投影算子形式在非Hamilton动力学中的应用
Application of the projection operator formalism to non-Hamiltonian dynamics
摘要：
从不完整或噪声数据中重构运动方程和降维是多自由度动力系统研究中的两个基本问题。对于后者广泛的努力，但有限的成功，推广Zwanzig-Mori投影形式主义，最初开发的哈密顿系统接近热力学平衡，一般非哈密顿系统缺乏详细的平衡。这种系统引入的一个困难是缺乏定义统计分布所需的不变测度。基于最近发现的一组随机微分方程定义的非Hamilton系统可以映射到一个Hamilton系统，我们开发了这样的一般投影形式主义。在所得到的广义朗之万方程中，一组广义涨落耗散关系将记忆核和随机噪声项连接起来，类似于服从细致平衡的哈密顿系统。这些关系的缺乏限制了以前的应用推广Langevin形式主义。研究结果可为进一步开展减少自由度的模型重建技术提供理论依据。我们首先使用一个解析可解的例子来说明形式主义和波动耗散关系。我们对一个具有终产物抑制的化学网络的数值实验进一步证明了该方法的有效性。我们认为，形式主义可以找到广泛的应用在科学建模。具体来说，我们讨论了生物网络的潜在应用。特别是，该方法提供了一个合适的框架，获得洞察网络属性，如鲁棒性和参数可转移性。
英文摘要：
Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter extensive efforts have been made but with limited success to generalize the Zwanzig-Mori projection formalism, originally developed for Hamiltonian systems close to thermodynamic equilibrium, to general non-Hamiltonian systems lacking detailed-balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-Hamiltonian system defined by a set of stochastic differential equations can be mapped to a Hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation-dissipation relations connect the memory kernel and the random noise terms, analogous to Hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation-dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability.
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