The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton–Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables.

中文翻译：

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从 GENERIC 到 GENERIC 的动态失配减少的推广

失拟统计归约首先由布鲁斯·特金顿 (Bruce Turkington) 开发和制定，是一种将 Liouville 方程用于概率密度（详细级别）并将其转换为投影量的简化动态（不太详细级别）的通用方法。本文对该方法进行了推广。哈密​​顿刘维尔方程被任意哈密顿演化结合梯度动力学（GENERIC）代替，玻尔兹曼熵被任意熵代替，动能被任意能量代替。梯度部分是由耗散势产生的广义梯度动力学。投影状态变量的简化演化显示为保留原始（详细级别）演化的 GENERIC 结构。耗散势通过求解 Hamilton-Jacobi 方程获得。