Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems

A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

We provide a simple hypocoercivity analysis for the effective Mori-Zwanzig equation governing the time evolution of noise-averaged observables in a stochastic dynamical system. Under the hypocoercivity framework mainly developed by Dolbeault, Mouhot and Schmeiser and further extended by Grothaus and Stilgenbauer, we prove that under the same conditions which lead to the geometric ergodicity of the Markov semigroup $e^{tK}$, the Mori-Zwanzig orthogonal semigroup $e^{tQKQ}$ is also geometrically ergodic, provided that $P=I-Q$ is a finite-rank, orthogonal projection operator in a certain Hilbert …

Faber逼近Mori-Zwanzig方程

Faber approximation to the Mori-Zwanzig equation

本文提出基于正交动力传播算子级数展开的新有效逼近方法，利用Faber级数对线性动力系统实现R超线性收敛的渐近最优逼近，并应用于随机波传播和贝叶斯晶格及任意拓扑图上的谐振子相互作用。

Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations

We develop a thorough mathematical analysis of the effective Mori-Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon recent work on hypoelliptic operators, we prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to a unique equilibrium state which admits an explicit representation. We apply the new theoretical results to the Langevin dynamics of a high-dimensional particle system with smooth interaction potential.

关于Mori-Zwanzig记忆积分的估计

On the estimation of the Mori-Zwanzig memory integral

本文基于Mori-Zwanzig（MZ）公式，开发了严格估计和可证明收敛的近似方法，研究了短记忆近似、t模型和分层（有限记忆）近似的误差界和收敛条件，并提出了可计算的MZ记忆积分上界，用于估计MZ记忆对动力学的贡献。

Combinatorial Mori-Zwanzig Theory

We introduce a combinatorial version Mori-Zwanzig theory and develop from it a family of self-consistent evolution equations for the correlation function or Green's function of interactive many-body systems. The core idea is to use an ansatz to rewrite the memory kernel (self-energy) of the regular Mori-Zwanzig equation as a function composition of the correlation (Green's) function. Then a series of algebraic combinatorial tools, especially the commutative and noncommutative Bell polynomials, are used to determine the exact Taylor series expansion of the composition function. The resulting co…

Nakajima-Mori-Zwanzig公式中的对偶和条件期望

Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation

我们开发了一个新的运营商代数制定的Nakajima-Mori-Zwanzig（NMZ）方法的投影。新理论建立在严格的数学基础上，它可以应用于经典和量子系统。我们表明，NMZ制定之间的可观测量的空间和状态空间中的对偶原理可以建立，类似于海森堡和薛定谔图片在量子力学。基于这种对偶性，我们证明了在自然假设下，出现在NMZ方程中的投影算子一定是条件期望。在各种实施例中说明了所提出的制剂。