In this thesis, I investigate various non-equilibrium systems and analyse statistics relating to thermodynamic and rheological observables. I start by theoretically considering a cyclically driven engine, analogous to those used in the formulation of classical thermodynamics, which instead uses an active fluid as its working medium. Due to the driving at the level of the active constituents, there is a constant energy flux in this system, some of which can be transferred to useful work even at a single thermostat temperature. I explicitly show that extensive work extraction is possible by varying two external control parameters in a cyclic fashion, for the case of an engine with harmonically repulsive walls. A procedure for optimising the cycle shape for maximal work output in the quasistatic limit is given, and various properties of the optimization problem for cycles driven in finite time are discussed.

I then move on to constructing a new elastoplastic model called N-element Hébraud-Lequeux to analyse fluctuations in the local injected power in amorphous systems under constant shear. Unlike many popular elastoplastic models, this one permits stochasticity in its local stresses and shear rates simultaneously, and so is suitable for investigating phenomena which requires independent fluctuations in both. Using this model, I show that a crossover occurs in the dominant mode of rare negative power fluctuations, which is analogous to an effect numerically observed in particle-based simulations of jammed systems.

Finally, we turn our attention to viewing elastoplastic models of the Hébraud-Lequeux (HL) type more generally as a subclass of stochastic resetting systems, which is an underappreciated fact, Despite their intrinsic non-linearities, I argue that the steady state of HL-type models can be cast as a more standard resetting problem and, using a path-integral approach, derive several results related to the temporal and stress statistics in this regime. These include a proof of the existence of a stress fluctuation theorem given conditions on the local resetting criterion, as well as a dynamical transition in the survival time of stress elements, among other results.