A new mathematical theory for stochastic flows composed of active particles making discrete decisions will open new avenues of research and build towards novel solutions to challenges in traffic and pedestrian management.
Active matter systems are composed of large numbers of individual elements that consume energy to move and interact. From flocking birds to driven colloids, many interesting phenomena in this field cannot be understood within the historically successful theories of fluid dynamics and equilibrium thermodynamics, making this an exciting and challenging field. In the vast majority of active matter models particles respond smoothly to their environments. A step-change in both real-world applicability and mathematical depth will be achieved by considering particles that move continuously but make discrete changes of state. In applications these state changes might represent agents making decisions or abruptly adjusting behaviour in response to others. To motivate the programme and maintain focus, we will develop our framework with reference to two key applications: traffic and pedestrian flows. Subject to both continuous random fluctuations and discrete demographic noise arising from the random timing of state changes, these active flows have a rich set of behaviours. The research programme proposed here will open a new field of study between traditional applied mathematics and probability, with methods applicable to mathematical research spanning evolutionary biology to robotics.