Statistical Mechanics and Dynamics of Driven and Active Systems

Systems driven out of equilibrium display a rich variety of patterns and surprising response behaviors. There exist different types of non-equilibrium processes, for instance a system that has been prepared in a non-Boltzmann initial state and is relaxing back to equilibrium, or a system that adopts a non-equilibrium steady state distribution when it is driven by an external field. In these different cases, the main characteristic that distinguishes these systems as non-equilibrium is that they are constantly dissipating heat, or likewise producing entropy. This entropy production is often the starting point for developing a systematic theory to describe such non-equilibrium processes.Entropy production can be related to the irreversible processes occurring within a system. Particularly strong statements can be made about non-equilibrium systems when a local equilibrium assumption can be made, that is, when smaller subsets of a large system can be considered to be in equilibrium. This turns out to be justified for a wide variety of systems under different conditions. When this holds, the entropy production can be written as a generalized thermodynamic force (often the gradient of some intensive variable of the system) multiplied by a flux. When the thermodynamic force is small, the fluxes can be written as linear combinations of the thermodynamic forces, connected by response coefficients–this is known as linear irreversible thermodynamics. The full extension of equilibrium thermodynamic concepts to dissipative processes beyond this linear regime, including the development of microscopic principles justifying irreversible thermodynamic theories (as equilibrium statistical mechanics justifies equilibrium thermodynamics), is still a work in progress.In this thesis, we work towards advancing the thermodynamic theory of non-equilibrium phenomena by studying models of driven-diffusive systems, growth processes, and active matter. We use developments from stochastic thermodynamics, large deviation theory, and irreversible thermodynamics to characterize the non-equilibrium phases and properties exhibited by these systems. We question to what extent equilibrium approximations are valid for predicting pattern formation in these systems and whether there exist general unifying features describing these non-equilibriums processes. In the process we develop trajectory sampling methods to investigate the statistics of dynamical order parameters distinguishing these phases. We show how the first and second laws of thermodynamics, including consistent expressions for entropy production, can be extended to active systems, where microscopic reversibility is broken at the level of individual particles. Additionally we derive fluctuation relations, exact analytical results for the fluctuations of entropy production in the form of equalities, for the entropy production in active systems. We also extend the Irving-Kirkwood procedure to active systems, deriving the balance laws of mass, momentum, and energy. Consequently we obtain expressions for the stress and couple stress tensors in the system as functions of the microscopic variables. This provides a foundation to extend the framework of irreversible thermodynamics to active systems