Discrete noise in stochastic active flows

一种新的数学理论，用于由做出离散决策的活性粒子组成的随机流，将开辟新的研究途径，并为交通和行人管理方面的挑战提供新的解决方案。活性物质系统由大量消耗能量来移动和相互作用的单个元素组成。从群鸟到被驱动的胶体，这个领域中许多有趣的现象无法在历史上成功的流体动力学和平衡热力学理论中理解，这使得这个领域令人兴奋和具有挑战性。在绝大多数的活性物质模型中，粒子对环境的反应是平稳的。通过考虑连续移动但状态离散变化的粒子，将实现现实世界适用性和数学深度的逐步变化。在应用中，这些状态变化可能代表代理做出决策或突然调整行为以响应他人。为推动计划及保持重点，我们会参照两个主要应用项目：交通流量及行人流量，发展我们的架构。受连续的随机波动和离散的人口噪声所产生的随机时间的状态变化，这些活跃的流量有一套丰富的行为。这里提出的研究计划将打开一个新的研究领域之间的传统应用数学和概率，适用于数学研究的方法跨越进化生物学机器人。

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.