Applied Mathematics Research Centre

The AMRC hosts a series of seminars on subjects in statistical physics and fluid dynamics.

**Entropic long range order for Potts antiferromagnets**

We prove entropic long range order for the $q=3$ Potts antiferromagnet on a class planar lattices at low temperatures. The proof is based on an enhanced Peierls argument (which is of independent interest even for the Ising model for which it extends the range of temperatures with proven long range order) combined with an additional percolation argument. The main ideas will be explained in the simplest zero temperature case that turns out to be a pure case of "order by disorder" with "entropic barriers' between phases. Based on a joint work with Alan Sokal and Jan Swart.

**Fluid Rings and Floating Plates**

TBA

**Planetary magnetic fields, ****concentrating on Jupiter**

Numerical dynamo models have had some success in reproducing important features of the Earth's magnetic field. Here we report on simulations of Jupiter's magnetic field using the anelastic approximation, which takes into account the large density variation across the dynamo region. The reference state used in these models is a Jupiter model taken from ab initio calculations of the physical properties of Jupiter's magnetic field (French et al. 2012), which makes the reasonable assumption that the interior is close to adiabatic. The French et al. work also gives an electrical conductivity profile which is adopted here.

Dynamo simulations depend on the dimensionless input parameters, partibularly the Ekman number, Rayleigh number, the Prandtl number and magnetic Prandtl number. Many different types of field have been found, some of which will be described. The most relevant models are those which produce a Jupiter-like strong dipole dominated field. These are found at low Ekman number, Rayleigh numbers well above critical, low Prandtl number and moderate magnetic Prandtl number. Another important issue is the driving heat flux source. Here we assume that Jupiter evolves through a sequence of adiabats, leading to a distributed entropy source throughout the planet, rather than basal heating from the small rocky core. The interaction between the magnetic field, the zonal flow and the convection appears to be crucial in determining the type of magnetic field found.

**Simulated Tempering and Magnetizing: Application to Crossover Scaling in ****the 2d 3-State Potts Model**

The recently introduced simulated tempering and magnetizing (STM) Monte Carlo method is a variant of generalized ensemble algorithms that aims at covering a mesh of simulation points in the two-dimensional temperature-field plane in a dynamically controlled manner. We applied STM Monte Carlo simulations to the two-dimensional three-state Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields. Our data set has been generated by STM simulations of several square lattices with sizes up to $160\times 160$ spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points. Careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size will be discussed.

**Solvable metric growing networks***Wednesday September 18, 2013, 15:00 h, DH Seminar room*

In considering the structure and dynamics of complex networks, we usually deal with degree distributions, clustering, shortest path lengths and other graph properties. Although these concepts have been analysed for graphs on abstract spaces, many networks happen to be embedded in a metric arrangement, where the geographical distance between vertices plays a crucial role. The present work proposes a model for growing network that takes into account the geographical distance between vertices: the probability that they are connected is higher if they are located nearer to each other than if they are further apart. In this framework, the mean degree of vertices, degree distribution and shortest path length between two randomly chosen vertices are analysed.