High Dimensional Phenomena

Gordon Blower carries out research into High Dimensional Phenomena, which have applications in statistical physics.

In statistical mechanics, lattices of vibrating atoms are used to model the behaviour of solids at the atomic level, while the overall state of the solid is described by thermodynamic quantities such as temperature and entropy.

The motion of each atom relative to the lattice is considered as random, and has distribution like a normal random variable. At high temperatures, there is more movement and hence higher variance. The state of the lattice can be specified by a huge vector giving the coordinates of all the points in the lattice, and the aim is to describe how the system changes as temperature in decreased.

At the atomic level, the distribution of each atom is described by entropy and Fisher's information. Logarithmic Sobolev inequalities show how entropy is bounded by information, and hence how fast the system converges to equilibrium. To relate the bulk properties of the solid to the atomic description, it is important to have logarithmic Sobolev inequalities that do not depend directly on the number of particles. Such inequalities should depend upon the local geometry, and how many particles are neighbours to each point in the lattice. In recent years there has been a significant advance in our understanding of such questions, and many new techniques have been introduced to study them, but some significant open problems remain.

Consider a ring of particles arranged like daisies in a daisy chain, so that each particle interacts with its nearest neighbours but with no others. An important question is to obtain logarithmic Sobolev inequalities for this array, with constants that do not depend directly on the number of particles. Another problem is to understand logarithmic Sobolev inequalities for double well potentials.

Research in this area involves analysis, especially thinking about problems geometrically, and basic probability theory. There are opportunities for PhD students to make progress by studying particular models or proving general results.