Exact solutions to the long-time statistics of nonequilibrium processes

Nonequilibrium statistical mechanics deals with noisy systems whose dynamics breaks time-reversal symmetry. Stochastic Markov processes form a mathematical framework for a unified theory of nonequilibrium phenomena, particularly at the microscale where fluctuations play a dominant role. The development of both theory and useful applications has been aided by minimal, exactly solvable models, where the logical connection between model feature and behaviour can be ascertained. In this thesis I consider a number of biologically inspired models in relation to two aspects of the long-time statistics of nonequilibrium processes: the attainment of a nonequilibrium steady state, and steady-state fluctuations using dynamical large deviation theory. For the first topic, I consider two models of particles moving stochastically on a ring under no-crossing interactions. The first is of two lattice run-and-tumble particles, each of which moves in a persistent direction interspersed by `tumble' reorientation events. I extend the previously known steady-state solution to a solution for all time, in the sense of obtaining a diagonalization of the Markov generator of the process. The spectrum exhibits eigenvalue crossings at exceptional points of the tumbling rate, which leads to a singular dependence of the relaxation time to the steady state on this parameter. In the second model of heterogeneous single-file diffusion, I solve for the steady state of N driven particles with individual diffusion properties. This reveals an inter-particle ratchet effect by which the particle current is disproportionately affected by slow-diffusing, rather than slow- or fast-driven, particles. The model generalizes to higher dimension without compromising the solution structure, if a key property of quasi-one-dimensionality is maintained. In both models, the relation of key model features to generalized notions of reversibility forms an overarching theme. For the second topic, I first consider a random walker on a linear lattice and the effect that adding internal states to the walker has on the emergence of singular behaviour in the fluctuations of either the velocity observable or the time spent at a given site. In particular, I use generalizations of the run-and-tumble particle and probe the trajectories associated with the different fluctuations regimes that this model can exhibit. I show that internal states can either have a drastic influence on the likelihood of a large deviation, or none at all. I then extend the dynamical large deviation formalism for diffusions to the case of reflective boundaries and current-like observables. In particular, this allows the large deviations of the particle current in the heterogeneous single-file diffusion to be obtained analytically. These are found to coincide with those of a single diffusive particle with certain effective parameters, in interesting contrast to comparable studies on the lattice