEn and PYs, based on either the original or the equal-value permutation, give contradictory results for congestive heart failure cases and healthy young and elderly heartbeats. PEn quantifies the complexity by calculating the amount of mean information, whereas PYs measures the probabilistic differences among symmetric nonequilibrium distributions, and these yield different or even contradictory outcomes. Our findings demonstrate the statistical associations between the Shannon entropy and qTIR, contribute to more reliable elucidation of the nonlinear dynamics of heartbeats, and improve our understanding of the complexity and nonequilibrium nature of complex systems.