Minimal Models of Nonequilibrium Phase Transitions

相变是物理系统的性质发生质的变化的过程，就像一壶沸水变成蒸汽或一根铁棒变成磁性一样。在这两个例子中，唯一的能量输入来自系统周围的环境，最终系统和环境达到平衡。长期以来，物理学家通过经典热力学和朗道理论来理解平衡态系统的相变。然而，在自然界中，我们经常遇到与环境不平衡的系统。例如，在生物系统中，生物体将食物中的能量转化为运动，因此可以在没有周围环境输入的情况下开始和停止运动。这些系统也可以表现出相变，比如当一群鸟开始作为一个整体飞行时，但与平衡的例子不同，目前还没有通用的框架来描述它们。在这个项目中，我们研究非平衡系统的最小数学模型。这些模型由简单的规则定义，有时可以精确求解，不像更现实和复杂的模型。研究这些模型的目的是双重的。首先，它们有时可以被用作真实的物理系统的近似。在这种情况下，可以定性地描述复杂系统的行为，而不需要大量的数值计算。其次，通过剥离一个模型的过多细节，我们通常可以揭示出导致一些复杂现象的更简单的潜在机制。然后，我们可以用一个最小模型来描述许多真实世界的系统。本计画的目标是研究这些最小模型，以增进我们对造成非平衡相变的机制的了解。

Phase transitions are processes in which the properties of a physical system change qualitatively, like when a pot of boiling water turns into steam or a bar of iron becomes magnetic. In both those examples, the only energy input comes from the environment surrounding the system and eventually the system and environment reach an equilibrium. Physicists have understood phase transitions of systems in equilibrium for a long time through the frameworks of classical thermodynamics and Landau theory. However, in nature we often encounter systems which are not in equilibrium with their environments. For example, in biological systems, organisms convert the energy from their food into motion and thus can start and stop moving without inputs from their surroundings. These systems can also exhibit phase transitions, like when a flock of birds starts flying as a single unit, but unlike the equilibrium examples, there is currently no general framework to describe them. In this project we study minimal mathematical models of nonequilibrium systems. Such models are defined by simple rules and can be sometimes be solved exactly, unlike more realistic and complex models. The purpose of studying these models is twofold. Firstly, they can sometimes be used as approximations of real physical systems. In that case, it is possible to qualitatively describe the behaviour of a complex system without the need for extensive numerical computations. Secondly, by stripping a model of excessive detail, we can often reveal more simple underlying mechanisms which are responsible for some complicated phenomenon. We can then describe many real-world systems by one minimal model. The goal of this project is to study such minimal models to develop our understanding of the mechanisms causing nonequilibrium phase transitions.