Towards an investigation of structures, dynamics, and phase transitions in disordered systems

It is the significant themes of structural description, dynamical analysis, and phase transition detection that form the foundation of research on disordered condensed matter systems. However, traditional physical methods often prove inadequate when faced with the complexity of disordered systems. This thesis integrates recent advances in graph theory, network science, and topological data analysis to provide a unified and interpretable framework for addressing these key challenges in complex systems. Firstly, we present a methodology for identifying phase transitions in condensed matter systems without relying on any preexisting physical assumptions. By constructing pseudo-order parameters based on graph limit theory and leveraging results from molecular dynamics simulations, our approach rigorously connects combinatorial mathematics with condensed matter physics. Applying this framework to Lennard-Jones liquids and the Kob-Andersen binary mixture enables us to determine the onset temperature of crystallization and assess glass-forming ability using purely structural metrics, rather than conventional order parameters in Landau theory. Secondly, we put forward a set of graph-based descriptors—such as node centrality and local clustering coefficients—traditionally utilized in network science, as alternative representations for condensed matter systems. When applied to the prototypical Lennard-Jones system, these graph-based features outperform various traditional descriptors and even deep learning (GNN) in predicting dynamical properties and phase transitions, as well as in classifying “soft” and “hard” particles according to their rearrangement trends. The results demonstrate that these descriptors offer superior transferability across different thermodynamic states, and highlight the potential for broad applications of network science concepts in condensed matter analysis. Finally, we develop a unified framework based on persistent homology to characterize both local and global structures in disordered systems. This approach simultaneously encodes the local environments of particles and global structures of systems, providing highly effective and interpretable predictors for machine learning and achieving the high performance in predicting the particle rearrangements and global phase structures. Notably, a single predictor allows a linear support vector machine to achieve nearly perfect three-phase classification. Inspired by this discovery, we define a nonparametric metric, the separation index, which not only achieves this classification without sacrificing significant performance but also establishes a connection between particle environments and the global phase structures. Overall, these studies offer a versatile and transferable toolkit for describing structure, analyzing dynamics, and detecting phase transitions in disordered systems. The methodologies presented in this thesis not only provide new methods for data-driven research and theoretical understanding in soft matter and related disciplines, but also have the potential to contribute to new perspectives of general complex systems.