Field Theories for Non-Reciprocal Phenomena, Oscillating Matter, and Neural Dynamics

For most of us, life is out of equilibrium. This fact is also true in most scientific endeavors, despite our best efforts. The canonical theories of statistical physics contain a very well-developed set of tools for understanding properties of equilibrium systems. These include the partition function, notions of various thermodynamic potentials -- free energy in particular -- and culminate in the ideas of universality and the techniques of the renormalization group (RG). Despite the immense richness and insight the equilibrium theories provide, the possibilities of different states or phases of matter outside of equilibrium are vast. On top of that, there is no guarantee that the same insights and methods, especially those of scaling, universality and RG will apply. Therefore, it is of great interest to explore these methods in the context of non-equilibrium systems. As paradigmatic manifestations of non-equilibrium, we consider three classes of systems in this dissertation. The first is non-reciprocal systems, where an action does \textit{not} cause an equal and opposite reaction. The second is systems with self-sustained oscillations which are made possible by either an internal source or constant influx of energy. Finally, we consider some models of neural dynamics pertaining to epileptic seizures. Drawing from the equilibrium theories, we consider these problems in a coarse-grained framework, which allows for the use of field-theoretic methods. Some common themes arise throughout these models, despite them describing seemingly unrelated phenomena. These include dynamical phases, where the steady state of the order parameter is not time-translationally invariant, and critical exceptional points (CEPs), where the statistical fluctuations of the order parameter are greatly enhanced compared to equilibrium critical points. This work contributes to the long-standing goal of establishing a theory of universality in non-equilibrium systems.