Fluctuations and Criticality in Heavy-Ion Collisions

Quantum Chromodynamics (QCD) predicts various extraordinary phases of matter whose underlying physics is described by the strong interaction. Identifying the QCD phase diagram, of which a critical point serves as the landmark, is one of the fundamental open subjects in modern physics. The properties of the QCD phase transitions and criticality can be studied by the relativistic heavy-ion collision experiments at various laboratory accelerator facilities, where a strongly interacting primordial matter is expected to form in extreme conditions on an event-by-event basis. Fluctuations are therefore indispensable for describing such stochastic process and become rather crucial for understanding the universal behavior when the system approaches the critical point. In this thesis, we will discuss the current methodology and challenge of discovering the QCD critical point, with an emphasis on the recent progress in quantifying the fluctuation-driven phenomena both in and out of equilibrium. The state-of-the-art formalism we developed constitutes an integral part of the theoretical framework for interpreting the experimental results from the ongoing RHIC Beam Energy Scan Program.