This book presents theoretical analyses and numerical simulations of fluctuations and crystallization in simple active matter systems. It begins with a comprehensive review of models of self-propelled particles, the common features of fluctuations in active fluids, and crystalline order in two-dimensional active solids. After presenting numerical results based on simple particle models, a fluctuating hydrodynamic theory derived via a bottom-up approach is developed to analyze density fluctuations in active fluids. This theory provides a unified explanation for anomalous density fluctuations, such as giant density fluctuations in active Brownian particles and hyperuniformity in chiral active particles, observed in both numerical simulations and previous experiments. The book further investigates the effects of chirality on the crystallization of active particles, demonstrating that chiral self-propulsion stabilizes crystalline order and can induce long-range order even in two dimensions, in contrast to typical behaviors in equilibrium systems. Overall, it offers a detailed numerical analysis of anomalous fluctuations and their impact on crystallization arising from nonequilibrium nature, as well as a theoretical framework that captures these phenomena for a broad class of active matter systems.
