Discrete noise in stochastic active flows

由作出离散决策的活性粒子组成的随机流的新数学理论将开辟新的研究途径，并朝着解决交通和行人管理中的挑战的新方案迈进。活跃的物质系统由大量消耗能量来运动和相互作用的单个元素组成。从鸟群到被驱赶的胶体，这一领域的许多有趣现象都无法在历史上成功的流体力学和平衡热力学理论中得到理解，这使这一领域成为一个令人兴奋和具有挑战性的领域。在绝大多数的活性物质模型中，粒子对其环境的反应是平稳的。通过考虑连续运动但离散改变状态的粒子，将在现实世界的适用性和数学深度方面实现阶跃变化。在应用中，这些状态变化可能代表代理做出决定或突然调整行为以响应他人。为了推动这项计划，并保持重点，我们会参考两项主要应用发展我们的架构：交通和行人流量。受连续随机波动和由状态变化的随机时间引起的离散人口噪声的影响，这些活跃的流动具有丰富的行为集。这里提出的研究计划将在传统应用数学和概率之间开辟一个新的研究领域，方法适用于从进化生物学到机器人学的数学研究。

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.