# Projection Operators and Memory Function Formalism ### Books * Zwanzig-Nonequilibrium Statistical Mechanics * Memory Functions, Projection Operators, and the Defect Technique (Lecture Notes in Physics) * (Advances in Chemical Physics) - Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62(1985). * Boon, Yip - Molecular Hydrodynamics (Chapter 1, 2) * Projection Operator Techniques in the theory of fluctuations by Bruce Berne in Modern Theoretical Chemistry. v.5 Statistical Mechanics, Part B Time-dependent processes * Dieter Forster.-Hydrodynamic fluctuations, broken symmetry, and correlation functions * Chapter 11 in Bruce J. Berne, Robert Pecora - Dynamic Light Scattering ### Papers * [Ottinger - General projection operator formalism for the dynamics and thermodynamics of complex fluids](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.57.1416) * [Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach_2013](https://doi.org/10.1007/s10955-013-0726-8) * [Derivation of Transport Equations Using the Time-Dependent Projection Operator Method_2002](https://doi.org/10.1143/PTP.107.525) * [Mode-Coupling Theory of the Glass Transition: A Primer_2018](https://doi.org/10.3389/fphy.2018.00097) * [Transport coefficients of non-Newtonian fluid and causal dissipative hydrodynamics_2008](https://doi.org/10.1103/PhysRevE.78.051107) * [A mode coupling theory for Brownian particles in homogeneous steady shear flow_2009](https://sor.scitation.org/doi/10.1122/1.3119084) * [Mori–Zwanzig formalism as a practical computational tool_2009](https://doi.org/10.1039/D1SM00521A) * [Many chain correlated dynamics in polymer fluids_1999](https://doi.org/10.1063/1.478660) * [On the non-stationary generalized Langevin equation_2017](https://doi.org/10.1063/1.5006980) * [The modified Langevin description for probes in a nonlinear medium_2016](https://doi.org/10.1088/1361-648X/29/6/064004) * [Equation of motion for coarse-grained simulation based on microscopic description_2007](https://doi.org/10.1103/PhysRevE.75.051109) * [Microscopic derivation of particle-based coarse-grained dynamics_2013](https://doi.org/10.1063/1.4795091) * [Extracting macroscopic dynamics: model problems and algorithms_2004](https://doi.org/10.1088/0951-7715/17/6/R01) * [Mori-Zwanzig theory for dissipative forces in coarse-grained dynamics in the Markov limit_2017](https://doi.org/10.1103/PhysRevE.95.013303) * [Microscopic derivation of particle-based coarse-grained dynamics: Exact expression for memory function_2017](https://doi.org/10.1063/1.4978572) * [Microscopic derivation of coarse-grained, energy-conserving generalized Langevin dynamics_2019](https://doi.org/10.1063/1.5096655) * [Systematic derivation of hybrid coarse-grained models_2019](https://doi.org/10.1103/PhysRevE.99.013303) * [Bottom-up approach to represent dynamic properties in coarse-grained molecular simulations_2018](https://doi.org/10.1063/1.5064369) * [The multi-dimensional generalized Langevin equation for conformational motion of proteins_2019](https://doi.org/10.1063/1.5055573) * [Construction of dissipative particle dynamics models for complex fluids via the Mori–Zwanzig formulation_2014](https://doi.org/10.1039/C4SM01387E) * [Perspective: Dissipative particle dynamics_2017](https://doi.org/10.1063/1.4979514) * [Hybrid description of complex molecules_2009](https://doi.org/10.1209/0295-5075/88/40008) * [Contraction of State Variables in Non-Equilibrium Open Systems. I_1980](https://doi.org/10.1143/PTP.63.1865) * [Generalized Langevin equation for nonequilibrium systems_2001](https://doi.org/10.1016/S0378-4371(01)00328-4) * [Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians_2019](https://doi.org/10.1103/PhysRevE.99.062118) * [Fluctuation–dissipation relations far from equilibrium: a case study_2021](https://doi.org/10.1039/D1SM00521A) * [Application of the projection operator formalism to non-Hamiltonian dynamics_2011](https://doi.org/10.1063/1.3530071) * [Computing generalized Langevin equations and generalized Fokker–Planck equations_2009](https://doi.org/10.1073/pnas.0902633106) * [Generalized Langevin Equation: An Introductory Review for Biophysicists](https://doi.org/10.1142/S1793048019300019) * [Mode-coupling theory](https://doi.org/10.1088/1742-5468/2005/05/P05013) * [Phys. Rev. E 104, 024121 (2021) - Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium (aps.org)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.104.024121)
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Gaussian process inference of stochastic differential equations

Approximate Gaussian process inference for the drift function in stochastic differential equations Phys. Rev. E 98, 022109 (2018) - Approximate Bayes learning of stochastic differential equations (aps.org) Gaussian Process Approximations of Stochastic Differential Equations (mlr.press) Moment-Based Variational Inference for Stochastic Differential Equations (mlr.press) Variational Inference for Stochastic Differential Equations - Opper - 2019 - Annalen der Physik - Wiley Online Library Sparse Gaussian Processes for Stochastic Differential Equations | OpenReview

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Awesome Fluid Mechanics

A curated list of repositories related to fluid dynamics. Please send pull requests or raise issues to improve this list. Contents Educational Notebooks Lecture Series

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Shapash

🎉 What's new ? :warning: Starting from Shapash v2.0.0: '.compile()' parameters must be provided in the SmartExplainer init: :warning: For clearer initiation, method's parameters must be provided at the init: xpl = SmartExplainer(model, backend, preprocessing, postprocessing, features_groups) instead of xpl.compile(x, model, backend, preprocessing, postprocessing, features_groups) Version New Feature Description

5/7/2022
Stochastic fast-slow systems

The evolution of a physical system goverened by a nonlinear differential equation often occurs on two time scales when the parameters affecting the dynamics is small or large. This separation of time scales vastly simplifies their study and is a fundamental requirement to all sorts of perturbation techniques in the parameter space that one could employ. The subclass of linear systems on the hand can be readily generalized to the case of linear partial differential equations like the Fokker-Planck or forward Kolmogorov equations, Euler/Burgers equation using Mori-Zwanzig's projection operator formalism which has the effect integrating out the fast degrees of freedom of the system (e.g. solvent) and projecting this averaging effect onto the slow variables (e.g. time evolution of the spatial coordinate of a protein molecule) whose dyamics now acquires a memory kernel plus noise. The appearance of the memory kernel makes the overall dynamics non-Markovian. The projection operator formalism is a very powerful technique which leads to reduced order modelling of the system where one only tracks (or models) the dynamics of the slow variables with clear separation of two time scales. More details on this formalism can be found in the following excellent presentation: {%youtube e8QFNh5u_1U %} For a detailed introduction to slow-fast dynamical systems take a look at Nils Berglund, Barbara Gentz - Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach. Many real systems, however, possess a continuum of time scales, with no clear separation. In the multiscale analysis of such systems one is concerned with the bridging of disparate scales by merging different mathematical models appropriate at different scales (such as quantum, molecular, and continuum). Finally one looks for a phenomenological description of the phenomena under study that is valid at all scales using the more powerful renormalization group theory. This paper discusses all these powerful theoretical concepts beautifully.

5/4/2022
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