|abstract ||The aim of this thesis is to study nucleation both numerically and analytically.
The approach followed is to start with very simple models. In chapter 2 we study the Ising model without an external magnetic field. This system does not feature nucleation, but at low temperatures it jumps back and forth between a state in which most spins are up, to one in which most spins are down. The dominant pathway at low
temperatures consists of the formation of a single pair of closed interfaces in
the shorter periodic direction, which perform a relative diffusive motion
around the longer periodic direction and annihilate after meeting each other
through the periodic boundary.
In chapter 3 we study the Ising model with an external field on a square
lattice. Initially the system is in a metastable state, with most of the spins
anti-aligned with the external field. It will stay in this metastable state for
an extended period of time, but eventually one of the small clusters of aligned
spins that arise due to fluctuations, will grow beyond the critical cluster
size, and take over the whole system. After this, most of the spins are aligned
with the external field, and the system is in its stable state.
In chapter 4 the same model is studied. The effective rates of growth and
shrinkage of clusters are now studied in detail. The mass of the nucleating
cluster is followed in time, and mapped to a random walker undergoing drift and
diffusion. The latter is described by the Fokker-Planck equation, in which in
our case the drift and diffusion coefficients depend on cluster size.
The method developed in chapter 4 is then applied to a completely different
phenomenon: the fluctuations of the geomagnetic dipole. Instead of to the
time-evolution of the size of the nucleus, we apply our method of analysis,
based on the Fokker-Planck equation, to the time-evolution of the strength of
the geomagnetic dipole, which has been measured accurately over the last
800,000 years from fossile records. This is described in chapter 5.|