标题： 有多少更多是不同的？ 显示英文标题

标题： How many more is different?

摘要： 从水分子小团簇中冰的形成到行军蚁群的群体袭击，集体的涌现行为在其规模上起着关键作用。 同时，普遍认为这些行为对个体的损失具有鲁棒性。 这种矛盾指出了系统研究数量如何影响集体行为的必要性。 我们通过关注在特定临界个体数时突然变化的集体行为来开始这项研究。 我们表明，对标准分岔分析的细微修改可以识别这些临界数，包括与离散性和噪声引起的转变相关的临界数。 通过将它们视为同一现象的实例，我们表明不同物理尺度和科学领域的临界数通常来源于随着数量变化而不同竞争反馈。 然后，我们利用这一想法，在过去关于集体行为的研究中找到被忽视的临界数，并探讨其结论的含义。 特别是，我们强调在临界数附近随机模型的确定性近似可能失败。 最后，我们区分这些定性变化与密度依赖的相变，并讨论我们的方法如何推广到更广泛的集体行为类别。 显示英文摘要

摘要： From the formation of ice in small clusters of water molecules to the mass raids of army ant colonies, the emergent behavior of collectives depends critically on their size. At the same time, common wisdom holds that such behaviors are robust to the loss of individuals. This tension points to the need for a more systematic study of how number influences collective behavior. We initiate this study by focusing on collective behaviors that change abruptly at certain critical numbers of individuals. We show that a subtle modification of standard bifurcation analysis identifies such critical numbers, including those associated with discreteness- and noise-induced transitions. By treating them as instances of the same phenomenon, we show that critical numbers across physical scales and scientific domains commonly arise from competing feedbacks that scale differently with number. We then use this idea to find overlooked critical numbers in past studies of collective behavior and explore the implications for their conclusions. In particular, we highlight how deterministic approximations of stochastic models can fail near critical numbers. We close by distinguishing these qualitative changes from density-dependent phase transitions and by discussing how our approach could generalize to broader classes of collective behaviors.