Non-equilibrium Statistical Mechanical Methods for Dynamical Systems and Stochastic Processes with Phase Transitions

This work deals with some key questions in non-equilibrium statistical physics, with a particular focus on transport phenomena in confined systems and the onset, therein, of phase transitions. Within this context, the thesis explores particle transport in specific systems based on mathematical billiards. The basic model features two identical regions of two-dimensional space urns connected by a channel, serving as table of a mathematical billiard. The billiard is populated by an ensemble of N non-interacting particles, incorporating a feedback mechanism that regulates particle transport. In this kind of models interesting non-equilibrium phenomena take place, such as uphill currents and first-order phase transitions. A key aspect of this work is the investigation of the ergodicity of these billiard systems, focusing on how geometric alterations, such as the transition from circular to polygonal urns, impact the system's dynamical properties. The results show that the geometry of the urns significantly influences the system's ergodic behavior, particularly in polygonal billiards where the ergodicity is dependent on the internal angles of the polygons employed as urns. A 1D urn model, based on stochastic billiard-like dynamics and obtained as a projection of higher-dimensional dynamics onto a one-dimensional ring, further develops these ideas and provides a deeper understanding of phase transitions in confined geometries, showing how feedback mechanisms and initial conditions shape the statistical properties and mixing behavior of particle ensembles evolving in confined billiard systems. A Markov chain stochastic model is derived from the original deterministic N particles system, inspired by the classical Ehrenfest urn model. This model will serve as a base for the application of some relevant concepts from stochastic thermodynamics. In particular, we leverage Work Fluctuation Theorems (WFTs), such as the Jarzynski Equality (JE), to explore the possibility to detect phase transitions in non-equilibrium systems with stochastic dynamics. In this sense, we employ both analytical and numerical methods to demonstrate how finite size effects and irreversibility in such systems may negatively affect the theoretically exact computation of averages of exponential observables (as happens in the JE). This apparent shortcoming of the theory will be used to detect phase transitions in the state space of stochastic systems. We will prove how the lack of information about the rarest states of a system can actually point to the selection of suitable subsets of the system's state space, which can in turn be used to detect phase transitions through the JE and similar averaging procedures. Altogether, these findings contribute to the better understanding of some key aspects of non-equilibrium statistical mechanics, such as transport phenomena and phase transitions, through the analysis of classically relevant aspects of the theory such as ergodicity, finite size effects and Work Fluctuation Theorems. Indeed this work offers novel and interesting perspectives on how anomalous transport and non-equilibrium phase transitions can be revealed and characterized through both deterministic and stochastic approaches.