标题： 全局耦合映射中的热力学极限的大偏差 显示英文标题

标题： Large deviations from the thermodynamic limit in globally coupled maps

摘要： 具有大量N个全局耦合映射的系统已成为高维动力学的相对简单的原型，显示出同步、簇形成和多稳态等许多有趣和典型的现象，并在约瑟夫森结阵列或生物物理模型等系统中具有潜在的应用。 存在大量关于全局耦合映射的数值研究。 尽管在无限N极限下对这类系统宏观行为的解释已经取得了很大进展，但当N趋近于无穷大时，仍需要关于有限-N系统渐近行为的坚实理论。 本文介绍了一种方法，可以通过该方法获得长期偏离热力学极限行为的渐近估计。 该方法基于准势的概念，最初由Freidlin、Wentzell和其他人开发，用于描述小随机扰动对动力系统长期行为的影响。 讨论了在当前背景下显式计算准势的问题以及可能的近似方案。 本文描述的所有概念都通过一个简单示例进行说明。 显示英文摘要

摘要： Systems of a large number N of globally coupled maps have become popular as a relatively simple prototype of high-dimensional dynamics, showing many interesting and typical phenomena like synchronisation, cluster formation and multistability, and having potential applications in systems like Josephson junction arrays or in biophysical models. There exists a wealth of numerical investigations of globally coupled maps. While much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit of infinite N, there is still need for a sound theory about the asymptotic behaviour of finite-N systems as N approaches infinity. This article introduces a method by which it is possible to obtain asymptotic estimates for long-term deviations from the thermodynamic limit behaviour. This method is based upon the concept of quasipotentials, originally developed by Freidlin, Wentzell, and others for describing the influence of small random perturbations on the long-term behaviour of dynamical systems. The problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed. All the concepts described in this article are illustrated with a simple example.