## Phase transitions in low-dimensional driven systems

##### Abstract

The study of non-equilibrium physics is an area of interest since, unlike for
their equilibrium counterparts, there exists no general framework for solving
such systems. In this thesis I investigate the emergence of structure and front
propagation in driven systems, a special type of system within the area of non-equilibrium
physics. In particular I focus on three particular one-dimensional
models each of which illustrate this in a different way.
The Driven Asymmetric Contact Process (DACP) describes a system where
activity is continuously generated at one end of a one-dimensional lattice and
where this activity is allowed to spread in one direction along the lattice. In the
DACP one observes a propagating wave of activity which appears to abruptly
vanish as the system undergoes a phase transition. Using a modified Fisher
equation to model the system reveals the continued existence of the propagating
wave, now contained within a decaying envelope. Furthermore this establishes
relations between properties of the travelling wave and Directed Percolation
critical exponents.
The Zero-Range Process (ZRP) is a much studied system exhibiting a
condensation transition. In the ZRP individual particles hop along a lattice
at rates which depend only on the occupancy of the departure site. Here I
investigate a modi cation of the ZRP where instead the majority of the particles
at a site depart during a single hopping event. For this, the Chipping model,
a condensate which propagates along the lattice is observed. It is found that
this condensation transition is present even for hop rates which fall foul of the
condensation requirements of the normal ZRP. Further it is observed that, unlike
for normal ZRP, condensation occurs even in the low-density limit. As a result I
suggest a condensation mechanism which depends only on the hop rates of low
occupancy sites. The Host-Solute-Vacancy model (HSV) is a three-species system designed
to model electromigration in a circuit. As the parameter space is navigated
the system undergoes what appear to be two separate phase transitions from a
randomly distributed state to a condensed state with either of two structures. To
investigate the model new measures for determining condensation are developed.
These show that, again, condensation occurs in the low-density limit. By a
reduction to a ZRP an effective hop rate of the system is measured. This effective
hop rate is found to beta function of the occupancy of a site as a fraction of the
total system size. To explain this behaviour I invoke a description whereby there
is a step in the hop rate as a function of occupancy.
Through these three examples I illustrate how minor modi cations to the
dynamics of known systems can result in a new and rich phenomenology. I draw
particular attention to the effect of asymmetry in the dynamics.