低维地图的物理遍历性和精确响应关系

L. Rondoni, G. Dematteis
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引用次数: 4

摘要

近年来,为了寻找涨落关系的物理相关公式和推导,引入了新的遍历概念。这些概念随后被用于时间连续确定性动力学的一般响应理论的发展。该理论的关键组成部分是耗散函数Ω,在非平衡系统中,物理兴趣可以用能量耗散率来识别,并用于精确确定相空间中系整体的演化。与基于物理上难以捉摸的相空间变化率的(广义)Liouville方程的标准解相比,这是一种进步。在此框架下产生的响应理论侧重于可观测值,而不是动力学的细节和相空间上的平稳概率分布。特别是,这个理论不依赖于度量传递性,这相当于标准遍历性。它依赖于初始平衡的性质,在初始平衡中,系统在被扰动离开该状态之前被发现。这个理论是精确的,不局限于线性响应,它适用于所有的动力系统。此外,它还为系综的松弛(如通常的响应理论)以及单个系统的松弛提供了必要和充分条件。我们将连续时间理论扩展到时间离散系统,我们用简单的映射来说明我们的结果,并将它们与其他最新的理论进行比较。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Physical Ergodicity and Exact Response Relations for Low-dimensional Maps
Recently, novel ergodic notions have been introduced in order to find physically relevant formulations and derivations of fluctuation relations. These notions have been subsequently used in the development of a general theory of response, for time continuous deterministic dynamics. The key ingredient of this theory is the Dissipation Function Ω, that in nonequilibrium systems of physical interest can be identified with the energy dissipation rate, and that is used to determine exactly the evolution of ensembles in phase space. This constitutes an advance compared to the standard solution of the (generalized) Liouville Equation, that is based on the physically elusive phase space variation rate. The response theory arising in this framework focuses on observables, rather than on details of the dynamics and of the stationary probability distributions on phase space. In particular, this theory does not rest on metric transitivity, which amounts to standard ergodicity. It rests on the properties of the initial equilibrium, in which a system is found before being perturbed away from that state. This theory is exact, not restricted to linear response, and it applies to all dynamical systems. Moreover, it yields necessary and sufficient conditions for relaxation of ensembles (as in usual response theory), as well as for relaxation of single systems. We extend the continuous time theory to time discrete systems, we illustrate our results with simple maps and we compare them with other recent theories.
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