Computational Studies of Dendritic Deposition and Trajectory Phase Coexistence

Many out-of-equilibrium systems display collective transitions in the behavior of particles akin to phase transitions. The field of nonequilibrium statistical mechanics seeks to develop new theories and methods to characterize these phenomena. In this thesis, we advance this aim by presenting computational studies of two different kinds of nonequilibrium transitions: the compact-to-dendritic (CTD) transition in the deposition of Brownian particles and trajectory phase coexistence (TPC) in stochastic dynamical systems. The CTD transition occurs when Brownian particles (like ions, colloids, or misfolded proteins) deposit from all sides onto a reactive cluster. While the cluster initially maintains a compact morphology, upon reaching a critical radius, it spontaneously develops dendritic branches. Although the size of the critical radius depends on the deposition conditions, this relationship is not well understood at a mechanistic level. Here, we show that contrary to previous evidence, the critical radius in Brownian dynamics simulations follows the behavior predicted by a continuum analysis. That is, dendrites emerge when the cluster circumference exceeds the length that particles can diffuse in the characteristic reaction timescale. Consequently, our results provide microscopic validation for continuum methods that are widely applied to study dendrite formation in electrodeposition and lithium metal batteries. Trajectory phase coexistence (TPC) arises when qualitatively different trajectory behaviors interconvert in a stochastic dynamical system. This type of coexistence plays a central role in theories of glassy dynamics. In this work, we focus on two different research areas related to TPC. First, we introduce an importance sampling method, Variational Ansatz for Rare Dynamics (VARD), for characterizing a system's rate function. VARD is technically and conceptually straightforward yet can still sample the large deviations of many-body models found in the literature. We then examine the meaning of kinks in the scaled cumulant generating function (SCGF). Although these singularities are often taken to be proof that TPC occurs, a more precise understanding of the connection between kinks and coexistence remains lacking. By characterizing the dynamics of two kinds of random walkers, we show that kinks actually result from diverging timescales in the dynamics and do not always indicate the presence of TPC.