森重文-茨万齐格形式主义：正交动力学方程弱解的存在性证明
Mori-Zwanzig formalism: An existence proof for weak solutions of the orthogonal dynamics equation
浏览论文内容
中文总结 AI 辅助
本文将吉冯等人对茨万齐格投影弱解存在性的证明推广到非平稳非哈密顿系统，该系统时间演化由拟收缩半群给出，建立了增长界与正则解唯一性，并以阻尼谐振子为例应用于茨万齐格投影。
中文摘要 AI 辅助
在经典统计物理学中，森重文-茨万齐格投影算子技术用于推导随机变量（可观测量）的广义朗之万方程。标准推导隐含地假设了所谓正交动力学方程解的存在性以及常数变易公式（戴森恒等式）的有效性。吉冯、哈尔德和库普弗曼指出，对于像茨万齐格投影这样的无限秩投影，存在性是一个微妙问题。作者在平稳哈密顿系统的背景下证明了茨万齐格投影弱解的存在性。本文将吉冯等人的存在性证明推广到时间演化由拟收缩半群给出的非平稳非哈密顿系统，建立了增长界以及充分正则解（若存在）的唯一性，最后以阻尼谐振子为例将结果应用于茨万齐格投影。
英文摘要
In classical statistical physics, the Mori-Zwanzig projection operator technique is used to derive generalized Langevin equations for a random variable (observable). Standard derivations implicitly assume the existence of solutions to the so-called orthogonal dynamics equation as well as the validity of the variation of constants formula (Dyson identity). It was pointed out by Givon, Hald and Kupferman that the existence is a subtle issue for infinite-rank projections such as Zwanzig's projection [D. Givon, O. H. Hald, R. Kupferman, Israel Journal of Mathematics, 145 (221-241), 2005]. The authors proved the existence of weak solutions for Zwanzig's projection in the context of stationary Hamiltonian systems. To this date, this is the only existence proof that allows for an infinite-rank projection, whereas the uniqueness and regularity remain open problems. In this article, we generalize the existence proof by Givon et al. to nonstationary non-Hamiltonian systems whose time evolution is given by a quasicontraction semigroup. We establish growth bounds as well as the uniqueness for sufficiently regular solutions (if existent). Finally, we apply our results to Zwanzig's projection using the damped harmonic oscillator as an example.
