Statistical mechanics of colloids and active matter in and out of equilibrium

Thermal and viscous forces compete for dominance at the microscopic length-scales which govern the behaviour of many soft or biological systems. We study three systems of increasing complexity with the central goal of understanding the statistical or hydrodynamic nature of their mechanics. First we study experiments that have been conducted on ferromagnetic colloidal rods. At equilibrium, the magnetically pinned rod is observed to randomly flip between two orientational states, which our theoretical analysis shows is due to a competition between entropic and Hamiltonian forces. We show analytically how entropic forces can arise by considering the coupling between observed and unobserved variables of a system. Experiments in which a rod is driven out of equilibrium by a rotating field display three phases of steady-state behaviour as a function of driving frequency. Using Brownian dynamics simulations we match the lower critical frequency to the experimentally obtained values, showing that thermal fluctuations play an important role in this regime and propose a simple argument to demonstrate that hydrodynamic interactions between the substrate and rod affect the upper critical frequency. We then turn to the biophysical topic of cell locomotion in viscoelastic media. In order to study how bacterial flagella interact with similarly-sized polymers in their environment, we construct a Stokesian dynamics model of a helical filament and bead--spring polymer. Simulating their interaction first for a pinned--rotating helix, then for a swimming helix, we demonstrate that large polymers become hydrodynamically entrained by the flagellum and coil around it, causing both pinned and swimming flagella to expend more work. For the swimming helix, this results in a reduction of swimming speed on average. Finally, we consider an active nematic fluid confined to a channel and show that the inclusion of a passive colloid induces a global state of coherent flow maintained by the intrinsic activity of the system. This flow is persistent, and transports the colloid with it along the channel. By this mechanism, a passive colloid is able to spontaneously induce its own transport through an otherwise quiescent fluid.