Over the last few decades the interests of statistical physicists have broadened to include the detailed quantitative study of many systems - chemical, biological and even social - that were not traditionally part of the discipline. These systems can feature rich and complex spatiotemporal behaviour, often due to continued interaction with the environment and characterised by the dissipation of flows of energy and/or mass. This has led to vigorous research aimed at extending the established theoretical framework and adapting analytical methods that originate in the study of systems at thermodynamic equilibrium to deal with out-of-equilibrium situations, which are much more prevalent in nature. This thesis focuses on a microscopic model known as the asymmetric exclusion process, or ASEP, which describes the stochastic motion of particles on a one-dimensional lattice. Though in the first instance a model of a lattice gas, it is sufficiently general to have served as the basis to model a wide variety of phenomena. That, as well as substantial progress made in analysing its stationary behaviour, including the locations and nature of phase transitions, have led to it becoming a paradigmatic model of an exactly solvable nonequilibrium system. Recently an exact solution for the dynamics found a somewhat enigmatic transition, which has not been well understood. This thesis is an attempt to verify and better understand the nature of that dynamical transition, including its relation, if any, to the static phase transitions. I begin in Chapter 2 by reviewing known results for the ASEP, in particular the totally asymmetric variant (TASEP), driven at the boundaries. I present the exact dynamical transition as it was first derived, and a reduced description of the dynamics known as domain wall theory (DWT), which locates the transition at a different place. In Chapter 3, I investigate solutions of a nonlinear PDE that constitutes a mean-field, continuum approximation of the ASEP, namely the Burgers equation, and find that a similar dynamical transition occurs there at the same place as predicted by DWT but in disagreement with the exact result. Next, in Chapter 4 I report on efforts to observe and measure the dynamical transition through Monte Carlo simulation. No directly obvious physical manifestation of the transition was observed. The relaxation of three different observables was measured and found to agree well with each other but only slightly better with the exact transition than with DWT. In Chapter 5 I apply a numerical renormalisation scheme known as the Density Matrix Renormalisation Group (DMRG) method and find that it confirms the exact dynamical transition, ruling out the behaviour predicted by DWT. Finally in Chapter 6 I demonstrate that a perturbative calculation, involving the crossing of eigenvalues, allows us to rederive the location of the dynamical transition found exactly, thereby offering some insight into the nature of the transition.

Lattice Density Functional Theory is a powerful method to treat equilibrium structural properties and non-equilibrium kinetics of condensed matter systems. In this thesis an approach based on Markov chains is followed to derive exact density functionals for interacting particles in one-dimension. First, hard rod mixtures on a lattice are considered. For the treatment of this system, certain sets of site occupation numbers are introduced. These sets reflect zero-dimensional or one-particle cavities in continuum treatments, which can hold at most one particle. The exact functional follows from rather simple probabilistic arguments. Thereby the derivation simplifies an earlier, more complicated treatment. A rearrangement of the functional casts it into a form according to lattice fundamental measure theory. This makes it possible to systematically setup approximate density functionals in higher dimensions, which become exact under dimensional reduction. In the next step, the theory is extended to hard rod mixtures with contact interactions. Finally, hard rods with arbitrary nearest-neighbor interactions extending over two rod lengths are studied. For those interactions, two types of zero-dimensional cavities need to be introduced. The first one is a one-particle cavity corresponding to a set of occupation numbers with at most one occupation number being nonzero. The second type is a two-particle cavity, which is a cavity that cannot hold more than two particles, that means at most two occupation numbers can be one in the corresponding set. In order to account for time-dependent kinetics, a lattice version of Time-Dependent Density Functional Theory is followed and applied to hard rods with contact interactions.

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