Tom Gray

Research

There are two (related) thrusts to my research. The first is Brownian motion in disordered potential energy landscapes , where an exponential distribution of barrer heights (or trap depths, depending upon how the landscape is constructed) can bring about a transition between diffusion and sub-diffusion. The behaviour of the effective diffusion coefficient around this transition is closely related to the shape of the energy barriers separating one potential energy minimum from another.

In order to extract accurately and precisely the exponent characterising sub-diffusive motion it is necessary for an ensemble of particles to explore a lot of the potential energy landscape, thereby sampling thoroughly the distribution of barrier heights (trap depths) and thus the distribution of first-passage times from minimum-to-minimum. Naturally this requires reaching the long-time limit. When the position of each particle is advanced incrementally using the Euler scheme based upon the over-damped Langevin equation, the simulations take a very long time to reach this limit. We are developing a coarse-grained simulation scheme which treats the potential energy minima as lattice sites and moves particles from site-to-site according to probabilities and timescales determined from the energy barriers surrounding that site.

The second thrust is studying thermally activated escape processes. Our goal is to derive formulae for calculating mean first-passage times in potential energy landscapes which offer improvement upon existing, largely approximate expressions. There is an exact formula for use in one dimension, but no general higher-dimensional counterpart because of the greater freedom in the shape of absorbing and reflecting boundaries. Sometimes the shape of the potential energy landscape and the absorbing boundary facilitate usage of the one-dimensional expression in two (and higher) dimensions. In other cases, where the shape of the potential in the non-escape directions changes along the direction of escape, modifications to this formula must be made.