Details of Grant 

EPSRC Reference:	EP/I004378/1
Title:	Constrained random geometries: phase boundary fluctuation and sub-ballistic motion
Principal Investigator:	Hammond, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:	New York University
Department:	Statistics
Organisation:	University of Oxford
Scheme:	Career Acceleration Fellowship
Starts:	01 October 2010	Ends:	20 August 2014	Value (£):	547,739
EPSRC Research Topic Classifications:	Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 09 Jun 2010 EPSRC Fellowships 2010 Interview Panel B Announced
Summary on Grant Application Form
As a mathematician working in the field of statistical mechanics, I am pursuing research problems that concern mathematical models that exemplify some important aspect of the physical world. The aim is find the simplest viable description of the typical large scale behaviour of such systems. For example, there are huge number of air molecules in a room, so that, microscopically, the behaviour of the air in the room is described by all of the positions and velocities of those molecules. This is a vast amount of information. In the large, however, what effectively describes the air is a few parameters, such as the temperature and the pressure, that are specified by averages of particles over small regions of space (that nonetheless contain very many particles). In the rigorous theory of statistical mechanics, we choose a suitable mathematical model of a physical system, and prove how the behaviour of such macroscopic quantities as temperature and pressure arises from the microscopic structure of the system.In this proposal, I am undertaking three related research projects, each of which reflects in some way this theme:I. Phase boundary fluctuation. If oil is injected into still water, it forms into a droplet that makes the total surface tension at the boundary as small as possible. On a finer scale, however, the boundary between the two substances may be random. In a recent series of papers, I have investigated, for a natural mathematical model of two such substances, the geometry of this random boundary. I am proposing to investigate what is universal about this random fluctuation: that is, which elements of this behaviour are shared with a diverse range of other systems. II. Trapping in disordered media.If a charged particle in an electric field moves in an environment populated by occasional obstacles, its progress is liable to be frustrated by traps formed by the obstacles. What is the geometry of the traps that waylay the particle, and to what degree do these traps slow down the walk? Alexander Fribergh and I are carrying out an extensive investigation of a mathematical model of this problem, in which a walker jumps generally in a preferred direction, but makes other random moves as well, on a grid in which some edges are impassible.III. Spatial random permutations.At very low temperatures, helium condenses into a remarkable substance that flows with extreme ease. A mathematical model of repelling random particles is naturally associated to such low-temperature gases. I am planning to investigate how these particles behave in a fashion that, while random, has large scale order, and how this order is related to the special properties of very cool gases.
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Organisation Website:	http://www.ox.ac.uk