The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.

中文翻译：

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具有广泛连续特征长度和时间的系统的多重缩放：纳米复合材料的结构转变

N 体系统的多尺度方法被推广以解决与集体行为相关的长时间和长度尺度的广泛连续体。基于不可数的时间变量集和指定系统主要特征的顺序参数 (OP) 的概念开发了一种技术。我们采用这种观点作为常用离散时间尺度和 OP 集的自然扩展，这在只有少数、广泛分离的尺度存在时是实用的。这种系统（在准平衡条件下）的时间尺度谱中存在的间隙被用来引入连续尺度并执行刘维尔方程的多尺度分析。为傅立叶分量 OP 的连续谱的随机动力学导出了函数微分 Smoluchowski 方程。还导出了 OP 的空间非局部朗之万方程的连续体。该理论是通过分析复合材料中的结构转变来证明的，就像病毒衣壳和分子回路一样。