In this thesis an overview is given of the renormalisation group as it is
applied to equilibrium systems; the methods of field theory are extended to
non-equilibrium systems, described by a Langevin equation in the stead of a
Hamiltonian; this analysis is applied to a well known model of surface growth
driven by molecular-beam epitaxy.
The renormalisation group is a celebrated technique in both hard and soft
condensed matter physics for probing the asymptotic behaviour of a model,
though in this thesis no examination is made of quantum effects. Several
distinct methods exist under the banner of renormalisation, most famously the
approaches of Wilson and field theory. The renormalisation group is explored
through a comparison of these approaches.
The approach of field theory, with its methods being applied to an equilibrium
system where a model is defined by a Hamiltonian, can be extended
to analyse non-equilibrium systems, where a model is described by a Langevin
equation. One class of the non-equilibrium condensed matter systems which
have received extensive attention is that of surface growth. For the last two
decades the Villain-Lai-Das Sarma equation has been used to understand conserved
surface growth processes such as molecular-beam epitaxy. However, the
theory has some aspects that seem incomplete. The mound formation observed
experimentally and numerically lacks a complete theoretical narrative for its
mechanism. Also, no clear picture has emerged over a disagreement in the literature
about the alleged exactness of scaling relations. Using field theory to
analyse the original derivation of the Villain-Lai-Das Sarma equation reveals
that terms responsible for mound formation are generated under renormalisation,
further these terms should have been included initially on symmetry
grounds. It is possible to recover several widely studied Langevin equations at
the trivial fixed point of the full theory, allowing a more complete theoretical
picture to be presented for conserved epitaxial surface growth.