Dry Active Matter on a Sphere

When flocks of birds migrate southwards imagine larger and larger swarms moving
together on a shrinking globe, until one gigantic flock covers the entire planet.
What becomes of their motion if they reach the pole? We propose to build and analyse
a model of active particles confined to move on the surface of a sphere. Unlike in the
plane where all particles can align with each and travel with the same velocity parallel
to each other, motion on the sphere is far more peculiar. This is a direct consequence
of the famous Hairy ball theorem that states that "you can't comb a hairy ball flat without
creating a cowlick". Translated into the language of active motion, it is not possible for the
entire flock to travel with a uniform velocity on the surface of a sphere. As a result, one expects
to observe a rich and complex set of motion patterns emerging from the intricate interplay
between the curvature and particles' alignment. It is our aim to identify, understand and classify
those motion patterns.

This project aims to use numerical simulations to study a model of self-propelled particles confined to move on the
surface of a sphere. As such, its direct and immediate impact will be most apparent in
the vibrant and rapidly growing physics community interested in understanding
behaviour of systems far from equilibrium. While there have been many recent advances in the field of active matter
research, there are still numerous fundamental questions that remain
open. To the best of our knowledge, to date there have been no attempts
to systematically study effects of curvature on the emergent collective
behaviour in active systems. This project strives to open a new direction of active matter
research, which would place an emphasis on a systematic study of
the effects of curvature on formation and evolution of dynamic patterns,
in hope to inspire new experimental and theoretical work into this
unexplored venue of the out-of-equilibrium physics.

In the UK, there are many research groups working on active systems.
Most notably, groups at University of Aberdeen (S. Henkes, F. Ginelli),
University of Bristol (T.B. Liverpool), Cambridge University (R. Goldstein),
Durham University (S. Fielding), University of Edinburgh (M. E. Cates, D. Marenduzzo, W. Poon) and
Oxford University (R. Golestanian, J. Yeomans). Internationally, the Syracuse Soft Condensed
Matter group has a long tradition in exploring effects of geometry
on various physical systems (Bowick, Marchetti). Experimental groups
at the University of Massachusetts - Amherst (Menon), at the University
of Chicago (Irvine) and at the New York University (Chaikin), to name
a few, have been performing intriguing experiments in passive systems
in curved geometries, while the group of Zvonimir Dogic at Brendeis
University (USA) has recently performed seminal experiments with active
nematics on the surface of a bubble, albeit effects of curvature have
not been addressed. All those groups could directly benefit from the results of this study.

In a broader sense, developing an understanding of how curvature affects
activity and out-of-equilibrium behaviour in general can have far
reaching impact on other fields of research, primarily in life sciences.
A living organism is probably the most common yet exceedingly complex
example of a system out of equilibrium. Complexity of life stems from
an intricate dynamic interplay of many individual components spanning
multiple length and time scales. It is an emergent phenomenon that
cannot be fully understood by studying individual components on their
own. This project hopes to point out importance
of geometry on systems far from equilibrium. Such effects might play
a crucial role in many aspects of life. For example, it is well known
that during the early development, cells in an embryo undergo large
structural and spacial changes. Embryos are almost never flat, but
rather have distinctly curved geometries and often undergo non-trivial
topological changes. It is very plausible to expect that the geometry plays an important
role in how cells move and organize.