Fig. 3
From: Nearest neighbor permutation entropy detects phase transitions in complex high-pressure systems
Phase transitions manifest as anomalies in entropy values. Panels (A), (B), and (C) show the values of k-nearest neighbor permutation entropy (\(H_P\), solid black line) calculated from spectra of carbon dioxide and distilled petroleum fraction mixtures collected during depressurization trials. Light gray lines denote the confidence bands defined by threshold values \(H_P^-\) and \(H_P^+\), whose exceedance indicates a phase transition. Gray dotted lines display the average entropy values (\(\mu _P\)) calculated within expanding windows. Transition pressures visually determined by experts are indicated by colored vertical lines alongside the type of initiating phase, while colored cross markers show transition pressures identified by entropy values outside the confidence bands. \(\textrm{CO}_2\) concentration levels and temperatures are annotated within the panels. In all panels, the entropy parameters are set to \(k = 265\) and \(d = 5\), with the threshold parameter \(\gamma = 2.25\).
