## Nonequilibrium dynamical transition in the asymmetric exclusion process

##### Abstract

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.