text:
F1 road cars: performance cars from the 2021 Formula One teams tested
We put the performance products of seven Formula One teams through their paces on track
by: Auto Express
It looks like 2021 is set to be a golden year for Formula One. Seven of the 10 entrants on the grid will be fully manufacturer-backed, while four – Ferrari, Honda, Mercedes and Renault – will supply the power units for the entire grid.
Each wants to become world champion, not only for the thrill of competition, but in order to forge links between motorsport success and the road cars that we can buy. We've lined up seven performance cars from those constructors to see how deep the associations lie.
The 5 BIG questions for Formula One's new 2021 season
Alpine A110 Colour Edition
Engine: 1.8-litre 4cyl turbo
Power: 288bhp
Transmission: Seven-speed auto, rear-wheel drive
0-62mph: 4.4 seconds
While the Alpine name and colours are new to the Formula One grid for 2021, behind the scenes stands a well established operation. Alpine's parent company Renault has rebranded the squad that formerly took its own name to reflect its performance car ambitions both on road and on track.
But while it's a relative newcomer to F1, Alpine has enjoyed success in other motorsport disciplines for decades. The original A110, which inspired the design of the striking modern-day model, was a star of rallying, winning the World Championship for Manufacturers in 1973. More recently, Alpine has enjoyed success in the LMP2 category of the World Endurance Championship.
Much like the F1 car, Alpine's current one-model road-car range uses Renault power; behind the driver of the A110 is a 1.8-litre four-cylinder turbo engine from the Mégane hot hatch. Allied to a lightweight, rear-driven chassis, this coupé offers one of the purest driving experiences available today.
Mercedes-AMG GT 4-Door Coupe
Price: £145,495
Engine: 4.0-litre V8 twin-turbo
Transmission: Nine-speed auto,four-wheel drive
Of all the manufacturers competing in F1, it falls to the brand that has dominated the sport since 2014 to give us mere mortals the closest idea yet of what Lewis Hamilton gets up to in his day job.
Due later this year, the AMG One hypercar uses an evolution of the hybrid V6 turbo power unit that has thrust Mercedes to every driver and constructors' title for the last seven years. To make the car more driveable on the road, the 1.6-litre V6's idle speed will drop, as will the rev limit to improve longevity, although it still stands at a heady 11,000rpm. The hybrid battery size will increase, too, giving the One a fully electric range of 16 miles.
The current big-hitters in the mainstream AMG range, such as the GT 4-Door Coupé here, make use of twin-turbocharged V8 power. And if you ever wonder about the size of the car Lewis Hamilton will aim to take his record-breaking eighth world title this year, its two-metre width and five-metre length are almost identical to the AMG GT 4-Door Coupé.
0-62mph: 2.85 seconds
Top speed: 211mph+
With 238 race wins, 758 podiums, 228 pole positions and 16 constructors' World Championships, by almost any measure Ferrari is the most successful team in F1 history. The Scuderia is the only constructor to have competed in every season of the championship since its inception in 1950, and in 2020 it passed 1,000 races entered, fittingly, on home soil in Mugello.
It's that rich history and wealth of success that have given Ferrari the know-how that can be transferred to its road cars. Past links have often been very close between road and race: the iconic Ferrari F50's 4.7-litre V12 was an evolution of the 3.5-litre unit used by the 641 grand prix car which came so close to sealing the 1990 driver's title at the hands of Alain Prost. Like the F1 car, the F50's unit was also bolted directly to the carbon chassis, acting as a structural member of the car.
More recently, Ferrari's knowledge of hybrid energy-recovery systems come from its competitive exploits. The same goes for its launch and traction control tech, plus the steering wheel-mounted Manettino drive selector found on cars like the Pista.
McLaren's Formula One team is riding a wave of rejuvenation. In 2020 it scooped third in the Constructors' World Championship – its best result since 2012. Things are looking up for 2021, too; a switch to Mercedes engines and the signing of the highly rated Daniel Ricciardo could make the coming year more exciting still.
The road car division is riding a similar wave. The 720S remains a thrilling supercar, while its new little brother, the Artura, sports an advanced V6 plug-in hybrid powertrain that shows the future direction of the brand.
One area where McLaren's road cars maintain a link to the race cars is in the steering systems – albeit for different reasons. The production division has stuck with hydraulic power steering because it believes it gives the driver more detailed feedback than any fully electrical set-up.
The F1 machinery maintains a hydraulic set-up because electric power steering – which could be developed into an adaptive driver aid – has been banned by the sport since the early 1990s.
Transmission: Eight-speed auto, rear-wheel drive
The 2021 Formula One season will see Aston Martin return to the grid for the first time since the 1960s. The new AMR21 hits the track with iconic green livery and four-time world champion Sebastian Vettel behind the wheel, alongside Canadian Lance Stroll. Lance's father and Aston Martin owner Laurence Stroll has ambitious plans for the team: World Championship glory within five years.
The road car program has received a recent boost with the DBX SUV, but will soon see a motorsport flagship arrive in the form of the Valkyrie. The mid-engined hypercar features a naturally-aspirated V12 from another iconic F1 name, Cosworth, while the wild chassis, aerodynamics package and extreme F1-style driving position are the work of one of Formula One's most successful engineers, Red Bull's Adrian Newey.
But we won't just see Astons racing on the grid in 2021; they will be keeping hold of proceedings, too. The Vantage coupé and DBX will fill the roles of safety and medical car duties respectively, alternating each race with Mercedes-AMG products.
Engine: 3.5-litre V6 twin-turbo/triple electric motors
Transmission: Nine-speed auto, four-wheel drive
Until Mercedes releases the ultra-exclusive AMG One to the public, it's arguable that Honda, power unit supplier of 2020 championship runner-up Red Bull and sister team Alpha Tauri, gives us mere mortals the closest link between F1 track and the road.
The dramatic shape of the hi-tech NSX houses a power unit which, on paper, features a similar configuration to the current F1 power unit regulations: both use a V6 turbo hybrid unit mounted just behind the driver. The similarities end there, though; the road car's 3.5 V6 has over double the F1 car's capacity, it sports an extra turbo, and has three electric motors (two on the front axle) to give four-wheel-drive traction. It's a combination that works to devastating effect on the road; 0-62mph takes just 2.9 seconds.
While Honda has announced that it will officially be pulling out of F1 at the end of the 2021 season, a development freeze from 2022 means that the two energy drink-backed outfits can continue to run the Japanese power units until new power unit rules come into force for 2025.
Alfa Romeo Giulia Quadrifoglio
While many now view Alfa Romeo as Italy's second F1 manufacturer, it boasts a heritage that dates back even further than Ferrari's.
Indeed, Enzo Ferrari himself raced for – and then managed – Alfa Romeo's motorsport programme back in the 1920s and 1930s before setting up his own outfit. Alfa Romeo won the first two Formula One Constructors' World Championships in 1950 and 1951, with drivers' world titles secured by Giuseppe Farina and Juan Manuel Fangio respectively.
Fast forward to 2021, and the links between Ferrari and Alfa Romeo remain – both on road and track. The Alfa Romeo C41 sources its power unit and gearbox from Ferrari, while on the road, the Alfa Romeo Giulia Quadrifoglio's marvellous V6 engine was developed with extensive input from the Prancing Horse. Step up to the extreme Giulia GTA-M and the F1 links continue; wind tunnel development of the aggressive aerodynamics package was conducted by the Sauber team that runs Alfa's F1 operations.
The other road car links on the F1 grid
While Williams is a customer of Mercedes-AMG and has used its power units since 2014, it hasn't designed and built any road cars from scratch like the seven major players. There are road car links, however: in 2013 Williams Advanced Engineering entered an agreement with Nissan to offer "specialist expertise of aerodynamics, simulation and material science", a collaboration which resulted in the extreme upgrade package for the GT-R Nismo. More recently it has worked on Aston Martin's electric Rapide E and the Lotus Evija.
Haas, meanwhile, holds a similar relationship with Ferrari to the deal Williams has with Mercedes; the US machine-tooling builder uses as many components from Ferrari as the rules allow. Since it joined the sport in 2016, it has relied upon Dallara for the design and construction of its chassis, and the Italian brand produced an extreme lightweight road car offering, the Stradale, back in 2018.
Crazy F1 special editions
Every once in a while, car manufacturers like to highlight their ties to Formula One through special editions and concepts – with some links more bizarre than others. Here are some of our 'favourites.
Fiat Seicento Schumacher
Power: 53bhp
0-62mph: 13.1 secs
A cute city car doesn't seem like a fitting tribute to a seven-time world champion. Similarities to the German's F1 Ferrari start – and end – with the fact that both need to be driven flat out everywhere.
Infiniti FX Vettel Edition
0-62mph: 5.6 secs
Perhaps a slightly more fitting tribute to a German world champion than the Seicento, Sebastian Vettel's name was applied to a special FX SUV. Carbon- fibre extras and a limited run of 150 units resulted in a £108,000 price tag.
Honda Civic Jordan
The Jordan F1 team – a distant relative of the team now known as Aston Martin – used Honda engines from 1998 to 2003. This special-edition Civic VTi-S was finished in the team's striking yellow paint finish.
Mercedes A 45 AMG Petronas 2015 WC Edition
Mercedes and partner Petronas celebrated consecutive F1 world titles with a special A 45 featuring a teal-coloured livery. It boasted the same power as the standard car, though.
Check out our 2021 F1 season preview to get the full lowdown on the cars, drivers and teams...
New Aston Martin Vantage F1 Edition 2021 review
The new Aston Martin Vantage F1 Edition is the brand's first car under new ownership, and it bodes well for the future
New Alpine A110 facelift launched with two new 296bhp variants
Alpine has released even more power from the new A110's engine, cutting the 0-62mph time to 4.2 seconds
Next Mercedes-AMG GT to be fixed roof only and more track-biased
Details have emerged of the next-generation Mercedes-AMG GT, which will not feature a Roadster variant due to new SL
Best used sports cars 2021
Best cars & vans
Our top used sports cars for those looking for a fun but usable second-hand car


text:
Sviatoslav Dziadykevych (born August 6, 1995) is a Ukrainian footballer who plays as a forward.

Playing career
Dziadykevych began at the academy level with FC Karpaty Lviv in 2012. In 2016, he played with FC Chortkiv in the Ukrainian Amateur Football League. He later signed with Nyva Ternopil in the Ukrainian Second League. In 2019, he played abroad in the Canadian Soccer League with FC Ukraine United.

References

1995 births
Living people
Ukrainian footballers
FC Krystal Chortkiv players
FC Nyva Ternopil players
FC Ukraine United players
Canadian Soccer League (1998–present) players
People from Zboriv
Association football forwards
Ukrainian Second League players
Sportspeople from Ternopil Oblast

redpajama_set_name: RedPajamaWikipedia


The world's largest passenger liner has just docked in Southampton after sailing in from France in advance of its inaugural voyage to Rotterdam.
It is a vast ship boasting a crew of 2,100 crew. It can host 6,780 guests.
From a plant lover's perspective, the Harmony of the Seas takes parks and gardens to a new level. On board there is an impressive 10,587 plants, 48 vine plants, and 52 trees.
I'm supposing that there are also huge beds and borders with soil and compost deep enough to support the roots of these plants?
What a great initiative. Imagine sitting under the shade of a tree as this vast boat crosses the equator. Pretty unique, I would assume.
But what about bio security? In today's ever challenging environment, growers and gardeners are constantly warned of the risks of contamination to our native plants from pests and diseases hosted on plant matter imported from abroad.
The vessel, owned by Royal Caribbean, will spend its life travelling the high seas, docking at exotic locations around the world. Every time it docks there's the potential for a parasitic threat - whether that's an insect or a pathogen - to be passed from shore to ship, and vice versa. Any potential contamination then has the ability to go wherever the ship goes.
Under normal circumstances one would imagine insects and bacteria dying very quickly without a suitable host. For example if a ship's guest wandered around a park or garden in Egypt, any pathogen picked up on a shoe is likely not to survive without a suitable host to incubate it.
But if any pathogen, during its transit from shore to ship, has an opportunity to jump onto a suitable host i.e. a tree or plant, then there's a possibility that that pathogen will thrive on its new host.
I can see many opportunities for potential plant diseases to reach land, at many ports of call, and infect local flora and fauna.
Defra publishes advisory information on the transit of plants into the United Kingdom.
DEFRA says: Plant health legislation controls the import and movement of certain plants, seeds and organic matter - such as soil - and certain plant products, including fruit, potatoes, vegetables, cut flowers, foliage and grain.
Controls differ according to the species - and whether or not they are classified as quarantine organisms - but could include the need for classification, a phytosanitary certificate, a plant passport and/or inspection requirements.
DEFRA is very clear that plants, soil and seedlings from all non-EU countries is tightly controlled or strictly forbidden.


By popular demand, PTS is proud to release our latest E-newsletter which will now be published every other month rather than once per quarter. With the volume of exciting news and new data center related solutions, the additional opportunities to provide mission critical news to our clients is welcomed. As mentioned in our May newsletter, PTS continues to expand upon our unique position as an integrated facility and IT consultancy by providing useful information for both disciplines.
Facility – Facility Business Strategy Consulting: Our latest strategic new service can be succinctly stated as "To COLO Or Not To COLO". PTS' new Data Center Facility Business Strategy Consulting Solution supports a strategic philosophy under consideration by many business and operational managers — when the time comes does it make sense to expand your existing facility, build a new facility, or move to a colocation facility? Make sure to check this section out.
IT – Storage & Data Deduplication: In a continuing effort to provide best-in-class storage and ancillary solution, we are highlighting two exciting products in our IT Solutions Showcase. The Compellent Fluid Data Storage and the ExaGrid Systems Disk-based Backup solutions are excellent enterprise-level solutions which achieve enhanced storage utilization and scalability and ultimately reduce overall storage-related operational and management costs in the data center.
In addition to the Solutions Showcases, we continue to release interesting new white papers and are busy with quotes in various online and print media. Visit the PTS white paper download section to learn more about new white papers and click here to see what's new for PTS in the news. The news section also highlights several exciting upcoming events including a Data Center Consolidation, Storage & Data Protection seminar in September co-sponsored with Compellent and ExaGrid.
Everyday we come across statements such as "Will We Still Be Building Data Centers in 10 Years?" or "If Your Data Center Space is Less Than 20K Square-Feet You Should be in a COLO Facility". Some of what you read in the trade journals and blogs is true in terms of costs, reliability, etc. However, it is critical to fully understand the impact of a "make versus buy" or, in this case, build/expand versus colocate decision.
PTS has recently launched a Data Center Facility Business Strategy Consulting offering which leverages our experience in data center analysis, surveys, design, build, commissioning, and colocation to support clients searching for an answer to the question "To COLO Or Not To COLO"!
Although there are a number of variables involved in a decision, alas the ultimate decision whether to expand, build new, or migrate to a colocation facility comes down to two basic strategic questions: 1. What is the philosophy within the enterprise for ownership? Specifically, where does the control lie? Are business managers comfortable with potentially losing control of mission critical resources? 2. How does the organization want to finance its IT operations? In other words, would it be better to invest in facility capital to build or expand existing facilities or would the enterprise rather expense or "rent" space?
The following diagram shows the Network Design Alternatives and the impact of control versus expense. PTS provides detailed analysis, investigation, financial justification, and strategic thought to help business managers make the best decision for their organizations.
To learn more about PTS Data Center Facility Business Strategy Consulting Click Here.
PTS continues to build upon our Storage & Data Protection Consulting Services aimed at providing the right storage solution for our client's needs. Therefore, the IT-side of this month's Solutions Showcase provides an overview of our newest storage and data deduplication solutions.
Compellent Fluid Data Storage solutions and ExaGrid Systems Disk-Based Backup with Dedeplication solutions add additional depth to the PTS storage portfolio.
With a powerful data movement engine, intelligent software applications, and an open, agile hardware platform, Fluid Data storage is an enterprise-class solution which actively and intelligently manages data at a more granular level to cut cost, time, and risk.
The Fluid Data storage intelligent software applications enables enterprises of all sizes to move beyond simply storing data to actively, intelligently managing data. Powerful network storage software with built-in intelligence and automation optimizes the provisioning, placement and protection of data throughout its lifecycle.
To learn more about Fluid Data Storage, Click Here.
The six core ExaGrid disk-based backup appliances include GRID computing software which allows them to virtualize into one another when plugged into a switch. As a result, any of the six appliance models can be mixed and matched into a single GRID system with full backup capacities up to 100TB (6 PB logical). Once virtualized, they appear as a single pool of long-term capacity. Capacity load balancing of all data across servers is automatic, and multiple GRID systems can be combined for additional capacity. Even though data is load-balanced, deduplication occurs across the systems so that data migration does not cause a loss of effectiveness in deduplication.
Cost-Effective and Flexible Solution with No "Forklift" Upgrades – no need to over-buy storage capacity up front – modular systems are easily combined in a virtualized GRID to smoothly scale up for larger capacities as needed with no painful "forklift" upgrades.
To learn more about the ExaGrid EX Series, Click Here.
Check out the latest white papers added to our website! We're sure there will be some interesting content for your reading pleasure.
For up-to-the-minute educational information don't forget to check out the PTS Blog – Data Center Design and consider joining the LinkedIn Computer Room Design Group. We do all we can to keep up-to-date on the latest industry trends and look to share this information with you!
Power Distribution Units (PDUs): Power monitoring and environmental monitoring to improve uptime and capacity planning (by Raritan). Corporate America is more dependent than ever on computing resources to handle everything from mission-critical financial transaction applications to storing vast amounts of patient information. Data center facilities are under increasing pressure to flawlessly add significant, reliable computing power whether the data center is experiencing rapid growth, consolidating from multiple locations or trying to extract more capacity out of an existing space.
This white paper explores the challenges of increasing computing power in both an inflexible and resource-constrained physical space, as well as centers that are experiencing high growth. Raritan also explores how companies can employ emerging support technology – like intelligent power distribution units (PDUs) – to help manage server uptime and capacity planning. Click here for the full white paper.
Economizer Fundamentals – Smart Approaches to Energy-Efficient Free-Cooling for Data Centers (by Emerson Network Power). The strategic business importance of today's data center requires a more complex and thoughtful approach to environmental control planning than ever before. Intelligent control to maintain the data center environment within acceptable ranges for temperature and humidity is essential for efficient data center operation, integral to protecting the significant financial investment in computer and network technology, and key to preventing downtime that can cost millions and drive away customers.
Economizers – with their promise of "free-cooling" – are attracting much attention for their ability to reduce energy usage. Economizers also lessen wear and tear on precision cooling equipment and lower operating costs. However, when considering the use of an economizer system, care must be taken to avoid introducing new problems that result in achieving the goal of reducing energy consumption at the expense of data center availability. Click here for the full white paper.
Leveraging a Persistent Hardware Architecture (by Compellent). Many enterprises looking to increase the flexibility, improve the scalability, enhance the availability and reduce the costs of data storage have turned to storage area networks (SANs). Yet traditional hardware vendors have failed to deliver on the promise of SANs. By relying on proprietary hardware designs, imposing restrictions on the types of technology that can be used and prohibiting customers from mixing and matching different drive and interconnect technologies, these vendors doom their products to early obsolescence and force their customers into costly forklift upgrades.
This white paper reviews the new Compellent Fluid Data storage design which uses an open, non-proprietary hardware architecture. This architecture enables businesses to continuously integrate new disk drive, switch, and network interface technologies without having to replace existing hardware or software. This persistent hardware architecture allows administrators to configure storage systems using a combination of select industry-standard components for optimum flexibility, scalability and availability. Click here for the full white paper.
Optimizing Backup and Data Protection in Virtualized Environments (by ExaGrid Systems). The promise of maximizing IT investments while minimizing complexity has resulted in widespread adoption of server virtualization. In fact, VMware users are now deploying nearly half of their new servers as virtual servers, rather than as physical servers. That said, rapid adoption of server virtualization has brought with it a variety of challenges in backup and data protection.
While server virtualization does result in additional backup and data protection challenges, many of these challenges can be mitigated via a disk-based backup solution with data deduplication. There are a variety of these types of solutions in the market, however, each with their own approach. This white paper analyzes the disk-based backup and data deduplication segment and provides readers with specific items for consideration prior to purchase. Click here for the full white paper.
Failure to manage is often the elephant in the room when troubleshooting data center problems. Asking six data center managers to define infrastructure management is a sure way to get six different answers. It is similar to the ancient story of the six blind men and the elephant. When each of the men approached the elephant, he came to the conclusion that the elephant was like a wall, a spear, a snake, a tree, a fan, and a rope. In the end, they were all wrong. Just as an elephant is more than the sum of its parts, a true data center management solution is more than the sum of the various infrastructure management tool sets.


text:
\section{Introduction}\label{sec:introduction}
In computational biology, fluid-dynamics engineering and soft matter physics the dynamics of complex systems is often modelled by means of the Langevin equation \cite{snook2006,zwanzig2001nonequilibrium,parish2017}. The basic idea of this approach is to simulate a set of "relevant" degrees of freedom while the effect of the remaining degrees of freedom is treated by means of friction terms and stochastic forces.
Projection operator formalisms are a set of methods that allow to derive these effective equations of motion, in principle, rigorously from the underlying microscopic dynamics \cite{Mori_65,Zwanzig_61,grabert2006projection,Vrugt2020,Hijon_2010,chorin:2000,ottinger2002}. After having been a very active research field from the 1960s to the 1980s, followed by three decades of somewhat lesser research activity, recently the development of models and of computer simulation methods by means of projection operator formalisms has regained much in popularity again \cite{espanol2009,MeyerVoigtmann_2017,Charbonneau2018,Krommes2018,vogel2020,izvekov2021,ayaz2021,jung2022,vroylandt2022,Ayaz2022,vrugt2022}. In particular, the use of time-dependent projection operators to model the dynamics of systems which are far from equilibrium has moved into the focus of the soft matter modelling community \cite{schilling2022,MeyerVoigtmann_2019,Izvekov,Kawai_2011,vrugt2019}.

In this article we discuss the the non-stationary, linear, generalized Langevin equation, i.e.~the type of Langevin equation which one obtains when applying a linear ("Mori-type"), time-dependent projection operator to microscopic dynamics with an explicitly time-dependent Liouvillian \cite{MeyerVoigtmann_2019}. In previous work, we have shown how to extract the non-stationary memory kernel of this equation from experimental data  or data from computer simulations \cite{Meyer_2020,Meyer_Wolf}. Here we will show how to replace the fluctuating force by a set of random processes such that the dynamics generated by the resulting stochastic Langevin equation corresponds to the dynamics of the deterministic non-stationary, linear, generalized Langevin on the level of moments of a given order. I.e.~we show how to construct a coarse-grained model for a system out of equilibrium that reproduces the dynamics of a given set of observables.

\section{Generalized Langevin equation}\label{sec:gle}
We briefly recall how to derive the generalized Langevin equation from microscopic dynamics, which are governed by a time-dependent Liouvillian \cite{Zwanzig_61, Mori_65,grabert2006projection,MeyerVoigtmann_2017,MeyerVoigtmann_2019,schilling2022}. As an example, the reader could imagine a polymer melt under external mechanical driving, for which one would like to construct a coarse-grained model based on results from an atomistic simulation. Here, we show the case of real-valued, vectorial observables (e.g.~the positions of united atoms in the case of the polymer melt). The complex-valued case is discussed in appendix \ref{sec:complex_observables}.

We introduce a phase space observable $\vec{A}:\Gamma\to\mathbb{R}^n$ and a phase space trajectory $\gamma(t;t_0,\Gamma_{t_0})$ with $\gamma(t_0;t_0,\Gamma_{t_0})=\Gamma_{t_0}$, where $\Gamma$ is some point in phase space and the parameters on the right-hand side of the semicolon denote the initial values of the trajectory. (Note that all derivations and results in this article hold true for explicitly time-dependent observables, too. This can be seen using the concept of an augmented phase space and following the line of arguments presented in ref.~\cite{MeyerVoigtmann_2019}.) The stream lines are denoted by $\dot{\Gamma}(t,\Gamma)$, where $\dot{\gamma}(t;t_0,\Gamma_{t_0})=\dot{\Gamma}(t,\gamma(t;t_0,\Gamma_{t_0}))$. Further, the time evolution operator $\mathcal{U}(t,t_0)$ is introduced as
\begin{equation}\label{eq:time_op}
	\vec{A}_t:=\mathcal{U}(t,t_0)\vec{A}:= \vec{A}(\gamma(t;t_0,.)) \, .
\end{equation}
By taking the time derivative on both sides and introducing the Liouville operator $\mathcal{L}(t):=\dot{\Gamma}(t,.)\cdot \partial_\Gamma$, one obtains the formal solution
\begin{equation*}
\mathcal{U}(t,t_0) := \exp_-\left( \int^t_{t_0}\text{d}\tau \mathcal{L}(\tau) \right) \, ,
\end{equation*}
where $\exp_-$ denotes the negatively time-ordered exponential, see appendix \ref{app:gle1}.

Further, we introduce the time-dependent cross-correlation matrix $(\vec{X},\vec{Y})_t$, the Mori projection operator $\mathcal{P}(t)$, and its orthogonal complement $\mathcal{Q}(t)$
\begin{align}
	\scalar{\vec{X}}{\vec{Y}}_t &:= \int \text{d}\Gamma \, \rho(t,\Gamma) \vec{X}(\Gamma)  \vec{Y}^\top(\Gamma) \, , \\
	\mathcal{P}(t)\vec{X}&:=\scalar{\vec{X}}{\vec{A}}_t \cdot \scalar{\vec{A}}{\vec{A}}_t^{-1}\cdot \vec{A}\label{eq:MoriProjector} \, , \\
	\mathcal{Q}(t)&:=1-\mathcal{P}(t) \, ,
\end{align}
where $\rho(t,\Gamma)$ denotes the phase space density. Note that the ordering of the matrices and arguments of the cross-correlation in \cref{eq:MoriProjector} is uniquely fixed by the idempotence of $\mathcal{P}(t)$ as opposed to the one-dimensional case, refs.~\cite{MeyerVoigtmann_2017,
MeyerVoigtmann_2019}.
Further, $\scalar{\vec{A}}{\vec{A}}_t$ in \cref{eq:MoriProjector} must be invertible.
The cross-correlation matrix satisfies
\begin{equation*}
    	\scalar{\vec{X}}{\vec{Y}}_t= \scalar{\mathcal{U}(t,t')\vec{X}}{\mathcal{U}(t,t')\vec{Y}}_{t'} \, ,
\end{equation*}
and is reduced to a genuine scalar product in the one-dimensional case.

The equation of motion for the trajectory $\vec{A}_t$ is obtained by applying the Dyson-Duhamel identity to the orthogonal contribution of the observable. The final result is the \textit{linear non-stationary generalized Langevin equation} (nsGLE)
\begin{align}
	\vec{\dot{A}}_t &=\mat{\omega}(t)\vec{A}_t + \int^t_{{t_0}}\text{d} s\,\mat{K}(t,s) \vec{A}_s \, +  \vec{\eta}_{t{t_0}}\, ,  \label{eq:gle2}
\end{align}
where the \textit{drift} $\mat{\omega}(t)$, the \textit{fluctuating forces} $\vec{\eta}_{ts}$ and the \textit{memory kernel} $\mat{K}(t,s)$ are defined by
\begin{align}
	\mat{\omega}(t) &:= \scalar{\mathcal{L}(t)\vec{A}}{\vec{A}}_t\scalar{\vec{A}}{\vec{A}}^{-1}_t \, , \label{eq:drift}\\
	\mat{K}(t,s) &:= -\scalar{\vec{\eta}_{ts}}{\vec{\eta}_{ss}}_s\scalar{\vec{A}}{\vec{A}}^{-1}_s
	  \label{eq:definitionMemoryKernel}\, ,\\
	\vec{\eta}_{ts} &:= \mathcal{Q}(s)\mathcal{G}_-(t,s)\mathcal{L}(t)\vec{A} \, , \label{eq:ff1}\\
	 	\mathcal{G}_-(t,s) &:= \exp_-\left(\int^t_{s}\text{d}\tau \mathcal{L}(\tau)\mathcal{Q}(\tau) \right) \, ,\label{eq:ff2}
\end{align}
see appendix \ref{app:gle} for details.
By construction, the cross-correlation between the fluctuating force and the observable vanishes, i.e.
\begin{equation}
	\scalar{\vec{\eta}_{tt'}}{\vec{A}}_{t'} = \mat{0} \, . \label{eq:mean_f_A}
\end{equation}
With this, the memory kernel may be written as
\begin{equation}\label{eq:fdt}
	\mat{K}(t,s) = -\scalar{\vec{\eta}_{tt'}}{\vec{\eta}_{st'}}_{t'}\scalar{\vec{A}}{\vec{A}}^{-1}_s \, ,
\end{equation}
where $t'$ may be chosen arbitrarily. This relation is known as the generalized or second \textit{fluctuation-dissipation theorem}\cite{grabert2006projection}.

Finally, by inserting eq.~(\ref{eq:gle2}) into the time derivative, choosing $t_0:=t'$, and applying eq.~(\ref{eq:mean_f_A}), we obtain the equations of motion for the auto-correlation function
\begin{align}
	\mat{C}(t,t')&:=\scalar{\vec{A}_t}{\vec{A}_{t'}}_{t_0}  \, ,\label{eq:def_two_time_correlation} \\
	\frac{\text{d}}{\text{d} t}\mat{C}(t,t') &= \mat{\omega}(t)\mat{C}(t,t') + \int^t_{t'}\text{d} s\, \mat{K}(t,s) \mat{C}(s,t') \, . \label{eq:eom_ac}
\end{align}

Nordholm and Zwanzig pointed out in ref.~\cite{nordholm1975} that powers of the same observable are an interesting exemplary set of observables for \cref{eq:gle2}. Let $X$ denote the observable of interest and choose $\vec{A}=\left(X,X^2,\cdots,X^m\right)^\top$ with $m\in\mathbb{N}$. Then the nsGLE for the first component reads
\begin{align}
  \label{eq:nonlinear}
	\frac{\text{d} X_t}{\text{d} t} &= \sum\limits_{i=1}^m\left(\mat{\omega}_{1,i}(t)X_t^i +\int\limits_{t'}^t\text{d}\tau\,\mat{K}_{1,i}(t,\tau)X_\tau^i\right) + \left(\boldsymbol{\vec{\eta}}_{1}\right)_{t't}.
\end{align}
Hence, one obtains an nsGLE containing drift and memory terms, which are non-linear in the observable. The terms may be regarded as a Taylor expansion in the observable and, thus, may provide a way to connect the exact equation of motion, \cref{eq:nonlinear}, to Landau Ginzburg models \cite{chaikin1995}, phase field models \cite{steinbach2009} and related approximative approaches that use Langevin dynamics in free energy landscapes, which are expressed as a series expansion in powers of the order parameter. We give a detailed discussion about equations of motions with a potential of mean force and (non-)linear memory in the observable in ref.~\cite{Glatzel_2021}. Here, we will not elaborate this special case further and continue with the derivation of general relations between the terms in the nsGLE.\\

\section{Calculation of the Memory Kernel}
Differentiating \cref{eq:eom_ac} with respect to $t'$ yields a Volterra integral equation for the memory kernel $\mat{K}(t,.)$ for each time $t$,
\begin{align*}
	\mat{K}(t,t') &= \mat{\omega}(t)\scalar{\vec{A}_t}{\vec{\dot{A}}_{t'}}_{t_0} \scalar{\vec{A}}{\vec{A}}^{-1}_{t'}-\scalar{\vec{\dot{A}}_t}{\vec{\dot{A}}_{t'}}_{t_0} \scalar{\vec{A}}{\vec{A}}^{-1}_{t'} \\
	& \quad + \int^t_{t'} \text{d} s \, \mat{K}(t,s)\scalar{\vec{A}_s}{\vec{\dot{A}}_{t'}}_{t_0}  \scalar{\vec{A}}{\vec{A}}^{-1}_{t'} \, .
\end{align*}
For continuous coefficient functions, there exists a unique continuous solution which can be solved by a Picard iteration \cite{burton}. Here, the numerical calculations of the memory kernel follow the algorithm described in ref.~\cite{Meyer_2020,Meyer_Wolf}. However, in these references, the algorithm is presented for a single, scalar-valued observable only, thus, the following adaptations to the case of vectors should be noted:\\
First, as mentioned in \cref{sec:gle}, the order of the individual terms in the projector (\cref{eq:MoriProjector}) is interchanged to get a proper projector for multiple observables. For a single observable the two definitions coincide.\\
Second, the matrices in ref.~\cite{Meyer_2020} are replaced with block matrices, where the individual blocks contain the corresponding quantities for the multiple observables. For example, because molecular dynamics simulations and their results are discrete in time, the (sampled) two-time correlation function (\cref{eq:def_two_time_correlation}) for a single observable can be represented as a matrix, where the row and column specify the two times of the correlation (cf. ref.~\cite{Meyer_2020}). Now, each of these values is replaced by the correlation matrix for multiple observables at the corresponding times. If $\mmat{C}$ denotes the block matrix for the two-time correlation function, we can write it in the following form
\begin{align}
	\mmat{C} &= \begin{pmatrix}
		\mat{C}(0,0) & \cdots&\mat{C}(0,T)\\
		\vdots & \ddots &\vdots\\
		\mat{C}(T,0)&\cdots&\mat{C}(T,T)
	\end{pmatrix}.
\end{align}
Third, the splitting into triangular matrices, as used in ref.~\cite{Meyer_2020}, is done only on the block level. For example, if we want to use the upper-right triangular block matrix of $\mmat{C}$, the blocks on the diagonal ($\mat{C}(0,0)$ etc.) are either kept or discarded (depending on the convention) as a whole.\\

\section{Generalized Langevin dynamics simulations}\label{sec:numerical_method}
The aim of a coarse-graining procedure is to obtain a model, which allows to generate the dynamics of an observable by directly solving the corresponding Langevin equation rather than simulating the underlying microscopic system.
Hence the typical tasks when coarse-graining, are first to determine the drift and memory kernel based on a given set of data from simulations or experiments, and then to generate a random noise, which mimicks the flucuating force.  \\
In this section, we analyze how to simulate generalized Langevin dynamics based on a given data set such that the simulations are statistically equivalent to the original data up to a given order on the level of moments. The main results are necessary and sufficient criteria.
\\
Once the drift $\mat{\omega}(t)$ and memory kernel $\mat{K}(t,s)$ are evaluated from a given data-set, we obtain the fluctuating forces $\vec{\eta}_{tt_0}(\Gamma_{t_0})$ for each trajectory $\vec{A}_t(\Gamma_{t_0})$ with initial value $\vec{A}(\Gamma_{t_0})$ from the nsGLE, eq.~(\ref{eq:gle2}). Vice versa, by integrating the nsGLE with the given fluctuating forces and initial value, we recover the trajectory $\vec{A}_t(\Gamma_{t_0})$.

Hence, for a given drift $\mat{\omega}(t)$ and kernel $\mat{K}(t,s)$, the phase space density $\rho({t_0},.)$ defines a function of time $t$ and initial value $\Gamma_{t_0}$ via,
\begin{equation}
	\vec{X}:\, (t,\Gamma_{t_0})\mapsto \vec{X}_t(\Gamma_{t_0})=(\vec{\eta}^\top_{tt_0}(\Gamma_{t_0}),\vec{A}^\top_{t_0}(\Gamma_{t_0}))^\top \, , \label{eq:stochastic_process}
\end{equation}
 which uniquely determines the initial values and time-evolution of the trajectory $\vec{A}_t(\Gamma_{t_0})$.
In an abstract mathematical sense, $\vec{X}$ is a \textit{continuous-time stochastic process}, where the sample space is the phase space \cite{lamperti1977,bremaud2014}. As a function of solely the sample space, one may consider $\vec{X}$ to be a \textit{random function} that maps some random initial value $\Gamma_{t_0}$ onto a function of time.

 We seek to simulate further trajectories that have dynamics "similar to the dynamics of $\vec{X}$".
 To this end, we construct a stochastic process $\vec{X}'$ in order to replace eq.~(\ref{eq:stochastic_process}), which approximates the true dynamics on the level of a given set of moments. The first step is to classify all stochastic processes of the form
\begin{equation}
	\vec{X}':\, (t,\Lambda)\mapsto \vec{X}'_t(\Lambda)=(\vec{\eta}'^\top_{tt_0}(\Lambda),\vec{A}'^\top_{t_0}(\Lambda))^\top \, ,
	\label{eq:stochastic_process2}
\end{equation}
that will preserve the mean and auto-correlation function of the trajectories, where $\Lambda$ is the outcome of some sample space specifying the fluctuating forces $\vec{\eta}'_{tt_0}(\Lambda)$ as well as the initial value $\vec{A}'_{t_0}(\Lambda)$. The resulting trajectory obtained by numerical integration is denoted by $\vec{A}'_t(\Lambda)$.

\subsection{Reproducing of the First and Second Moment of the Observables}\label{sec:second_order}

Let us begin with the mean value $\langle \vec{A}_t \rangle$, where $\langle . \rangle$ denotes the expected value of some random variable. For given functions $\mat{\omega}(t)$ and $\mat{K}(t,s)$, the mean value $\langle \vec{A}_t\rangle$ solves the following system of Volterra integro-differential equations
\begin{equation*}
\left. \begin{cases}
\frac{d}{dt} \langle \vec{A}_t \rangle = \mat{\omega}(t)\langle \vec{A}_t\rangle + \int^t_{t_0} ds \, \mat{K}(t,s)\langle \vec{A}_s \rangle + \langle \vec{\eta}_{tt_0} \rangle  \\
\text{initial value: }\, \langle \vec{A}_{t_0} \rangle
\end{cases}  \right\} \, .
\end{equation*}
For each initial value $\langle \vec{A}_{t_0}\rangle$, these kind of equations possess a unique solution for any continuous functions $\langle \vec{\eta}_{tt_0}\rangle$, $\mat{\omega}(t)$ and $\mat{K}(t,s)$, see ref.~\cite{burton} for details.

If we intend to use \cref{eq:gle2} to generate new trajectories, we use the original drift matrix $\mat{\omega}(t)$ and memory kernel matrix $\mat{K}(t,\tau)$ and  randomly draw the initial values of the observables $\vec{A}'_{t_0}$ and the values of the fluctuating force during the process $\vec{\eta}'_{tt_0}$. From the considerations above we know that the mean values of the generated trajectories equal the original ones, $\langle\vec{A}'_t\rangle\equiv\langle\vec{A}_t\rangle$, if and only if $\langle\vec{A}'_{t_0}\rangle = \langle\vec{A}_{t_0}\rangle$ and $\langle\vec{\eta}'_{tt_0}\rangle\equiv\langle\vec{\eta}_{tt_0}\rangle$. In shorthand notation, these two conditions can also be written as  $\langle\vec{X}'_t\rangle\equiv\langle\vec{X}_t\rangle$.

\begin{widetext}
Now let us continue in the same fashion with the auto-correlation function. Inserting the equations of motion (\cref{eq:gle2}) into the time derivative of the auto-correlation function yields again a system of Volterra differential equations:
\begin{align}
&\left. \begin{cases}
\frac{d}{dt} \langle \vec{A}_t  \vec{A}_s^\top \rangle = \mat{\omega}(t)\cdot \langle \vec{A}_t  \vec{A}_s^\top \rangle + \int^t_{t_0} dr \, \mat{K}(t,r)\cdot\langle \vec{A}_r  \vec{A}_s^\top \rangle + \langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle\\
\text{initial value: }\, \langle \vec{A}_{t_0}   \vec{A}_s^\top \rangle
\end{cases}  \right\} \label{eq:ac1}
\end{align}
Here, the dynamics of $\langle \vec{A}_t  \vec{A}_s^\top \rangle$ for arbitrary but fixed $s$ are completely determined by the correlation function $\langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle$ and the initial value $\langle \vec{A}_{t_0}   \vec{A}_s^\top \rangle$. We want to understand under which conditions of our simulation a) the correlation function $\langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle$ and b) the initial value $\langle \vec{A}_{t_0}   \vec{A}_s^\top \rangle$ (for arbitrary $s$) show the same behavior as in the original simulations of the microscopic system:
\begin{itemize}
    \item[a)] We start with the correlation function $\langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle$ and derive an equation of motion for it by calculating the derivative with respect to $s$ and using the nsGLE \cref{eq:gle2} again. The result reads:
\begin{align}
&\left. \begin{cases}
\frac{d}{ds}\langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle = \langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle \cdot \mat{\omega}(s)^\top + \int^s_{t_0} dr\,  \langle \vec{\eta}_{tt_0} \vec{A}_r^\top \rangle \cdot \mat{K}(t,r)^\top + \langle \vec{\eta}_{tt_0}  \vec{\eta}_{st_0}^\top\rangle\\
\text{initial value: }\, \langle \vec{\eta}_{tt_0}  \vec{A}_{t_0}^\top \rangle
\end{cases}  \right\} \label{eq:ac2}
\end{align}
Hence, the dynamics of $\langle \vec{\eta}_{tt_0}  \vec{A}_s^\top \rangle$ for fixed $t$ is, again, completely determined by the initial value $\langle \vec{\eta}_{tt_0}  \vec{A}_{t_0}^\top \rangle$ and the two-time correlation of the fluctuating force $\langle \vec{\eta}_{tt_0}  \vec{\eta}_{st_0}^\top\rangle$.

\item[b)] Next, we ask under which conditions the initial value $\langle \vec{A}_t   \vec{A}_s^\top \rangle\rvert_{t={t_0}}$ (for arbitrary $s$) has the same form as in the original microscopic simulations. Again, we use a time derivative with respect to $s$ and the nsGLE  and obtain:
\begin{align}
&\left. \begin{cases}
\frac{d}{ds}\langle \vec{A}_{t_0}  \vec{A}_s^\top \rangle = \langle \vec{A}_{t_0}  \vec{A}_s^\top \rangle\cdot \mat{\omega}(s)^\top + \int^s_{t_0} dr\, \langle \vec{A}_{t_0} \vec{A}_r^\top \rangle\cdot \mat{K}(t,r)^\top + \langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top\rangle\\
\text{initial value: }\, \langle \vec{A}_{t_0}  \vec{A}_{t_0}^\top \rangle
\end{cases}  \right\} \label{eq:ac3}\, .
\end{align}
Here we can see that $\langle \vec{A}_{t_0}  \vec{A}_s^\top \rangle$ is completely determined by the initial value $\langle \vec{A}_{t_0}  \vec{A}_{t_0}^\top \rangle$ and the correlation function $\langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top\rangle$.
\end{itemize}
\end{widetext}

Let us summarize the results above in the following way:\\
If the dynamics of our observables are described by an equation of the structure of the nsGLE (\cref{eq:gle2}), the two-time correlation functions are uniquely determined if $\langle \vec{A}_{t_0}  \vec{A}_{t_0}^\top\rangle$, $\langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top \rangle$, and $\langle \vec{\eta}_{tt_0} \vec{\eta}_{st_0}^\top\rangle$ are given for all times $t$ and $s$. In compact form, these three correlation functions can be expressed as $\langle \vec{X}_t\vec{X}_s^\top\rangle$. Here, $\langle \vec{A}_{t_0}  \vec{A}_{t_0}^\top\rangle$ and $\langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top \rangle$ determine $\langle \vec{A}_{t_0}  \vec{A}_s^\top\rangle$ via \cref{eq:ac3} and similar $\langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top \rangle$ and $\langle \vec{\eta}_{tt_0} \vec{\eta}_{st_0}^\top\rangle$ determine $\langle \vec{\eta}_{tt_0} \vec{A}_s^\top\rangle$ via \cref{eq:ac2}. With all these quantities uniquely determined, also the two-time correlation functions $\mat{C}(t,s)=\langle \vec{A}_t \vec{A}_s^\top\rangle$ are uniquely determined through \cref{eq:ac1}. The special case where $\langle \vec{A}_{t_0}  \vec{\eta}_{st_0}^\top\rangle\equiv 0$, as is the case for the nsGLE with  \cref{eq:mean_f_A}, is discussed in \cref{sec:simplified_procedure}.

We conclude that new trajectories generated with the nsGLE (\cref{eq:gle2}) also reproduce the exact two-time auto-correlation functions, $\langle\vec{A}'_t\vec{A}'_s\rangle\equiv\langle\vec{A}_t\vec{A}_s\rangle$, if and only if $\langle\vec{X}'_t\vec{X}'_s\rangle\equiv\langle\vec{X}_t\vec{X}_s\rangle$. These results may be used to simulate trajectories that obey the same statistics up to second order as we will illustrate in \cref{sec:simplified_procedure}.

\subsection{Reproducing Higher Moments of the Observables}\label{sec:general_procedure}
In section \ref{sec:second_order}, we have seen that the simulated trajectories obey the same statistics up to second order, if and only if the stochastic process $\vec{X}'$ from \cref{eq:stochastic_process2} preserves the first and second moment of the true process $\vec{X}$ from \cref{eq:stochastic_process}. In fact, the same line of argumentation applies to arbitrary orders. To this end we define the $m$-th moment by
\begin{align}
	\mat{C}(t_1,\dots,t_m)&:=\langle \vec{A}_{t_1}\otimes\dots\otimes \vec{A}_{t_m}\rangle \, .
\end{align}

\begin{widetext}
Again, we use the nsGLE (\cref{eq:gle}) to express the $m$-th moment by the unique solution of an initial value problem. Then, we recursively repeat the procedure until all terms are determined by the $m$-th moment of the random process $\vec{X}$. We obtain
\begin{align}
&\left. \begin{cases}
\frac{d}{dt_1} \mat{C}(t_1,\dots,t_m) = \langle \vec{\eta}_{t_1t_0} \otimes \vec{A}_{t_2} \otimes \cdots \otimes \vec{A}_{t_m} \rangle + (\mat{\omega}(t_1) [\otimes\mat{\mathbf{1}}]^{m-1} )\cdot \mat{C}(t_1,\dots,t_m) + \\
\qquad + \int^t_{t_0} ds \, (\mat{K}(t,s)[\otimes\mat{\mathbf{1}}]^{m-1})\cdot \mat{C}(s,t_2,\dots,t_m)  \\
\text{initial value: }\, \langle \vec{A}_{{t_0}}\otimes \vec{A}_{t_2}\otimes\dots\otimes \vec{A}_{t_m}\rangle
\end{cases} \right\} \, . \label{eq:M_th_moment_1}
\end{align}
As before, we can show that the solution to this integro-differential equation is uniquely determined by the following set of integro-diffential equations:
\begin{align}
\left.\begin{cases}
\frac{\text{d}}{\text{d} t_2} \langle \vec{X}_{t_1}\otimes \vec{A}_{t_{2}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle = \langle \vec{X}_{t_1}\otimes\vec{\eta}_{t_{2}t_0}\otimes \vec{A}_{t_{k+1}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle + \\
\qquad+ (\mat{\mathbf{1}}\otimes\mat{\omega}(t_2) [\otimes\mat{\mathbf{1}}]^{m-2}) \cdot \langle \vec{X}_{t_1}\otimes\vec{A}_{t_{2}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle + \\
\qquad+  \int^t_{t_0} ds \, (\mat{\mathbf{1}}\otimes \mat{K}(t_k,s) [\otimes\mat{\mathbf{1}}]^{m-2} )\cdot \langle \vec{X}_{t_1}\otimes \vec{A}_{s}\otimes \vec{A}_{t_{3}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle  \\
\text{initial value: }\, \langle \vec{X}_{t_1}\otimes \vec{A}_{{t_0}}\otimes \vec{A}_{t_{2}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle
\end{cases} \right\} \, , \label{eq:M_th_moment_2}
\end{align}
This notation allows us to write the generalization of both \cref{eq:ac2,eq:ac3} in the single expression of \cref{eq:M_th_moment_2}. If $m=2$, \cref{eq:ac2,eq:ac3} are actually equivalent to \cref{eq:M_th_moment_2}.\\
Now, we can do the same procedure iteratively and after $k$ iterations we obtain:
\begin{align}
\left.\begin{cases}
\frac{d}{dt_k} \langle \vec{X}_{t_1}\otimes\cdots\otimes \vec{X}_{t_{k-1}}\otimes \vec{A}_{t_{k}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle = \langle \vec{X}_{t_1}\otimes\cdots\otimes \vec{X}_{t_{k-1}}\otimes \vec{\eta}_{t_{k}t_0}\otimes \vec{A}_{t_{k+1}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle + \\
\qquad+ ([\mat{\mathbf{1}}\otimes]^{k-1} \mat{\omega}(t_k) [\otimes\mat{\mathbf{1}}]^{m-k} )\cdot \langle \vec{X}_{t_1}\otimes\cdots\otimes \vec{X}_{t_{k-1}}\otimes \vec{A}_{t_{k}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle + \\
\qquad+  \int^t_{t_0} ds \, ([\mat{\mathbf{1}}\otimes]^{k-1} \mat{K}(t_k,s) [\otimes\mat{\mathbf{1}}]^{m-k} )\cdot \langle \vec{X}_{t_1}\otimes\cdots\otimes \vec{X}_{t_{k-1}}\otimes \vec{A}_{s}\otimes \vec{A}_{t_{k+1}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle  \\
\text{initial value: }\, \langle \vec{X}_{t_1}\otimes\cdots\otimes \vec{X}_{t_{k-1}}\otimes \vec{A}_{{t_0}}\otimes \vec{A}_{t_{k+1}}\otimes \cdots \otimes \vec{A}_{t_m} \rangle
\end{cases} \right\} \, , \label{eq:M_th_moment_k}
\end{align}
\end{widetext}

In principle, we can now recursively solve \cref{eq:M_th_moment_k} for $k=m,m-1,\dots,1$ in order to obtain the unique solution $\mat{C}(t_1,\cdots,t_m)$, if we are given the $m$-th moment of the original dynamics $\langle \bigotimes^m_{k=1} \vec{X}_{t_k}\rangle$ for all times. Hence, if and only if the $m$-th moments of $\vec{X}'_t$ in the stochastic process equal the original $m$-th moments of $\vec{X}_t$, the $m$-th moments of the observables are reproduced. More explicitely,
\begin{equation*}
	\left\langle \bigotimes^m_{k=1}\vec{A}'_{t_k}\right\rangle \equiv  \left\langle \bigotimes^m_{k=1}\vec{A}_{t_k}\right\rangle \Leftrightarrow \left\langle \bigotimes^m_{k=1}\vec{X}'_{t_k}\right\rangle \equiv  \left\langle \bigotimes^m_{k=1}\vec{X}_{t_k}\right\rangle \, ,
\end{equation*}
for any $m\in\mathbb{N}_+$. Therefore, if we choose $\vec{X}'$ such that $\left\langle \bigotimes^m_{k=1}\vec{X}'_{t_k}\right\rangle \equiv  \left\langle \bigotimes^m_{k=1}\vec{X}_{t_k}\right\rangle$ for all times and $m=1,\dots,M$, we obtain simulations that obey the same statistics up to $M$-th order by numerical integration of the nsGLE (\ref{eq:gle2}) for any given functions $\mat{\omega}(t)$ and $\mat{K}(t,s)$.

\subsection{Simplified procedure}\label{sec:simplified_procedure}

While the general procedure is independent from the construction of the projection operator formalism, we can now exploit \cref{eq:mean_f_A} in order to bypass statistical dependencies between the fluctuating forces and initial conditions within the second order approximation.

Let us demand $\langle \vec{A}_{t_0}\rangle=0$, i.e. we subtract the average initial value $\vec{A}_t\to \vec{A}_t-\langle \vec{A}_{t_0}\rangle$, which is easily reversed after the numerical integration is carried out. Further, we assume that $\vec{\eta}'_{tt_0}$ and $\vec{A}'_{t_0}$ are uncorrelated random variables. According to \cref{eq:mean_f_A}, we have $\langle \vec{\eta}_{st_0}  \vec{A}_{t_0}^\top\rangle\equiv0$. Hence, if $\langle \vec{A}'_{t_0}\rangle =0$, we automatically satisfy
\begin{equation*}
\langle \vec{\eta}'_{st_0} \vec{A}'^\top_{t_0}\rangle = \langle \vec{\eta}'_{st_0}\rangle  \langle \vec{A}'^\top_{t_0}\rangle = 0 = \langle \vec{\eta}_{st_0} \vec{A}^\top_{t_0}\rangle \, ,
\end{equation*}
for all times $s$. From this, we conclude the following statement. Let $\langle \vec{A}_{t_0}\rangle=0$ and let $\vec{\eta}'_{tt_0}$, $\vec{A}'_{t_0}$ be uncorrelated random variables. Then, $\langle \vec{A}'_t\rangle\equiv \langle \vec{A}_t\rangle$ and $\langle \vec{A}'_t \vec{A}'^\top_s\rangle\equiv\langle \vec{A}_t \vec{A}_s^\top\rangle$, if and only if $\langle \vec{\eta}'_{tt_0}\rangle\equiv \langle \vec{\eta}_{tt_0}\rangle $, $\langle \vec{A}'_{t_0}\rangle=0$ (or $\langle \vec{\eta}'_{tt_0}\rangle\equiv0$), $\langle \vec{\eta}'_{tt_0} \vec{\eta}'^\top_{st_0}\rangle\equiv \langle \vec{\eta}_{tt_0} \vec{\eta}^\top_{st_0}\rangle $, and $\langle \vec{A}'_{t_0}  \vec{A}'^\top_{t_0} \rangle=\langle \vec{A}_{t_0}  \vec{A}^\top_{t_0} \rangle$.

This suggests the following procedure. Assume we are given a set of trajectories $\vec{B}_t$. First, we compute $\vec{A}_t := \vec{B}_t-\langle \vec{B}_{t_0}\rangle$ for all trajectories. Afterwards, we determine $\mat{\omega}(t)$, $\mat{K}(t,s)$, $\langle \vec{A}_{t_0}  \vec{A}^\top_{t_0} \rangle$, $\langle \vec{\eta}_{tt_0}\rangle$ and $\langle \vec{\eta}_{tt_0} \vec{\eta}_{st_0}^\top \rangle$. For each simulation $\vec{A}'_t$, we draw the initial value $\vec{A}'_{t_0}$ from any distribution with mean zero and second moment $\langle \vec{A}_{t_0}  \vec{A}^\top_{t_0} \rangle$. Then, we integrate the equations of motion, where the fluctuating forces $\vec{\eta}'_{tt_0}$ are drawn from any distribution with mean $\langle \vec{\eta}_{tt_0}\rangle$ and auto-correlation function $\langle \vec{\eta}_{tt_0} \vec{\eta}_{st_0}^\top \rangle$. In the end, the desired simulations are given by $\vec{B}'_t = \vec{A}'_t+\langle \vec{B}_{t_0}\rangle$. Numeric evidence for the necessity of subtracting the mean in the beginning and adding it again in the end is presented later in \cref{ssec:exampleSingleObservable} (see \cref{fig:short_simulations_no_offset_comparison,fig:short_simulations_with_offset_comparison}). There, we also discuss a case where such a procedure is not necessary (see \cref{fig:long_simulations_comparison}).

\section{Numerical Results for Exemplary System}
In this section we will discuss algorithmic/numerical details in \cref{ssec:generatingTrajectories}, introduce a model system in \cref{ssec:exemplarySystem}, present some results for a single observable to illustrate the importance of the addition/subtraction of the mean in the beginning/end in \cref{ssec:exampleSingleObservable}, show the results of a simulation with two observables in \cref{ssec:exampleTwoObservables}, and show an example of how this method can be used to generate new trajectories for system parameters where no original data is available in \cref{ssec:interpolationSchemes}.

\subsection{Generating Trajectories with the Non-Stationary Generalized Langevin Equation}\label{ssec:generatingTrajectories}
In order to simulate trajectories with the nsGLE (\cref{eq:gle2}), we need to generate fluctuating forces with a certain mean $\langle\vec{\eta}_{tt_0}\rangle$ and auto-correlation function $\langle\vec{\eta}_{tt_0}\vec{\eta}_{st_0}\rangle$.
As the simulations are discrete in time, we apply the numerical method to the fluctuating forces $\vec{\eta}_{tt_0}(\Gamma_{t_0})$, where $t=1,\dots,N$ enumerates $N$ moments in time on an equidistant grid. As we care only about the first two moments in this section, a multi-dimensional Gaussian distribution is the natural choice, which is also easy to handle from a computational perspective. For non-stationary processes, we have to allow for arbitrary mean values and covariances, thus, the fluctuating forces are drawn from a $(nN)$-dimensional Gaussian distribution
\begin{align*}
	\vec{\eta}&:=(\vec{\eta}^\top_{1t_0},\dots,\vec{\eta}^\top_{Nt_0})^\top(\Gamma_{t_0}) \in \mathcal{N}(\vec{\mu},\mat{\Sigma}) \, , \\
	\vec{\mu} &= \langle \vec{\eta} \rangle  \, , \\
	\mat{\Sigma} &= \langle (\vec{\eta}-\vec{\mu})(\vec{\eta}-\vec{\mu})^\top\rangle \, .
\end{align*}
In order to obtain the correct distribution of inital values of the observables, the initial values $\vec{A}_{t_0}$ for the $i$-th simulation are simply taken from the $i\% N_{\text{data}}$-th trajectory from the given set of $N_{\text{data}}$ original trajectories. The numerical integration is carried out by the classical Runge-Kutta method with $dt=2$, where we use the Simpson rule to compute the integral from the nsGLE. Odd times are restored using the Verlet scheme after each Runge-Kutta step at the cost of one order in accuracy, i.e. the local error is in $O(1/N^4)$, assuming that $\vec{\eta}_{tt_0}$, $\mat{\omega}(t)$ and $\mat{K}(t,s)$ are accurate and sufficiently smooth.

\subsection{Exemplary System: Dipole Gas}\label{ssec:exemplarySystem}
As a simple example to illustrate the methods derived above, we consider the dynamics of the total dipole moment of a dilute gas of magnetic dipoles (cf. ref.~\cite{Meyer_2021}). The system consists of $N_\text{p}=125$ particles in a three dimensional box with periodic boundary conditions. The individual particles interact via a steric Weeks-Chandler-Anderson interaction with an interaction potential of the form
\begin{align}
	V_{\text{WCA}}(r) &= \begin{cases}
	4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}+1\right] & \text{if }r<2^{1/6}\sigma\\
	0 & \text{else}
	\end{cases}
\end{align}
and an exact magnetic dipole-dipole interaction \cite{Landecker1900} with the forces $F_{ij}$ and torques $G_{ij}$ on particle $j$ due to particle $i$
\begin{align}
	\vec{F}_{ij} &= \frac{3\mu_0}{4\pi r^4}\left(\left(\vec{n}\times\vec{m}_i\right)\times\vec{m}_j+\left(\vec{n}\times\vec{m}_j\right)\times\vec{m}_i\right.\nonumber\\
	&\phantom{=}\left. -2\vec{n}\left(\vec{m}_i\cdot\vec{m}_j\right)+5\vec{n}\left(\vec{n}\times\vec{m}_i\right)\cdot\left(\vec{n}\times\vec{m}_j\right) \right)\\
	\vec{\mathcal{G}}_{ij} &= \frac{\mu_0}{4\pi r^3}\left(3\left(\vec{m}_i\cdot\vec{n}\right)\left(\vec{m}_j\times\vec{n}\right)+\left(\vec{m}_i\times\vec{m}_j\right)\right)\,.
\end{align}
In these equations $\epsilon$ and $\sigma$ are the usual Lennard-Jones parameters, $r$ is the distance between the two particles, $\vec{n}$ is the 3D vector pointing from particle $i$ to particle $j$, $\mu_0$ is the permeability of free space, and $\vec{m}_{k}$ is the 3D dipole moment. All the interactions are evaluated in the nearest image convention. Further, we include an external homogeneous magnetic field $\vec{B}$, which couples to the magnetic dipoles via an additional torque
\begin{align}
    \vec{\mathcal{G}}_j &= \vec{m}_j\times\vec{B}\,.
\end{align}
For the integration of the equations of motion the velocity-Verlet method is used.

\subsection{Relaxation of a Single Observable}\label{ssec:exampleSingleObservable}

\begin{figure}[hb!]
    \centering
    \includegraphics{long_simulations_comparison.pdf}
    \caption{Relaxation of the total magnetic dipole moment along the $x$ direction as a function of time. The gray curves are obtained from MD simulations, the black curves from the nsGLE without the initial shift. The nsGLE simulations start at time $t_0=0.5\tau_\text{LJ}$ when there is almost no spreading of the values of $m_x$.}
    \label{fig:long_simulations_comparison}
\end{figure}

\begin{figure}[hb!]
    \centering
    \includegraphics{short_simulations_no_offset_comparison.pdf}
    \caption{Dynamics of the total magnetic dipole moment along the $x$ direction as a function of time. The gray curves are obtained from MD simulations and the black curves from the nsGLE as described in sec.~\ref{sec:simplified_procedure} and \ref{ssec:generatingTrajectories}, but without the initial shift. Hence, statistical dependencies between the fluctuating forces and initial values are neglected. The nsGLE simulations start at time $t_0=25\tau_\text{LJ}$ when there is a considerable amount of spreading of the values of $m_x$.}
    \label{fig:short_simulations_no_offset_comparison}
\end{figure}

\begin{figure}
    \centering
    \includegraphics{short_simulations_with_offset_comparison.pdf}
    \caption{Dynamics of the total magnetic dipole moment along the $x$ direction as a function of time. The gray curves are obtained from MD simulations, the black curves from the nsGLE with an initial shift $\vec{A}_t\mapsto\vec{A}_t-\langle\vec{A}_{t_0}\rangle$. The nsGLE simulations start at time $t_0=25\tau_\text{LJ}$ when there is a considerable amount of spreading of the values of $m_x$. Due to the shift, the trajectories generated by the nsGLE still reproduce the original dynamics well.}
    \label{fig:short_simulations_with_offset_comparison}
\end{figure}
First we investigate the relaxation of the dipole moment along one direction. For this purpose we initialize the system with all particles positioned on a simple cubic lattice and all magnetic dipole moments aligned along the positive $x$-direction. Further, an external magnetic field pointing along the negative $x$-direction is added. Hence, the dipoles minimize their potential energy with respect to the external field if they rotate $180^\circ$. To get the system out of the unstable equilibrium state, the particles are initialized with some random (angular) momenta, that are small compared to the values they take during the rest of the simulation.\\
All particles have a mass $m$ and a moment of inertia of $I=m\sigma^2/4$. They are placed in a box of size $(6\sigma)^3$ and the strength of the dipole-dipole interaction is set through the relation $\mu_0\vec{m}_i^2=\sigma^3\epsilon/8$. The strength of the external magnetic field is $|\vec{B}|=0.01\epsilon/|\vec{m}_i|$. In the followings plots, the time is given in units of $\tau_\text{LJ}=\sqrt{m\sigma^2/\epsilon}$. In total, 1000 simulations of duration $T_\text{max}=50\tau_\text{LJ}$ are run. Here, the coarse-grained observable is the $x$-component of the total dipole moment $\vec{m}=\sum_i\vec{m}_i$. Ten example trajectories are shown as grey lines in \cref{fig:long_simulations_comparison,fig:short_simulations_no_offset_comparison,fig:short_simulations_with_offset_comparison}.\\
In \cref{fig:long_simulations_comparison}, 10 example trajectories generated with the nsGLE are shown in comparison to the original MD trajectories. Here, the observable has not been shifted according to the procedure described in sec.~\ref{sec:simplified_procedure}. Nevertheless, the generated trajectories agree with the original MD ones very well. Recall that the shift in the beginning and end is done to ensure that $\langle \vec{\eta}_{tt_0} \vec{A}^\top_{t_0}\rangle= \mat{0}$ for all $t$. However, if the spread of the observable in the beginning is very small, as is the case in \cref{fig:long_simulations_comparison}, the value of the observable is almost the same $\vec{A}_{t_0}\approx\left\langle\vec{A}_{t_0}\right\rangle$ and we find that the mean of the fluctuating force has to vanish:
\begin{align}
    \mat{0} &\equiv\left\langle \vec{A}_{t_0}\vec{\eta}^\top_{tt_0}\right\rangle\\
    &\approx \left\langle\vec{A}_{t_0}\right\rangle \left\langle \vec{\eta}^\top_{tt_0}\right\rangle\\
    \Rightarrow \quad \forall_t \,:\quad  \left\langle\vec{\eta}_{tt_0}\right\rangle &\approx \vec{0}\label{eq:ff_is_zero}  \, .
\end{align}
Hence, in this case, the necessary condition $\langle \vec{\eta}_{tt_0} \vec{A}^\top_{t_0}\rangle \equiv 0$ is approximately satisfied for the generated trajectories as well, even without shifting the trajectories. Indeed, in \cref{fig:long_simulations_comparison}, the average of the fluctuating force term is very small for almost all $t$, hence, the trajectories generated with the nsGLE look very good even without the shifts.

In \cref{fig:short_simulations_no_offset_comparison} we show a counter example. There, the nsGLE simulations are started at time $t=25\tau_\text{LJ}$ when there is already a considerable spreading of the trajectories. Accordingly, the trajectories generated with the nsGLE without the shifts  do not agree with the original ones at all. However, if one adds the shifts, as done in \cref{fig:short_simulations_with_offset_comparison}, the trajectories generated with the nsGLE and the original ones show the same behaviour again.

\subsection{Dynamics of a Set of Two Observables}\label{ssec:exampleTwoObservables}
\begin{figure}
    \centering
    \includegraphics{exampleTrajectories.pdf}
    \caption{Exemplary trajectories of the total dipole moment in $x$-direction (bold lines) and $y$-direction (thin lines) for a period of $P=35\tau_\text{LJ}$. The continuous lines are obtained from the MD simulations, the dashed ones are obtained using the nsGLE. The colour indicates the set of initial values of the observables used for the integration.}
    \label{fig:exemplaryTrajectories}
\end{figure}
\begin{figure}
    \centering
    \includegraphics{ff_average70.pdf}
    \caption{Components of the average of the fluctuating force for a period of $P=35\tau_\text{LJ}$.}
    \label{fig:ff_average70}
\end{figure}

In this section, we use the formalism derived above to describe the dynamics of two correlated observables, namely the total dipole moment along the $x$- and $y$-direction. To obtain interesting dynamics for both observables, we apply the following modifications:\\
We place the particles in a larger simulation box of size $(50\sigma)^3$ and initialize all trajectories with a constant magnetic field $\vec{B}=0.01\epsilon\vec{e}_x/|\vec{m}_i|$. After a short simulation time of $100\tau_\text{LJ}$ with a Nos\'e-Hoover-chain thermostat (target temperature of $T=1\epsilon$, assuming the Boltzmann constant $k_\text{B}=1$), the thermostat is turned off and the external field (with unaltered strength) starts rotating in the $xy$ plane
\begin{align}
    \vec{B}(t) &= B\left(\cos\left(2\pi\frac{t-t_0}{P}\right)\vec{e}_x+\sin\left(2\pi\frac{t-t_0}{P}\right)\vec{e}_y\right),
\end{align}
where $P$ is the period of the rotation. At the time the rotation starts, we start sampling the total magnetic dipole moment $\vec{m}$. The time-step for the simulation is $\delta t=0.005\tau_{LJ}$. For every period $P\in\{25\tau_\text{LJ},30\tau_\text{LJ},32.5\tau_\text{LJ},35\tau_\text{LJ}\}$ we sample 1000 trajectories.

In the following, the observables of interest are the total dipole moment in $x$- and $y$-direction $\vec{A}=(m_x,m_y)^\top$. Some exemplary trajectories from the molecular dynamics (MD) simulations as well as some trajectories generated with the nsGLE can be seen in \cref{fig:exemplaryTrajectories}.

The quantities shown in \cref{fig:ff_average70,fig:CorrelationComparison} are calculated from data that is shifted in the beginning and end.
In \cref{fig:ff_average70} the average of the fluctuating force is shown, which is non-zero for almosta all times.
In \cref{fig:CorrelationComparison} a comparison of the two-timecorrelation function calculated from the original MD data and from the nsGLE data can be seen. As expected, they agree perfectly.
\begin{figure}
    \centering
    \includegraphics{CorrelationComparison.pdf}
    \caption{Slices of the two-time correlation function from the original MD data (continuous colored lines) and the nsGLE data (dotted black lines). Here, the $xx$-component (corresponding to the autocorrelation of $m_x$) is shown.}
    \label{fig:CorrelationComparison}
\end{figure}

\subsection{Interpolation Schemes}\label{ssec:interpolationSchemes}
If the method presented here only allowed to reproduce dynamics, which had been obtained by means of a simulation of the original, miscroscopic system, it would not be of much use. It turns into a proper coarse-graining method, once the drift and memory kernel can be extrapolated (or interpolated) to conditions other than those originally simulated. Here we show how to obtain such an interpolation.

In many cases, the memory kernels of the nsGLE show a simple exponential or power law behavior. In these cases, one might try to parametrize the shape of the memory kernel in terms of system parameters (e.g. the period of an external field) and use the methods introduced in this article in order to efficiently generate data for new parameter sets. The memory kernels for our example system do not have such a simple shape. However, using some physical intuition we can interpolate from two existing data sets (here, periods of the external field of $P=30\tau_\text{LJ}$ and $P=35\tau_\text{LJ}$) to generate trajectories for new system parameters (here, a period of the external field of $P32.5\tau_\text{LJ}$).

For the interpolation we use the intuition, that the most significant timescale of the process is determined by the period of the external field. Thus, we interpolate between values corresponding to the same ratio $t/P$. We use the following linear interpolation for the memory kernel
\begin{align*}
    \mat{K}^{(2)}(t,s) &:= (1-\lambda) \mat{K}^{(1)}(t P_1/P_2,s P_1/P_2)+ \\
    &\quad +\lambda\mat{K}^{(3)}(t P_3/P_2,s P_3/P_2) \, ,
\end{align*}
in order to interpolate between the periods $P_1$ and $P_3$, where $P_2=(1-\lambda)P_1+\lambda P_3$. The interpolation for the coefficients of the fluctuating force distribution is carried out analogously.

In \cref{fig:kernelInterpolation} some exemplary slices of the $xx$ component of memory kernels for different periods are shown both for the original MD data and the interpolation. As can be seen, the mean of the memory kernel for $P=30\tau_\text{LJ}$ and $P=35\tau_\text{LJ}$ is close to the real memory kernel for $P=32.5\tau_\text{LJ}$. However, the interpolation from $P=25\tau_\text{LJ}$ and $P=35\tau_\text{LJ}$ to $P=30\tau_\text{LJ}$ is quite far off from the actual memory kernel for $P=30\tau_\text{LJ}$ and, hence, this is definitely beyond the regime where a simple linear interpolation can be done.

Analogously we determine $\mat{\omega}(t)$ and $\left\langle\vec{\eta}_{tt_0}\right\rangle$ by means of interpolation. In \cref{fig:realVsGeneratedInterpTraj} some exemplary trajectories from an MD simulation with $P=32.5\tau_\text{LJ}$ are shown for comparison with the trajectories obtained by propagating the nsGLE with the interpolated memory kernel, drift and noise. The nsGLE with the interpolated coefficient functions produces a bundle of trajectories that have an average and  fluctuations very similar to the MD simulation.

Finally, we compare the two-time correlation functions of the real trajectories with the one from the trajectories generated with the nsGLE. To check if the interpolation of the coefficient function in the nsGLE has any advantage over directly interpolating either between the correlation functions or between the original trajectories, we also plot the correlation function obtained by an average of the two-time correlation functions for $P=30\tau_\text{LJ}$ and $P=32.5\tau_\text{LJ}$ (line labeled "direct av" in \cref{fig:comparisonttCorrs})
as well as the two-time correlation function obtained from simply averaging trajectories (labeld "av traj" in \cref{fig:comparisonttCorrs}). By averaging trajectories we refer to the procedure, where one picks randomly one trajectory with $P=30\tau_\text{LJ}$ and one with $P=35\tau_\text{LJ}$ and calculates their mean (again at fixed $t/P$). In \cref{fig:comparisonttCorrs} we see that for $t_1=P/4$ and $t_1=P/2$ there is almost no difference between the interpolation schemes. For $t_1=P$ the results from the interpolated nsGLE  are a much better than the results of the other schemes.
\begin{figure*}
    \centering
    \includegraphics[]{K11.pdf}
    \caption{Exemplary slices of the $xx$ component of memory kernels obtained from MD simulations (solid lines) and obtained from interpolation (dashed lines). The different colors correspond to different periods of the external field. The dashed blue line (for $P=32.5\tau_\text{LJ}$) is obtained by taking the mean of the solid green ($P=30\tau_\text{LJ}$) and the solid orange ($P=35\tau_\text{LJ}$) line. The dashed green line (for $P=30\tau_\text{LJ}$) is obtained by taking the mean of the solid red ($P=25\tau_\text{LJ}$) and the solid orange ($P=35\tau_\text{LJ}$) line.}
    \label{fig:kernelInterpolation}
\end{figure*}

\begin{figure}
    \centering
    \includegraphics{exampleTrajectoriesAndGeneratredTrajectories.pdf}
    \caption{Exemplary trajectories of the total dipole moment in $x$-direction (bold lines) and $y$-direction (thin lines) for a period of $P=32.5\tau_\text{LJ}$. The continuous lines are obtained from the MD simulations, the dashed ones are obtained using interpolated quantities together with the nsGLE.}
    \label{fig:realVsGeneratedInterpTraj}
\end{figure}

\begin{figure*}
    \centering
    \includegraphics{C11.pdf}
    \caption{Exemplary slices of the $xx$ component of the two time correlation function for $P=32.5\tau_{\text{LJ}}$. The solid black curve depicts the two time correlation function obtained from running the MD simulations, the dashed orange curve shows the two time correlation function obtained from integrating the nsGLE with averaged quantities. The dash-dotted red line is obtained from averaging the two time correlation functions for $P=30\tau_\text{LJ}$ and $P=35\tau_\text{LJ}$ and the dotted blue line is obtained from averaging the trajectories from the $P=30\tau_\text{LJ}$ and $P=35\tau_\text{LJ}$ simulations.}
    \label{fig:comparisonttCorrs}
\end{figure*}

\section{Conclusion}\label{sec:conclusion}
We have discussed how to replace the fluctuating force of the non-stationary generalized Langevin equation by a stochastic process such that the resulting dynamics reproduces the original, deterministic dynamics on the level of a given set of moments. Combined with a previously published derivation of the non-stationary generalized Langevin equation \cite{MeyerVoigtmann_2019} and with a previously published method to extract the memory kernel from simulation data of the underlying microscopic model \cite{Meyer_Wolf}, the procedure described here allows to construct and to simulate coarse-grained models for systems under time-dependent external driving.

\section*{Acknowledgement}
The authors acknowledge funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) in Projects No. 430195928 and 431945604, and useful didactic remarks from Graziano Amati.

\section{Data Availability}
The data that support the findings of this study are available from the corresponding author upon request.

redpajama_set_name: RedPajamaArXiv


text:
Decathlon 2000 › News › Unforgettable moments of Helsinki 1994
Unforgettable moments of Helsinki 1994 (0)
Janek Salmistu
Let's remember unforgettable European Champs 19 years ago.
"Dancing in the rain".
It's August 14, 1994. It's the second day of decathlon on the Olympic Stadium in Helsinki 1994. It's the eighth event – pole vault. A man of great experience, the Frenchman Alain Blondel is struggling with 5.20. He's cleared 5.00, decided to pass 5.10 and is facing his third attempt on 5.20. Blondel is concentrating hard. But just at that very moment it starts raining. No, not just raining but pouring down. Despite the downpour Blondel clears the height.
There are only two men left in pole vault. The Frenchman and Estonian Erki Nool. Average and weak vaulters weren't caught in the rain, they managed to score their points, but top vaulters finish 20 or 30 cm below their usual results. Blondel hasn't got a championships medal yet. Not a single one.
The bar is set to 5.40 and Blondel has a chance to show his competitors what he's capable of. He wants his performance to be perfect. He's standing on the runway, trying to dry the pole, but it's no use as it's still pouring down. There seems to be no end of the rain – it's been getting even worse. One can hardly see the runway or the bar on TV. Only a real professional would be able to use any kind of technique in those conditions. Blondel's personal best is 5.40. If only he could clear 5.40, he might be able to clear 5.60 as well.
Blondel is standing at the far end of the runway, holding the pole. Thousands of anxious people both on the stadium, packed in raincoats, and in homes, in front of their TV sets are watching his every move. And there he goes! He goes over the bar, lands on the soaked pit, water splashing in each direction and performs a winner's somersault.
Nool, who had earned a miss at the same height, is eyeing his competitor's performance. Nool is shaking his head, drops of water flying all around. The cameramen are filming him up close. His whole posture seems to say that it's not possible that Blondel cleared the height. With this jump Blondel demonstrated that he is not only a tough competitor but also a winner at long last. He is the European Champion!
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Erki Nool's biography
Götzis 2010: "Singing in the rain!"
Erki Nool pleased with progress
Erki Nool won Heptathlon in Reval Hotels Cup
Alain Blondel
Erki Nool
In 1997, Estonian sports enthusiasts founded the Erki Nool Fan Club, and about 200 members of this group accompany him to the various international competitions in which he participates.
Erki Nool Reval Cup 1999
Search: Alain Blondel Erki Nool rain
athens2004 8
Every decathlon has a story - Gabby Pieraccini
Decastar 2013 Full size photos through the years He's coming to Götzis! Interview with Harry Marra super-coach with super-athlete Let it snow, the Canadian winter-games in Götzis 2013 Short Interview with Gunnar Nixon Talence Decastar 2013 - Day 2 - the last set Talence Decastar 2013 - Heptathlon - Day 1 Unforgettable moments of Helsinki 1994 Unforgettable moments of Willi Holdorf

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Drivers rely on unorganized pocket size charging infra and technologically challenged power source. Due to poor current battery, high cost of Li-on battery, unorganized and lack of public charging stations, they face huge challenge in business because of cost, huge wait time for charging and hence earning.
Mobilytics Energy is a technology driven EV charging and swapping infrastructure provider company, which supply its Partners to set up, and manage stations on their location site while ensures accessibility and availability of docks for EV users. Through our technological stake, such, Battery Management System (BMS) and IoT enabled infrastructure, we are standardizing charging stations for 2 million e-rickshaws of India in a B2B2C model.
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text:
Novopokróvskaia - Новопокровская  és una stanitsa del territori de Krasnodar, a Rússia. Es troba a la plana de Kuban-Azov, al peu del Caucas, a la vora del riu Korsun, un dels constituents del riu Ieia. És a 165 km al nord-est de Krasnodar.

Pertanyen a aquesta stanitsa els possiolki de Gorki, Lesnítxestvo i el khútor de Ieia.

Història
Fou fundada l'1 de setembre de 1827 per colons de les gubèrnies de Vorónej, Kursk i Khàrkiv amb el nom original de Novo-Pokróvskoie, Karassun. El 1848 la vila obtingué l'estatus de stanitsa.

A començaments del segle XX tenia 9.136 habitants, dues esglésies i dues escoles, entre altres instal·lacions. Fins al 1920 pertanyia a l'otdel de Kavkàzskaia de la província de Kuban. El 1924 fou designada centre administratiu del raion homònim.

Vegeu també
 Ilínskaia

Stanitses del territori de Krasnodar


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Fly in Style
Swiss Air Lines releases the dreamy interiors for its new 777 jets.
By Phoebe Gavin
July 27, 20154:43 PM
Flyers around the world promptly lost it after a French company patented schematics for an airplane seating arrangement so tight that passengers were simultaneously touching and facing each other. As if to tell us everything is going to be OK, London-based design company Priestmangoode has released a new video render of the interior for Swiss Air Lines' Boeing 777 cabins. Behold:
In addition to looking lovely, the new design is marvelously high tech. In first and business classes, the window blinds are electronically controlled and the seating firmness is adjustable. All passengers will have access to a self-service snack bar between meal service and a wireless connection.
Nine of the new wide-body planes will be added to Swiss' long-haul rotation in January 2016, servicing routes between Switzerland and Los Angeles, Hong Kong, Bangkok, San Francisco, São Paulo, Tel Aviv, New York, and Montreal.


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schoolcouncil
The purpose of each School Council is to help the school provide the best possible education for its students.
The school council:
is accountable to the Board of Governors and also to parents and the wider community for the way it carries out its functions;
seeks to develop effective working relationships with the principal, staff and parents; ESF management; other schools and the wider community;
acts at all times in accordance with the requirements laid down in the Ordinance and Regulations;
is a collective entity and acts as a group. No one councillor or category of councillors has any right to act individually except when the school council has delegated authority to do so.
The strength of a school council lies in the talents and commitments of its members, and in their ability to work together as a team for the good of the school.
The BHS Council has 10 members, comprised of the following:-
a Chairman, appointed by the Board, who is neither an employee of ESF nor the parent of a student at the School
2 persons from the general community who are neither employees of ESF nor parents of students at the School, appointed by the Board;
2 persons elected by members of the teaching staff of BHS from among their own number;
2 parents who are not employees of ESF, one elected by the parent members of our PTA committee and one elected by the parents of the School;
1 person elected by the support staff of BHS from among their own number;
the Chief Executive Officer of ESF, or her appointed delegate;
the Principal.
The Council has meetings 5-6 times each year.
School Council Members 2019-2020
School Council Members Biography
Malcolm has lived in Hong Kong since 2002. He is a commodities and financial markets professional employed by Pinyin Capital. He has been involved in structuring innovative capital market transactions across Asia Pacific for sovereign wealth funds, hedge fund, corporate and family offices accounts for the past 20 years. Outside of work Malcolm is heavily involved with ESF as a parent and in his duties at Beacon Hill and when time permits is a very keen golfer and tennis player.
Brenda Cook, Principal
Brenda is a passionate educator who has been in leadership positions in ESF for over 17 years. Brenda joined Beacon Hill School in August 2019 after having been the principal of Glenealy School (ESF) since January 2010. She is a strong advocate for children and believes in educating the whole child and nurturing and developing all individuals to be the best that they can be.
Coming from a large extended family, Brenda loves to interact with people and make strong and lasting connections with students and their families. "
Alan Milliken, Director- Strategic Performance and Quality Assurance , ESF
Alan Milliken joined us as Director of Strategic Performance and Quality Assurance. He comes from the UK and he has held a number of roles within education. He has led schools as a Principal and has worked across schools leading curriculum, performance, resources and wider learning communities. He has worked closely with education systems, schools, Principals in the past to support them with their approaches to quality assurance and continuous improvement. He is very much looking forward to working and living in Hong Kong.
Susan Ho, PTA Representatives
Hi, this is my second year in the pta and the first as it's Chair. I was born and brought up in Edinburgh. Shortly after graduating in Mechanical Engineering I came to Hong Kong where I worked in various building projects here and in Macau. I have a daughter in Year 6 and a son in Year 2. I hope to continue the task of my predecessors in promoting parent teacher cooperation and raising funds to benefit the school.
Rosa Tse, Parents Representatives
Hi, I am Rosa Tse. I was born in Hong Kong and had my education in Canada since primary school.
I obtained my undergraduate degree in University of Toronto, then my LLM, PGDE and Master of Education Psychology in Hong Kong. I have been working in the education sector for over 15 years and am also a Director of a Child Behavioural Learning Centre. I have two daughters at BHS, one in Year 3 and the other in Year 5. Throughout my 5 years in the BHS community, I have participated as a PTA parent member and this is the second year as Vice Chair. I enjoy playing different roles in the BHS family and it is my honour to continue to contribute as a Parents Representative on the BHS School Council.
Anita Ko, Community Representatives
Anita joined the Beacon Hill School Council in 2014. She has 20 years of experience in strategic communications, corporate brand building and perception management. Anita is currently responsible for corporate branding strategy develo*ent and implementation for the Hong Kong Jockey Club, the largest community benefactor in Hong Kong. Committed to youth develo*ent, Anita was an active mentor of Christian Actions' Youth Mentoring Programme from 2011 to 2014, offering advice and guidance to teenagers from less privileged families. Her two sons are students at Quarry Bay School and South Island School.
Joe Lau, Community Representatives
oe Lau, CPA, is currently the Head of Finance of Credit Suisse Private Banking for North Asia based in Hong Kong. He is responsible for the region's financial management, strategic planning and controlling functions. He joined Credit Suisse in 2006 from J*organ Asset Management, where he was the Financial Controller. Prior to that, he spent fives years in public accounting with K*G Silicon Valley Office in the United States and PwC Hong Kong. He received his Bachelor Degree in Finance from the University of Hong Kong and he is a Certified Public Accountant in the United States. Joe's leisure activities include traveling, fitness training, watching his son playing ice hockey and spending time with friends and family.
Tim Teahan, Teacher Representatives
Hi I am Tim Teahan and I am proud to be the staff representative on the Beacon Hill School Council. I have been a teacher at Beacon Hill for 9 years and have been the PE teacher for the last 7 years. I have worked for the ESF for 15 years and been teaching since 1995. I am originally from New Zealand (go All Blacks!) and moved to Hong Kong in 1997. My two sons attend ESF Schools. My interests are geocaching and sports.
Charlotte Fennelly, Teacher Representatives
My name is Charlotte Fennelly and I am delighted to be one of the teacher representatives on the BHS School Council. I was born and educated in the UK where I worked in education for three years prior to successfully applying to work for ESF in 2002. I spent four years working at Kennedy School and have worked at Beacon Hill School for the past eight years. I care passionately about BHS and have been heavily involved with the BHS community through my role on the PTA, now I look forward to continuing to support the students through my role on the School Council.
Kannis Tsui, Support Staff Representative
Hello! My role is as a Media and Technology EA. I have personally been in the ESF stream, from year 1 all the way to year 13. I enjoy working with the children at BHS, and assisting them with all aspects of their learning.