平均场极限下全局耦合映射的可观测李雅普诺夫指数

中文摘要 AI 辅助

大型耦合系统可能在微观层面是混沌的，而在宏观层面却表现出稳定的集体行为。区分这两种不稳定性形式一直是物理学文献中的一个问题，并且已经提出了几种方法。在本文中，我们通过引入并研究可观测李雅普诺夫指数来解决这个问题，该指数衡量的是通过选定可观测函数（而非全相空间）观察时，初始条件扰动的增长或衰减速率。对于全局耦合映射，我们考虑有限时间可观测李雅普诺夫指数，并通过先取无穷系统极限再取长时间极限来定义其平均场对应物。我们的主要结果表明，对于一大类对称宏观可观测函数，这些指数由自洽传递算子（控制总体分布演化的非线性算子）的线性化决定。然后，我们将此结果应用于弱耦合一致扩张映射，并证明在平稳平均场状态附近，它们的平均场李雅普诺夫指数为负。因此，尽管微观动力学是混沌的，扰动在宏观层面却会衰减。我们的结果为区分大型相互作用系统中的微观混沌与宏观混沌提供了一个严格的框架。

英文摘要

Large coupled systems may be chaotic at the microscopic level while exhibiting stable collective behaviour at the macroscopic level. Distinguishing between these two forms of instability has been a problem in the physics literature, and several approaches have been proposed. In this paper, we address this problem by introducing and studying \emph{observable Lyapunov exponents}, which measure the rate at which perturbations of initial conditions grow or decay when viewed through a chosen observable, rather than in the full phase space. For globally coupled maps, we consider finite-time observable Lyapunov exponents and define their mean-field counterpart by first taking the infinite-system limit and then the long-time limit. Our main result shows that, for a broad class of symmetric macroscopic observables, these exponents turn out to be determined by the linearisation of the self-consistent transfer operator, which is the nonlinear operator governing the evolution of the population distribution. We then apply this result to weakly coupled uniformly expanding maps and prove that their mean-field Lyapunov exponents are negative near a stationary mean-field state. Thus, although the microscopic dynamics is chaotic, perturbations decay at the macroscopic level. Our results provide a rigorous framework for distinguishing microscopic from macroscopic chaos in large interacting systems.

发表机构

• King’s College London（伦敦国王学院）

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