Abstract

Much is now known about the collective dynamics of Active Brownian Particles (ABPs) whose self-propulsion directions have no alignment interactions and so evolve independently of one another. (This holds even if that actual self-propulsion is hindered by collisions with other particles.) Collectively ABPs show no orientational order, but do show a liquid-vapour or liquid-solid type phase separation, even in the absence of attractive forces. This is called motility-induced phase separation (MIPS) and is shared by a number of other scalar models. Phase separation is also an important part of the physics in flocking models (describing active particles whose self-propulsion directions tend to co-align), where the transition to an orientationally ordered state of "collective motion" can be strongly coupled to density changes.

Returning to ABPs, suppose we now bias (or filter) the dynamics to select time evolutions with atypically low collision rate. We then find a collective motion phase which does after all have orientational order. This is because the least unlikely way for collisions to be avoided is if the particles happen to be all moving in the same direction. Conversely, selecting an atypically high collision rate promotes MIPS in parameter ranges where it would not otherwise happen. In general, the biased dynamics maps onto the unbiased dynamics of some other system, known as the 'optimally controlled' system, whose interactions are not those of the original system. This suggests a strategy design the microscopic interactions of an active particle system to achieve some desired collective behaviour. Specifically: start from a baseline system, bias towards the desired behaviour, and find the control forces. Then choose these to be the the real interactions. Unfortunately the control forces are, in general, infinitely complicated, but simple classes of interaction (e.g. 2 body or mean field) can still be optimized to best approach the target behaviour. This is done by minimizing an upper bound on the large deviation rate function, which quantifies the unlikeliness of the atypical dynamics in the vanilla system. I will try to explain how this works for the collective motion phase of the ABP system with reduced collision rate.