Miniproject 2: The effect of topological defects in drops of active matter

(supervised by Dr. Gareth Alexander)

Active matter, as a material constituted of a large number of particles that locally consume energy
in order to perform directed movement, exhibits fascinating collective behaviour. In particular, for
active material confined to a drop experimental and theoretical investigations revealed the occurrence of internally generated flows and self-propelled motion of the entire drop, that were accompanied by long-range order of the active particles analogous to that in liquid crystals. Thus, it is natural to ask how the type of orientational order, and associated defects thereof, can enable the transmission of local activity onto large scales and lead to motility of the drop. In this paper we consider a three dimensional drop of active matter on a flat surface. We impose different orientational order with topological defects and calculate how it affects the flow of active fluid inside the drop using a hydrodynamic model, that is modified by an active stress term. We find, that an asymmetry in the orientation field is inherited by the flow, suggesting a possible motility of the drop in the case of appropriate boundary conditions at the contact surface. For one example we are able to calculate the speed of the drop.

Miniproject 1: Dynamics of the non-Markovian TASEP

(supervised by Dr. Stefan Grosskinsky, Dr. Rosemary Harris)

The totally asymmetric exclusion process (TASEP) is a simple transport model, where hardcore
particles hop on a one-dimensional lattice uni-directionally. In the Markovian model each
particle hops with a constant rate and the waiting time between two hop attempts is distributed
exponentially. However, many processes, e.g. in biology, are inherently non-Markovian resulting from
internal degrees of freedom of the particles. We perform stochastic simulations for a non-Markovian
TASEP with different waiting time distributions, that incorporate a time delay in the jump process
of each particle. For a delayed exponential distribution, and for a power-law distribution with a large
delay with fixed mean, we find higher currents than in the Markovian TASEP and higher densities
at maximum current, such that the current-density relation is skewed to the right. The increase in
the current results from an anti-correlation in the lattice site occupation, i.e. each particle typically
has a free site to its right and is able to hop with an enhanced probability. We explain the resulting
triangular shape of the current-density relation in the limit of large delay theoretically, as well as
the delay-dependence of the current at very high densities. For the delayed exponential distribution
we derive a full theoretical prediction for the current-density relation in dependence of the delay.
Our result however needs improvement for the intermediate density region, where it deviates from
the simulation data.