This is actually a very simple soup that is supposed to be consumed by Dutch Farmers.I got the recipe from here.This is a very very easy vegetable soup and the way it is served is also very innovative.I only substituted the Gouda cheese in the recipe with Swiss cheese.
Chop the vegies into bite sized pieces.Melt butter in a heavy bottomed soup pot or pressure cooker and add the vegetables and onion and saute until the onion is browned.Add little salt and pepper and the veg broth(I made broth from here).Bring to boil and then let it simmer for 20 minutes until the veggies are tender for about 30 minutes or if using pressure cooker cover and cook till 2-3 whistles.Check and adjust the salt and pepper.Now the soup is ready to be served.
Preheat the broiler of the oven.Ladle the soup in oven proof bowl,top it with a sliced bread piece,cover the bread with cheese sliced/shredded.Broil the big soup bowl or individual soup bowls for around 2 minutes in the centre rack of the oven or atleast 6 inches from the heat till the cheese melts.Serve.
Now you can do the warming/broiling in the microwave also.I tried both and except for pre-heating time ,broil time is almost same.In the microwave start with 30 seconds +.
The soup is very prefect for winter and very easy to make.No other spices apart from salt & pepper are added.The flavour of the soup comes from the broth,which is a must here.The thickness and consistency can be adjusted by varying the amount of broth.The amount of veggies can be varied as per taste and availability.The above estimate is for 2-3 big servings.
Sounds simple and looks delightful!
wow.. what a lovely combo.. cheese bread with soup. nothing more comforting that this..thanks for your beautiful comments in my blog CL.
Looks awesome, loved that cheesy bread slice..delicious soup..
Cool great warming soup for the chill night.


text:
Could Apple Pay protect your information at the gas pump?
The new Wawa off of Jones Loop in Punta Gorda has the feature at each individual pump.
Tuesday, December 31st 2019, 7:45 PM EST by Claire Lavezzorio
PUNTA GORDA, Fla. - Could an alternative way to pay protect your information at the gas pump?
ABC7 is looking into whether Apple Pay is a safer option.
Professor of Finance at FGCU Dr. Tom Smythe says it's a good option.
"To the extent that you're using technology that doesn't require you to swipe or put your chip into a machine, there's increased security associated with that," said Smythe.
Like anything, there are concerns hackers will find a new way to get ahold of your information that's transmitted by near-field communication technology.
"To think that we've found a magic bullet at this point is probably unrealistic," said Smythe.

redpajama_set_name: RedPajamaCommonCrawl


Join the National Czech & Slovak Museum & Library at the Backpocket Pilot Pub (415 3rd St SE, Cedar Rapids) for History on Tap!
Special guest Mark Stoffer Hunter, historian at The History Center, will share stories about local bars owned by Czech families. Find out where they were, what they served, and who was enjoying a pivo or two at these establishments!
The event is free, but if you desire some delicious craft beer, that must be purchased from Backpocket Pilot Pub during the program. Must be 21+ to buy beer.
Please note: this program will take place in the loft overlooking the Backpocket Pilot Pub seating area. There are stairs leading up to this area, but no elevator is available to reach the loft.


Oktoberfest is back! And so is the root beer keg. Join us on Sunday evenings during the month of October for a chance to connect with other high schoolers.
It's a party – all month long! From a root-beer keg to fun themed food, we'll be celebrating Oktoberfest on Sunday nights at Fusion at Main.


South Africa Negotiating CbC Exchange Agreements with Panama, Singapore, and the U.S.
More details have become available with respect to the new Netherlands-South Africa tax treaty and protocol on income and capital of 10 October 2005.
The Netherlands Ministry of Foreign Affairs has stated in a press release that the Netherlands and South Africa signed a new income and capital tax treaty on 10 October 2005. Once in force, the new treaty will replace the current Netherlands-South Africa income tax treaty of 15 March 1971. The intention of the new treaty is to bring matters into line with the fiscal developments in both states. The text of the new treaty is not yet available, but the press release indicates the details summarized below regarding the new treaty.
The South African Ministry of Finance announced that a new tax treaty between South Africa and the Netherlands will be signed on 10 October 2005. Once signed and in force, the new treaty will generally replace the South-Africa income tax treaty signed on 15 March 1971. Further details of the treaty will be reported subsequently.
South Africa and Turkey signed a first-time income tax treaty on 3 March 2005.
The first-time income tax treaty between Bulgaria and South Africa, signed on 29 April 2004, entered into force on 27 October 2004. The treaty generally applies from 1 January 2005.


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©The Wire | 2018
Here's Why Nepal PM K.P. Sharma Oli Is Now Keen to Mend Ties With India
It just so happens that both countries can no longer afford to prolong disagreements, whether over the border or otherwise.
K.P. Sharma Oli. Photo: Twitter/@PM_Nepal
Kamal Dev Bhattarai
Kathmandu: Nepal Prime Minister K.P. Sharma Oli, who had until recently faced pressure from party leaders to quit as prime minister and party chairman, seems to have managed intra-party rifts for the time being, thus securing his position for the remaining two and half years of the government's five-year term.
An intra-party taskforce has come up with a suggestion that Oli focuses on government matters and party chairperson, Pushpa Kamal Dahal Prachanda, holds executive rights in party affairs.
A statement released by Prachanda last week has indicated that Oli will remain as prime minister for the full five-year term. As a solution to the intra-party rift, Oli's role in party affairs will be curtailed, federal and provincial cabinets may be reshuffled to accommodate leaders from rival camps and the party will scrutinise the functioning of the government more.
Now that disagreements within the party are done with, Oli's next priority is to improve ties with India. On August 15, Oli held a phone conversation with Indian Prime Minister Narendra Modi to greet the latter on the occasion of India's 74th Independence Day. The conversation initiated by Oli ended the period of total communication breakdown that had erupted after boundary disputes between the two countries.
India's Independence Day offered a moment to be seized, and dialogue between the two countries resumed. Both leaders share a good rapport. Two days after the talks at prime ministerial level, Nepal and India held a mandated meeting of oversight mechanism, to review development projects funded by India.
Also read: India-Nepal War of Words Continues Over Offers, Non-Offers of Border Talks
The mechanism was formed four years ago to identify the bottlenecks in development projects and settle them fast.
Nepal has also extended the deadline for Small Development Projects (SDPs) with investment from the Indian Embassy by three years. This had been pending since Nepal entered its federal setup. The SDPs are regarded as one of India's most successful development projects in the neighbourhood and began in Nepal in 2003.
India has also provided 10 ventilators to the Nepal Army.
These developments come as a huge sigh of relief for those who want to see cordial relationships between the two countries. Both sides seem visibly keen to normalise bilateral relations.
Why is Oli keen to repair relations with India?
First, opposition parties have started raising the pitch on how Oli has been failing to talk with India on border issues.
Nepal's Prime Minister Khadga Prasad Sharma Oli (L) shakes hands with his Indian counterpart Narendra Modi in New Delhi, India, February 20, 2016. Credits: Reuters/Adnan Abidi/File Photo
An intra-party panel formed to resolve internal disputes has reportedly suggested that Prime Minister Oli talk with India immediately on border isssues. All opposition parties including Madhes-based parties had notably supported the constitution amendment on the map that had been initiated by government.
Now, they are of the view that Oli should demonstrate his diplomatic skills to settle disputes with India.
Opposition parties are also blaming him for damaging ties with India with his provocative statements. Even foreign policy observers have been vocal with their belief that Oli has threatened people-to-people relations between the two countries. Even a section of leaders of the ruling Nepal Communist Party (NCP) have expressed disappointment with some of Oli's statements.
Also read: Nepal PM Oli Says 'Ayodhya' Was in Nepal; Kathmandu Issues Clarification
This seems to have had an effect on Oli.
Second, Oli is quite aware that the border issue is time consuming to resolve and that it should not affect overall bilateral relations. There are several issues between two countries which demand regular and constant touch between the two, like the COVID-19 crisis for one.
Nepal is heavily dependent on India for the supply of day-to-day essentials.
Finally, along with criticism over how he handled the COVID-19 crisis, Oli has faced flak for the manner of his dealings with major powers like India, China and United States.
Due to the intra-party rift, the future of America's US $ 500 million grant to Nepal under the US Millennium Challenge Corporation (MCC) is uncertain. As per the agreement between two countries, the agreement should have come into effect from June but that deadline was missed and no fresh decision has been made. To get this grant endorsed by the parliament, Oli will have to settle the dispute within his party, although opposition parties are ready to support the government in this matter.
Though Oli is known to lean towards China, there has not been much progress when it comes to selection of projects under China's Belt and Road Initiative.
Also read: When China Chips Are Down, 'Diamonds' in India's 'Necklace' of Allies Lack Sparkle
It is not like India-Nepal ties were particularly cordial even before the map issue. Oli is perceived as having failed to convince India to receive the report of the Eminent Person's Group (EPG) report which was finalised in 2018. India in 2018, refused to receive the EPG report citing the busy schedule of Prime Minister Narendra Modi. This is still pending. Oli is thus keen to get his foreign policies on track.
India is also likely to reciprocate to Oli's call for resumption of dialogue between two countries. China's growing presence and engagement with South Asian countries even in the time of the COVID-19 crisis is likely to push India towards its 'neighbourhood first' policy.
In his Independence Day speech, Modi had said, "Neighbour is not just one with whom there is sharing of borders but also one with whom our heart stays connected."
Also read: The Kalapani Imbroglio: Has India Pushed Nepal Too Far?
Recently, Indian Foreign Secretary Harsh Vardhan Shringla visited Bangladesh amid the COVID-19 crisis and he is likely to visit other South Asian countries as well. Since assuming office, Shringla has not visited Nepal. India also cannot afford to ignore Nepal as there are several development projects funded by India which need to be expedited. Modi himself is facing criticism within India that his neighbourhood first policy has not yielded the desired result.
In conclusion, there is no alternative to both Nepal and India other than to settle issues related to trade, transit, water resources, floods, development projects, the submission of the EPG report and the border. These problems might prove difficult to tackle in the future.
Kamal Dev Bhattarai is a Kathmandu-based journalist.


Lady Gaga stunned in a red gown at American Horror Story Premiere red carpet. Lady Gaga red dress was stunning, which featuring slim a-line silhouette, one-shoulder neckline with pleated detailing, and a thigh-high slit at the skirt. The floor length evening prom gown would be a perfect special occasion dress or a prom party dress.


Nelson Mandela: May his words always be an inspiration to us. Rest in Peace Madiba. Thank you for everything you have done for this World. You put a seed of belief into people's mind and hopefully this will grow even bigger in the Year's to come so there will be more Peace and Happiness on Earth!
Thank you Lonneke, great idea to highlight the death of Nelson Mandela.


text:
\section{Structural and dynamical consistency between models at different resolution}\label{sect:hypoth_theor_back}

The simulation of an arbitrary atomistic system relies on the numerical integration of its Newton's equations of motion,
\begin{eqnarray} \label{NewtonEOM}
m_i \ddot{\mathbf{r}}_i=\mathbf{F}_i \ \ \ \mathrm{with}\ \ \ \mathbf{F}_i=-\frac{\partial u(\bf r) }{\partial \mathbf{r}_i},
\label{eq:newton_eq}
\end{eqnarray}
where ${\bf r}_i$, $i=1,...,n_A$ are the Cartesian coordinates of the constituent atoms in the system and  $u({\bf r})$ is the potential energy taking into account non-bonded (e.g. van der Waals, electrostatic...) as well as bonded (e.g. bonds, angles, dihedrals...) interactions \cite{gonzalez2011AAforcefield_rev,fuentes2015AAforcefield}.

The MZ formalism rephrases these equations in terms of a set of collective, CG variables ${\bf R}_I$, $I=1,...,N_B<n_A$ undergoing a {\it slow} dynamics. The ${\bf R}_I$ are related to the original coordinates through a vector-valued {\it mapping function} $\map({\bf r})$, namely
\begin{eqnarray}\label{eq:mapping}
&&\mathbf{R}_I = \map_I(\mathbf{r}) = \sum_{k\in S_I}\frac{m_k}{M_I} \mathbf{r}_k, \\ \nonumber
&&\mathbf{P}_I = \sum_{k\in S_I}\mathbf{p}_k,
\end{eqnarray}
where ${\bf R}_{I},\ {\bf P}_I$ and $M_I = \sum_{k \in S_I} m_k$ are respectively the coordinates, momenta, and masses of the $I^{th}$ CG bead, ${\bf r}_k$, ${\bf p}_k$  and $m_k$ are respectively the coordinates, momenta, and masses of the $k^{th}$ atom, and $S_I$ is the set of atoms that map onto bead $I$.

This mapping, together with further manipulations  and simplifications of Eq.~\ref{NewtonEOM} \cite{schilling2021coarse,di2019systematic,hijon2010mori} (see Supporting Information for a short recap), yields the generalised Langevin equations for the time evolution of the CG degrees of freedom, which read:
\begin{eqnarray}\label{eq:MZ_simpl_GLE}
&&\dot{\bf R}_I = \frac{\bf P_I}{M_I}, \\ \nonumber
&&\dot{\bf P}_I = -\frac{\partial U({\bf R})}{\partial  \bf R_I}-\int_{0}^{t}dt' K(t-t^\prime) \frac{\bf P_I}{M_I}  + {\bm \xi}_I(t),
\end{eqnarray}
where $U({\bf R})$ is the effective potential of the CG system induced by the mapping, while $K(\tau) $ is defined as the memory kernel.  The term $\bm \xi(t)$ originates from the projection of the fast degrees of freedom onto a rapidly-varying force acting on the slow, CG coordinates, and is typically modelled as an autocorrelated noise linked to the memory kernel $K(\tau)$ {\it via} the fluctuation-dissipation theorem~\cite{kubo1966FDT},
\begin{eqnarray}\label{FDT}
&&\langle  \xi^{\alpha}_I(t+\tau) {\xi}^{\beta}_J(t) \rangle = k_BT K(\tau) \delta^{\alpha\beta}\delta_{IJ},
\end{eqnarray}
where the indices $I,J=1,...,N_B$ run over the CG particles while $\alpha,\beta=x,y,z$ over the spatial dimensions.

The first step in the parametrisation of a CG model according to Eq.~\ref{eq:MZ_simpl_GLE} is the determination of the effective CG potential $U(\bf R)$. This task is pursued by systematic coarse-graining \cite{noid2008multiscale,potestioEntropy2014,giulini2021system}, which aims at developing potentials acting among effective interaction sites, typically representative of groups of atoms as in Eq.~\ref{eq:mapping}. The objective of this approach is that the equilibrium probability distributions of structural observables (e.g. two-body correlation functions), computed on the ensemble of CG structures sampled by the model, coincide with those obtained from the higher-resolution, all-atom structures {\it observed in terms of the CG degrees of freedom} \cite{noid2008multiscale,rudzinski2011coarse,shell2016coarse,lebold2019dual,dannenhoffer2019compatible,giulini2021system}. Irrespectively of the method, algorithm, or instrument employed to parametrise the interactions, the aforementioned objective provides not only a guideline in the construction of coarse-grained models, but also a quantitative criterion to perform such parametrisation and to assess the validity of its result.

In fact, quantitative agreement between structural correlations (in principle at arbitrary order) among the CG degrees of freedom of the reference and those of the model can be achieved if and only if a set of {\it consistency conditions}~\cite{noid2008multiscale} is satisfied, these being:

\begin{enumerate}
\item Each CG coordinate $\{\mathbf{R,P}\}$ is a well defined linear combination of a subset of the underlying atomistic coordinates $\{\mathbf{r,p}\}$, as it is the case in Eq.~\ref{eq:mapping};
\item The CG equilibrium distribution $P_{R}(\mathbf{R})$ associated to the effective potential $U(\bf R)$ is equal to the atomistic equilibrium distribution after mapping, $p_{R}(\mathbf{R})$.
\end{enumerate}
Assuming a system at thermal equilibrium in the canonical ensemble, $p_{R}(\mathbf{R})$ is defined as
\begin{equation}
p_{R}(\mathbf{R})=\frac{1}{Z}\int d{\bf r} \ e^{-\beta u(\bf{r})}  \delta( \map({\bf r})-{\bf R}),
\end{equation}
where $Z$ is the canonical configurational partition function. A CG potential $U({\bf R})$ correctly reproduces the reference system's structural properties if it is equal to the {\it multi-body potential of mean force} (MB-PMF) $W({\bf R})$, that is
\begin{eqnarray}\label{eq:consist:a}
U({\bf R})=W({\bf R}) = -k_BT\ln[p_{R}(\mathbf{R})]+\emph{const}.
\end{eqnarray}
Although exactly determining the MB-PMF is an extremely difficult task, Eq.~\ref{eq:consist:a} can be considered the constitutive equation of coarse-graining, in that it provides an exact prescription of what a CG model is required to satisfy in order to entail all those features of the underlying high-resolution model that can be preserved at that level of resolution. This, however, only pertains {\it structural} properties at equilibrium. Recently, Di Pasquale and coworkers have shown~\cite{di2019systematic} that the parametrisation of the CG interaction potentials and that of the GLE-related quantities (the memory kernel and the noise term in Eq.~\ref{eq:MZ_simpl_GLE}) are independent from each other: this is a key property that allows one to solve the structural problem by fixing the interaction potentials with some standard coarse-graining procedure, and subsequently fine-tune the memory and noise terms that are responsible for the dynamical behaviour of the system \cite{izvekov2006modeling,li2015incorporation,li2017computing,yoshimoto2017construction,han2018mesoscopic,wang2019implicit}.

To parametrise these terms we here turn our attention to the Euler-Lagrange formalism, or more appropriately to the stationary action principle, which allows us to characterise the global properties of a \emph{single trajectory as a whole}. We assume the underlying reference, high-resolution, or for short {\it atomistic} system to be composed by point-like particles that follow classical (Newton's) equations of motion in the {\it microcanonical} ensemble; as such, the system can be described {\it via} an action functional that is minimal in correspondence of the trajectory effectively followed by the system. Making use of the Mori-Zwanzig formalism, this trajectory is projected onto a set of slow, collective variables subject to a reduced (coarse-grained) force field plus a friction and a noise term; the latter are in general not {\it Langevin-like}, in that the noise is not delta-correlated, but rather it has a slowly-decaying, finite-time correlation function that is proportional to the memory kernel of the friction term. Conversely, the properties of the dynamics as observed in terms of the slow collective variables have to be consistent with the underlying microcanonical time evolution. We thus assume that the same structure in terms of a stationary-action principle can be preserved to describe the time evolution of the slow variables, provided that the interactions and the friction/noise terms are appropriately parametrised. Relying on these assumptions, we hereafter describe a consistency condition for the dynamics, and test its validity and effectiveness.

\section{Euler-Lagrange formulation of dynamical consistency}\label{sect:formulation}

In the Lagrangian formulation of classical mechanics, the equations of motion of the system can be derived from the minimisation (or extremisation) of the action functional $S$:
\begin{eqnarray}\label{eq:euler-lagr}
&&S[{\bf r}(t), \dot{\bf r}(t)] = \int_0^T dt \ L(t) = \int_0^T dt\ L({\bf r}(t), \dot{\bf r}(t)),\\
&&L({\bf r}(t), \dot{\bf r}(t)) = K(\dot{\bf r}(t)) - u({\bf r}(t)),\\
\label{eq:euler_lagr_newt}
&&\delta S = 0 \Rightarrow \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\bf r}_i}\right) = \frac{\partial L}{\partial {\bf r}_i},
\end{eqnarray}
where $K = \sum^{n_A}_{i=1} m_i \dot{\bf r}_i^2 / 2$ is the kinetic energy of the constituent particles, $u({\bf r})$ is the interaction potential acting among them, and appropriate boundary conditions are imposed on the configuration of the system at time $t = 0$ and $T$.

Let us now assume that the coordinates $\bf r$ of the system can be separated, {\it via} a linear transformation akin to Eq. \ref{eq:mapping}, in two groups: the fast coordinates $\bf q$ and the slow coordinates $\bf Q$. The Lagrangian becomes
\begin{eqnarray}\label{eq:euler-lagr-sep-1}
&&L( {\bf Q}(t), \dot{\bf Q}(t), {\bf q}(t), \dot{\bf q}(t)) =  K_Q(\dot{\bf Q}(t))  + \\ \nonumber
&&+ K_q(\dot{\bf q}(t)) - U_Q({\bf Q}(t)) - u_q({\bf q}(t)) - u_c({\bf q}(t), {\bf Q}(t)),
\end{eqnarray}
where the first two terms are respectively the kinetic energies of the fast and slow variables, the second two terms pertain to the interactions {\it within} each of these two groups of coordinates, while the last term contributes the interaction energy between fast and slow degrees of freedom.

Based on this Lagrangian, the action functional $S$ in Eq.~\ref{eq:euler-lagr} can be written as (for notational simplicity we omit the time dependence of the fast and slow variables)
\begin{equation}\label{eq:euler-lagr-sep-2}
S[ {\bf Q}, \dot{\bf Q}, {\bf q}, \dot{\bf q}] =  S_Q[ {\bf Q}, \dot{\bf Q}] + S_q[{\bf q}, \dot{\bf q}] + S_c[{\bf q}, {\bf Q}],
\end{equation}
with
\begin{eqnarray}\label{eq:euler-lagr-sep-3}
&& S_Q[ {\bf Q}, \dot{\bf Q}] = \int_0^T dt\ \left[K_Q(\dot{\bf Q}) - U_Q({\bf Q}) \right],\\
&&S_q[{\bf q}, \dot{\bf q}] = \int_0^T dt\ \left[K_q(\dot{\bf q}) - u_q({\bf q})\right], \label{eq:euler-lagr-sep-4}\\
&&S_c[{\bf q}, {\bf Q}] = - \int_0^T dt\ u_c({\bf q}, {\bf Q}) \label{eq:euler-lagr-sep-5}.
\end{eqnarray}
Eqs.~\ref{eq:euler-lagr-sep-2}-\ref{eq:euler-lagr-sep-5} display that the minimisation of $S$ could be performed separately for the two sets of variables if the coupling term $S_c$ were zero: it is the presence of such term that makes the time evolution of the fast and slow degrees of freedom mutually dependent. On the other hand, we can now assume that slow variables follow the \emph{actual} trajectory that minimises the action, i.e. ${\bf Q}(t) = {\bf Q}^\star(t)$. By doing this, $S$ can be considered a functional of the trajectory ${\bf q}(t)$ of the fast variables only, namely
\begin{equation}\label{eq:euler-lagr-sep-6}
S[ {\bf Q}^\star, \dot{\bf Q}^\star,{\bf q},\dot{\bf q}] = S_Q[ {\bf Q}^\star, \dot{\bf Q}^\star] + S_q[{\bf q}, \dot{\bf q}]  + S_c[{\bf q}, {\bf Q}^\star]. \\
\end{equation}
The action $S$ is minimised when computed over the physical trajectory $({\bf q}^\star(t), {\bf Q}^\star(t))$ of both the fast and slow variables; however, since the potential and kinetic energies where ${\bf Q}$ appears are already calculated onto such trajectory, it is now sufficient to perform the minimisation of Eq.~\ref{eq:euler-lagr-sep-6} with respect to the path followed by the fast variables ${\bf q}$. In doing so, we note that the term $S_Q$ in Eq.~\ref{eq:euler-lagr-sep-6} is constant with respect to the fast variables, and can be thus ignored in the differentiation of the action with respect to them. The integrand of the coupling term $S_c$, on the other hand, is now a function of the sole ${\bf q}(t)$ and time, and represents a background field for the dynamics of the fast variables.

This rather general framework is now specialised and applied to a particular case, with the particular extension to a time non-local action functional. Specifically, we will identify the slow variables in the coarse-grained degrees of freedom, and assume that their time evolution ${\bf Q}(t)$ has been solved, so that the minimisation of the action with respect to the remaining fast variables will provide us with the complete picture.

Implementing the identification of the slow variables with the coarse-grained ones and making use of Eq.~\ref{eq:mapping}, we thus have
\begin{eqnarray} \label{GLE}
&&{\bf Q}_I(t)={\bf R}_I(t) = \map_I({\bf r}(t)),\\ \nonumber
&&\dot{\bf Q}_I(t)=\dot{\bf R}_I(t) = \map_I(\dot{\bf r}(t)),
\end{eqnarray}
where $I=1,...,N_B$ and $N_B$ is the number of CG particles. Subsequently, we introduce an action $\Sigma$ representing an \emph{effective} analogue of the action $S$ in Eq.~\ref{eq:euler-lagr-sep-2}:
\begin{eqnarray}\label{eq:Lagrangian_subs}
\Sigma[{\bf R},\dot{{\bf R}};{\bm \zeta}] &=& \int_0^T dt\ \left[K_R(\dot{\bf R}(t)) - U({\bf R}(t))\right] +\\ \nonumber
&+& \int_0^T dt\ A[{\bf R}, \dot{\bf R}; t,{\bm \zeta}].
\end{eqnarray}
Here, $K_{R}$ and $U$ are, respectively, the kinetic and (effective) potential energy of the CG particles, and the term $A[{\bf R}, \dot{\bf R}; t, {\bm \zeta}]$ couples these slow variables with the fast ones, the latter being now represented by \emph{a set of properties} ${\bm \zeta}=[\zeta_1,...,\zeta_M]$. The function $A$ thus recapitulates all the information contained in the two latter terms of the r.h.s. of Eq.~\ref{eq:euler-lagr-sep-2}.

We now assume the dynamics followed by the fast variables, and their impact on the slow ones, to fall among those cases that can be treated through the Mori-Zwanzig formalism. The functional form of the action $\Sigma$ can then be determined by requiring that the equations of motion one obtains through its minimisation with respect to the slow variables coincide with the GLE describing the dynamics of the CG system. Specifically, we assume that the time evolution $I$-th CG particle, all particles having the same mass $M$, is dictated by
\begin{eqnarray}\label{eq:CG-eom}
&&M \ddot{{\bf R}}_I = {\bf F}_I({\bf R}(t)) - \int_{0}^t dt' K(t - t^\prime, {\bm \zeta}) \dot{\bf R}_I(t^\prime) + {\bm \xi}_I(t,{\bm \zeta}),  \nonumber \\
&&{\bf F}_I({\bf R}(t)) = - \frac{\partial U({\bf R})}{\partial {\bf R}_I}\bigg\rvert_{{\bf R}(t)}, \
\end{eqnarray}
where the memory kernel $K(\tau,{\bm \zeta})$ and the noise ${\bm \xi}_I(t,{\bm \zeta})$ are now \emph{parametric functions} of the properties ${\bm \zeta}$ that account for the effect of the fast variables, and satisfy the fluctuation-dissipation relation in Eq.~\ref{FDT}.

The presence of the memory term in Eq. \ref{eq:CG-eom} imposes some care in the construction of the action $\Sigma[{\bf R},\dot{{\bf R}};{\bm \zeta}]$ in Eq.~\ref{eq:Lagrangian_subs}: indeed, as these depend on the history of all velocities of the CG particles up to the current time, $\Sigma$ has to be written as the integral of a time non-local function. Critically, in such a case the usual differentiation with respect to positions and velocities at fixed time that appears in the ``traditional'' Euler-Lagrange approach of Eq.~\ref{eq:euler_lagr_newt} can no longer be performed. To tackle this problem, Ferialdi and Bassi \cite{ferialdi2012TNL} recently developed a generalised framework in which the equations of motion generated by minimising a time non-local action $\Sigma$ are defined as
\begin{eqnarray}\label{eq:TNL_Euler_Lagrange_eq}
\frac{\delta \Sigma [ {\bf R}, \dot{{\bf R}}; {\bm \zeta}]}{\delta {\bf R}_I(s)}-\frac{d }{d s}\frac{\delta \Sigma [  {\bf R}, \dot{{\bf R}}; {\bm \zeta}]}{\delta \dot{{\bf R}_I}(s)}=0,
\end{eqnarray}
where $s\in[0,T]$ and ${\delta}/{\delta {\bf R}(s)}$ and ${\delta}/{\delta \dot{{\bf R}}(s)}$ are now functional derivatives with respect to the positions and velocities of the $I$-th effective CG site, respectively. Note that, in the case of a time-local Lagrangian function, Eq. \ref{eq:TNL_Euler_Lagrange_eq} reduces to the usual form of Eq. \ref{eq:euler_lagr_newt}.

Based on this extension of the Euler-Lagrange formalism, we here propose an effective time non-local action $\Sigma$ associated to the GLE of the CG system:
\begin{eqnarray}\label{eq:CG_act_expl}
&\Sigma[{\bf R}, \dot{\bf R};{\bm \zeta}]  =  \int_0^T dt \left [  K_R(\dot{\bf R}(t)) - U({\bf R}(t))+  \right . \\
 &+ \left . \sum_{J=1}^{N_B} \dot{\bf R}_J(t)\cdot\int_{0}^t dt' \  Q(t - t',{\bm \zeta}) \dot{\bf R}_J(t') + \right . \nonumber \\
 & + \left .\sum_{J=1}^{N_B} \bm{\xi}_J(t,{\bm \zeta})\cdot{\bf R}_J(t)\right], \nonumber
\end{eqnarray}
where $Q(\tau,{\bm \zeta})$ is related to the memory kernel $K$ in Eq.~\ref{eq:CG-eom} by $\frac{d}{d\tau}Q(\tau,{\bm \zeta})=K(\tau,{\bm \zeta})$. By plugging the action $\Sigma$ of Eq.~\ref{eq:CG_act_expl} in Eq.~\ref{eq:TNL_Euler_Lagrange_eq}, one obtains the generalised Langevin equations of motion presented in Eq.~\ref{eq:CG-eom} for the dynamics of the CG particles, under the requirement---representing an approximation---that the time evolution of the low-resolution system depends only on its past, and not on its future; for the sake of brevity we will here omit the full derivation, all details being provided in the Supporting Information.

As anticipated, it is our assumption that the CG trajectory ${\bf R}(t)$ actually follows a dynamics for which the action $S$ in Eq.~\ref{eq:euler-lagr-sep-2} is extremal. Given the effective action $\Sigma[{\bf R}, \dot{\bf R}; {\bm \zeta}]$ defined in Eq.~\ref{eq:CG_act_expl}, we treat the trajectory of the CG variables as fixed, in that it is determined by the reference, all-atom coordinates ${\bf r}(t)$ projected onto the CG degrees of freedom through the mapping $\map({\bf r}(t))$, see Eq.~\ref{eq:mapping}. Subsequently, we look for the fast dynamics properties ${\bm \zeta}$ that minimise $\Sigma$, where the properties enter such action {\it via} its memory and noise components. This operation is justified by the assumption that the memory/noise terms of the GLE are actually representative of the fast degrees of freedom, so that minimising $\Sigma$ with respect to the properties ${\bm \zeta}$ is equivalent to minimising it with respect to the trajectories of those variables. Our \emph{consistency equation for the dynamics} of the CG model thus reads
\begin{eqnarray}\label{eq:const-dyn}
\frac{\partial \Sigma[{\bf R}, \dot{\bf R}; {\bm \zeta}]}{\partial {\bm \zeta}}\bigg\rvert_{\bm \zeta = \bm \zeta^{\star}} = 0.
\end{eqnarray}
The solution $\bm \zeta^{\star}$ to Eq.~\ref{eq:const-dyn} will provide \emph{optimised} parameters characterising the influence of the fast dynamics on the time evolution of the low-resolution system; in the case under consideration, these consist of optimised memory kernel and noise terms $K(\tau,{\bm \zeta^{\star}})$ and ${\bm \xi}_I(t,{\bm \zeta^{\star}})$. These ingredients can be subsequently employed for \emph{generating} the CG trajectory via the associated GLE, see Eq.~\ref{eq:CG-eom}, the statistical properties of which can be finally compared to those predicted by the original, mapped trajectory $\map({\bf r}(t))$ to quantify the accuracy of the approach.

In practice, the effective action $\Sigma$ in Eq.~\ref{eq:CG_act_expl} can be recast in the form of a sum over discrete times, enabling its calculation on the trajectory obtained by projecting the results of an atomistic molecular dynamics simulation of the system onto their CG counterpart. A parametric functional form of the memory kernel and the associated noise is imposed through an {\it Ansatz} based on physical considerations, see Sec.~\ref{sec:kern}; the minimisation in Eq. \ref{eq:const-dyn} is then carried out with respect to these parameters so as to determine their optimised values entering the GLE of the CG system. A set of constraints is further introduced in the minimisation protocol in order to stabilise the numerical solution to the particular optimisation problem that emerges from our Ansatz for the kernel; all details about these technical steps are provided in the Supporting information.

\section{Discussion}\label{sect:application}

The paramount importance of water for practically all natural and artificial processes, as well as its remarkable and unique physical properties \cite{eisenberg2005structure, tomobe2015_VAF}, has constantly fuelled the development of a vast library of atomistic and coarse-grained models of this molecule \cite{sanz2004phase,bizzarri2002molecular,anderson1995_CFD,olbers2012ocean,mark2001atomistic_water,johnson2007_water_model_simple,wu2010CG_water_model_complex,riniker2011_water_model_complex_2}. Given the structural simplicity and phenomenological complexity of water, we focussed on it to put our coarse-graining method at test. Specifically, we proceeded as follows: first, microcanonical all-atom simulations of a water system at an average temperature of $T=298$~K were performed, which provided the reference time evolution for the CG modelling. Subsequently, by mapping each water molecule onto a CG bead located in the molecule's center of mass, see Fig.~\ref{fig:water_system}, we moved to the construction of a force field for the low-resolution system. We relied on a pairwise-additive approximation, and determined the CG pair potential $U_2(R)$ between two water beads via the iterative Boltzmann inversion method \cite{reith2003IBI,rosenberger2016IBI} starting from the atomistic radial distribution function (RDF) $g(R)$ among the water molecules' centers of mass. Finally, we optimised the parameters of the GLE, namely the kernel $K(t,\bm \zeta)$ and the noise $\bm \xi(t,\bm \zeta)$ that appear in Eq.~\ref{eq:CG-eom}. As outlined in the previous section, this was achieved through the minimisation of the action $\Sigma[{\bf R}, \dot{\bf R};{\bm \zeta}]$ in Eq.~\ref{eq:CG_act_expl} w.r.t. the properties $\bm \zeta$, where $\Sigma$ was calculated over the all-atom trajectory of the water system projected onto the CG degrees of freedom.
An exhaustive description of the technical details involved in each of these steps is presented in Sec.~\ref{sec:methods} and in the Supporting Information.

Given the pair potential and the optimised kernel and noise, an in-house molecular dynamics integrator implementing a GLE solver was employed to perform simulations of the CG water system, see Supporting Information. We further simulated the CG model by relying on a plain Langevin equation (LE), so as to better quantify the improvement introduced by our GLE-based approach in reproducing the dynamics of the all-atom reference. Structural as well as dynamical properties of water were investigated: specifically, we computed the RDF $g(R)$ among the water molecules' centers of mass and, most importantly, their diffusion coefficient and velocity autocorrelation function (VACF). The results of these analyses in the three different setups---all-atom, CG LE and CG GLE simulations---are reported in Table~\ref{table-MSD} and Fig.~\ref{figure-Kernel_VACF}.

\begin{figure*}[ht]
\includegraphics[width=\textwidth]{fig3.pdf}
\caption{\textit{Main panel}: Velocity autocorrelation function $C(t)$ of the water molecules' center of mass obtained in the three different setups considered in this work: ``Atomistic'' refers to the results obtained from the fully atomistic SPC-E water system simulated in the microcanonical ensemble; ``LE" refers to the results obtained from the CG model simulated via a plain Langevin equation; Finally, ``GLE" results were computed from the CG simulation that integrates the GLE with the optimised memory kernel and noise. In contrast to the CG results obtained via a plain LE, the CG system simulated via the GLE reproduces the atomistic VACF with notable accuracy. \textit{Inset}: Comparison of the radial distribution functions $g(R)$ among the water molecules' centers of mass calculated in the three different setups. We observe that the CG potential obtained through IBI reproduces the atomistic results for the $g(R)$ in both the LE and GLE cases.}
  \label{figure-Kernel_VACF}
\end{figure*}

\begin{center}
\begin{table*}[ht]
\begin{tabular}{|p{4.5cm}|c|c|c|}
\hline
 & Atomistic & LE & GLE \\ \hline
Diffusion coefficient from MSD $[\mathrm{cm^2/s}  \times 10^{-5}]$ & $2.44 \pm 0.11$  &$2.40 \pm 0.06 $        & $2.42 \pm 0.02 $\\ \hline
Diffusion coefficient from VACF $[\mathrm{cm^2/s}  \times 10^{-5}]$ & $1.76 \pm 0.03$  & $1.99 \pm 0.04$        & $2.01 \pm 0.05$\\ \hline
Temperature {[}K{]}   & $298.09 \pm 1.18$                & $297.52 \pm 0.54 $         &$ 298.02 \pm 1.43$      \\ \hline
\end{tabular}
\caption{\textit{Top and middle row}: diffusion coefficient in the three different setups: Atomistic model, CG model simulated via a plain LE, and CG model simulated via the GLE with the optimised kernel and noise. \textit{Bottom row}: temperature of the system in the three setups. All simulations were ran at the same thermodynamical state point of volume and number of particles, see text.}
\label{table-MSD}
\end{table*}
\end{center}

First, we verified that the structural properties of the low-resolution model as obtained in both the CG setups were consistent with those of the reference, high-resolution one. That this is the case can be inferred from the inset of Fig. \ref{figure-Kernel_VACF}, where we report the RDF $g(R)$ for the three cases of interest: all-atom simulation, CG simulation through the LE, and CG simulation with the GLE. The associated curves overlap within line thickness, showing that, as expected~\cite{di2019systematic}, the structural equilibrium features of the CG system are decoupled from its kinetics.

We then moved to the computation of dynamical quantities, starting with the diffusion coefficient $D$ that was obtained from the slope of the mean square deviation (MSD) of the molecules' center of mass according to the relation MSD$(t) = 6Dt$. We observe that the coarse-grained LE and GLE results for $D$ are in good agreement with the atomistic reference, see Table~\ref{table-MSD}. Such an agreement is not surprising, as the reproduction of the all-atom diffusion coefficient was more or less explicitly enforced in the dynamics of the two CG models: more directly in the case of the LE, where we appropriately tuned the associated dissipation coefficient (see Sec.~\ref{sec:mdsim}), and somewhat implicitly in the GLE, where we introduced a constraint that relates the integral of the memory kernel to $D$ in the action minimisation workflow (see Supporting Information).

The most important dynamical property to quantify the improvement introduced by the GLE is thus the VACF $C(t)$, with
\begin{equation}
C(t)=\frac{ \sum_{t_0} \sum_{I=1}^{N_B} {\bf V}_I(t+t_0) {\bf V}_I(t_0) }{\sum_{t_0}\sum_{I=1}^{N_B} {\bf V}_I(t_0) {\bf V}_I(t_0) },
\end{equation}
where ${\bf V}_I(t)$ is the velocity of the $I^{th}$ CG water molecule center of mass at time $t$, $t_0$ being the reference starting point. Critically, this time-dependent observable was never involved in either the LE or GLE parametrisation workflow.

Fig.~\ref{figure-Kernel_VACF} shows that the VACF of the CG model simulated in the plain LE setup is markedly different from the all-atom one: more specifically, it decays to zero over significantly smaller time scales, and further fails to give rise to the negative region that is observed in the atomistic $C(t)$ before this reaches saturation. On the contrary, the CG VACF obtained by relying on the GLE with the optimised memory kernel and noise reproduces the high-resolution reference with remarkable accuracy. Interestingly, for small and intermediate time scales CG results for $C(t)$ appear to constitute a smoothened analogue of their atomistic counterpart; at the same time, we note the presence of a small, spurious oscillation that arises for long time scales in the GLE VACF and that is instead absent in the all-atom case. This comparison thus demonstrates that the proposed method allows the self-consistent parametrisation of a generalised Langevin equation that, once employed to simulate a coarse-grained model, endows it with a dynamics that is able to match the one associated to the underlying, original all-atom representation in a sensibly more quantitative manner.

It is now interesting to compute the diffusion coefficient, rather than from the mean square displacement, directly from the VACF making use of the relation \cite{Zwanzig_book}
\begin{equation}\label{vacfdiff}
D = \frac{1}{3} \int_0^\infty C(t) dt.
\end{equation}
We report the results associated to the three cases of interests in Table~\ref{table-MSD}. As for the all-atom simulation, we observe that the numerical value of the diffusion coefficient of water obtained via Eq.~\ref{vacfdiff} is different from the one derived from the mean square displacement, a discrepancy that is commonly present in the estimate of Green-Kubo-like coefficients via $C(t)$ \cite{Reilly1971VAF}. The same mismatch between MSD and VACF results for $D$ separately holds, albeit to a lesser extent, also in the case of the CG model simulated in the LE and GLE frameworks. We note that calculating the diffusion coefficients of the three different setups via the corresponding VACFs worsens the agreement between CG and all-atom predictions that was instead observed in the MSD case, see again Table~\ref{table-MSD}; at the same time, LE and GLE VACF results for $D$ are still perfectly compatible. Importantly, Fig.~\ref{figure-Kernel_VACF} displays that such an agreement between the dynamic properties of the two CG models is only apparent, as the underlying velocity autocorrelation functions substantially differ, with the LE $C(t)$ critically lacking consistency with the underlying high-resolution reference.

\section{Conclusions}\label{sec:conclusions}

In conclusion, the presented approach offers a novel conceptual framework to gain insight into the problem of the loss of dynamical consistency between high-resolution models and their coarse-grained counterpart, and thus contributes a new tool to mitigate it. In this work we introduced a time non-local action integral that depends on the coarse-grained coordinates explicitly, as well as on the residual fast coordinates in an effective, parametric manner. On the one hand, the minimisation of this action with respect to the coarse-grained variables results in the generalised Langevin equation governing the coarse-grained system; on the other hand, by computing this action on the coarse-grained trajectory obtained from a reference, all-atom simulation, one can carry out its minimisation with respect to the parameters representing the effect of the fast variables on the slow ones. This latter operation results in consistency conditions that link the time scales of the atomistic and CG models.

We followed this strategy to optimise the memory and noise components of the GLE to be employed in the simulation of a CG model. We chose a reasonable {\it Ansatz} for the kernel, and optimised its parameters through a derivative-free minimisation of the CG action. The numerical results obtained for a CG representation of a system consisting of water molecules confirmed that the theoretical framework and the proposed method substantially improve the accuracy with which the low-resolution model reproduces the velocity autocorrelation function of the all-atom reference. This improvement shows that the numerical procedure illustrated and applied in the present work can thus be fruitfully employed in the parametrisation of CG models that would reproduce not only the equilibrium, structural properties of a system, but also its dynamics.

Altogether, these results support the hypothesis that the reference system's fast dynamics, which is integrated out in the process of coarse-graining, satisfies a least action principle where the slow degrees of freedom constitute a given background. This provides comforting evidence that dynamical problems in the field of coarse-graining can be addressed in the framework of a global action, thereby contributing a novel instrument to understand the origin of the time-scale distortions that affect effective models of soft matter.

\section{Methods}\label{sec:methods}

\subsection{Coarse-grained effective potential: the iterative Boltzmann inversion method}

In principle, the multi-body potential of mean force (MB-PMF) $W({\bf R})$ acting among the CG effective interaction sites defined in Eq.~\ref{eq:consist:a} completely accounts for all equilibrium properties of the low-resolution model. In practice, however, $W({\bf R})$ can be computed explicitly only in few very simple cases \cite{diggins2018optimal}; hence, approximations are needed. Several approaches aim at approximating the MB-PMF with a sum of (state-dependent) effective pair potentials, that is,
\begin{equation}\label{eq:PMFpair}
W({\bf R}) \simeq  U({\bf R})= \sum_{I<J}U_2({\bf R}_I,{\bf R}_J),
\end{equation}
where, in homogeneous and isotropic systems,
\begin{equation}
U_2({\bf R}_I,{\bf R}_J) = U_2(|{\bf R}_I-{\bf R}_J|)=U_2(R_{IJ}). \label{eq:twobodypot}
\end{equation}
In this work, we determine the pair potential $U_2$ between two CG water beads by relying on the popular iterative Boltzmann inversion (IBI) method \cite{reith2003IBI,rosenberger2016IBI}. The IBI workflow starts from an initial guess $U_2^0$ for the pair interaction, e.g. given by
\begin{eqnarray}
U_2^{0}(R) = -k_BT\ln(g^{ref}(R)),
\end{eqnarray}
where $g^{ref}(R)$ is the reference atomistic radial distribution computed on the CG sites; the potential is then iteratively updated according to the following rule:
\begin{eqnarray}
\label{eq:IBI}
U_2^{n+1}(R) = U_2^{n}(R) + k_B T \ln \left( \frac{g^{n}(R)}{g^{ref}(R)} \right ),
\end{eqnarray}
where $g^{n}(R)$ is the radial distribution function obtained from a simulation of the CG model that employs the potential $U_2^{n}(R)$. As the number of iterations $n$ increases, $g^{n}(R)$ gets closer and closer to the reference RDF, and the update in Eq.~\ref{eq:IBI} becomes progressively smaller; the procedure is interrupted when the desired degree of convergence of the $U_2$ is achieved.
In this work, the IBI procedure was carried out using the VOTCA package \cite{ruhle2009versatile}.

\subsection{Molecular dynamics simulations}\label{sec:mdsim}
{\it Reference atomistic simulation.} MD simulations were performed for $1001$ three-points, rigid SPC/E water molecules in a cubic box with periodic boundary conditions by relying on the GROMACS 2019 software suite~\cite{van2005gromacs,abraham2015gromacs}. The Lennard-Jones intermolecular potential acting between two oxygen atoms was cut-off at $0.8$ nm, while the electrostatics of the system was computed through the Particle-mesh Ewald (PME) method. The LINCS algorithm was employed to preserve the holonomic constraints of the oxygen-hydrogen intramolecular bonds. In all simulations, the time step was set to $\Delta t=2$~fs.

A set of preliminary runs enabled the equilibration of the system around an average temperature and pressure of $T=298$~K and $p=1$~bar, respectively. Specifically, we initially performed a simulation of $0.2$~ns in the NVT ensemble by relying on the Nos\'{e}-Hoover thermostat; subsequently, a second equilibration in the NPT ensemble of $0.2$~ns, achieved by combining the Nos\'{e}-Hoover thermostat with a Parrinello-Rahman barostat, provided additional control over the system pressure. An equilibrated configuration extracted from the NPT run, whose linear box size was $L\simeq3.09$~nm, was then employed as initial condition for a microcanonical (NVE) production simulation of $0.5$~ns, where we integrated the system's equation of motion through a velocity Verlet algorithm. This simulation provided the reference data for the construction of the CG model as well as for extrapolating the thermodynamic properties of interest, namely the system temperature ($T=298$~K, see Table~\ref{table-MSD}), diffusion coefficient, velocity autocorrelation function $C(t)$, and radial distribution function $g(R)$ among the water molecules' centers of mass.

{\it CG model.} Starting from the all-atom results obtained for the $g(R)$, the effective pair potential $U_2(R)$ acting between two CG water molecules was parametrised by relying on the iterative Boltzmann inversion method (IBI) \cite{reith2003IBI,rosenberger2016IBI}, see Eqs.~\ref{eq:PMFpair}-\ref{eq:IBI}, making use of the VOTCA software \cite{ruhle2009versatile}. The initial configuration of the CG system was generated by mapping an equilibrated, all-atom NVE snapshot onto point-like beads located on the molecules' centers of mass, see Fig.~\ref{fig:water_system}. The simulations necessary for the IBI method to compute the CG potential were then conducted in NVT conditions, with $T=298$~K; in doing so, we relied on the equivalence between the canonical and microcanonical ensembles of the all-atom system, so that NVE results for the $g(R)$ are equal to their canonical counterpart when computed at the same thermodynamic state point.

{\it CG simulations.} Given the IBI pair potential $U_2(R)$ acting among water beads, simulations of the resulting CG model were performed in two different setups: a framework in which the time evolution of the low-resolution system was dictated by a plain Langevin equation, as well as one in which the GLE approach with the optimised memory kernel and noise was employed. The aim of the LE simulation was to provide a reference behaviour against which quantifying the ability of our method to mitigate the loss of dynamical consistency that arises when passing from an all-atom to a CG description of the system.

Both LE and GLE simulations involved 1001 CG beads in a cubic box of $L\simeq 3.09$~nm side with periodic boundary conditions, and were performed with a time step $\Delta t=2$~fs and at a temperature $T=298$~K. The initial configuration of the system was provided by a CG snapshot equilibrated in the NVT ensemble and extracted from the IBI procedure at convergence. The LE simulation relied on an in-house code that implemented the numerical integration scheme proposed in Ref.~\cite{goga2012efficient}, where we tuned the damping parameter so that the diffusion coefficient calculated in the CG water system coincides with its all-atom counterpart. As for the GLE, a modified velocity Verlet algorithm was employed to integrate the equations of motion, see Sec.~VI of the Supporting Information, using a in-house code available in a Zenodo repository  with DOI 10.5281/zenodo.6037951.

\subsection{Action minimisation and determination of the GLE parameters}
\label{sec:kern}

To implement the numerical minimisation of the effective CG action functional, we define the memory kernel in terms of a function depending on a set of parameters ${\bm \zeta}$ that are representative of the effect of the fast degrees of freedom as well as the ones the optimisation acts upon, see Eqs.~\ref{eq:CG_act_expl} and~\ref{eq:const-dyn}. Specifically, we define the kernel as follows:
\begin{equation}
\label{eq:kernelform}
K(t,{\bm \zeta})=K(t,a,{\bm b})=g(t,a)b(t,{\bm b})=e^{-a t^2}b(t,{\bm b}),
\end{equation}
that is, a Gaussian prefactor $g(t,a)$ followed by a modulation term $b(t,{\bm b})$, the latter being a positive function of time such that $b(t_k)=b_k $ at time $t_k=k\Delta t$. It follows that the set of parameters $\bm \zeta $ consists in the set of values $[a, b_0, b_1,..,b_{M-1}]$. The kernel in Eq. \ref{eq:kernelform} is thus implemented in the definition of the action, Eq. \ref{eq:CG_act_expl}, and the minimisation is performed to find the specific set of parameters $a$ and ${\bm b}= [b_0,b_1,...,b_{M-1}]$ that make the discretised functional stationary. In addition to the optimised kernel, knowledge of these parameters further enables the parameterisation of the noise terms in the GLE. The detailed description of this procedure, including the definition of the constraints and regularisation terms employed in its course, is provided as Supporting Information.

\section{Data availability}
This work consisted in three main computational steps: the atomistic simulation and the parametrisation of the effective potential, performed with the GROMACS software~\cite{van2005gromacs,abraham2015gromacs}; the optimisation of the memory kernel; and the LE/GLE simulations and related analyses. The last two steps were carried out making use of in-house MATLAB codes. All input files and scripts needed to obtain these data are available in the Zenodo repository with DOI 10.5281/zenodo.6037951.

\section{Acknowledgments}

The authors thank Leonardo Ricci for a critical reading and insightful comments. This project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant 758588). RP acknowledges support from the Italian Ministry of Education, University and Research (MIUR) through the FARE grant for the project HAMMOCK (Grant R18ZHWY3NC).

\section{Author contributions}

RP conceived the study and proposed the method. PL performed the simulations, wrote the software, and collected the data. PL and RM developed the method. All authors contributed to the analysis and interpretation of the data. All authors drafted the paper, reviewed the results, and approved the final version of the manuscript.

\section{Competing interests}

The authors declare no competing interests.

\section{Supporting Information}

In the Supporting Information (\emph{i}) we report a brief summary of the Mori-Zwanzig formalism; (\emph{ii}) we prove that the effective, time non-local CG action introduced in this work, when minimised with respect to the CG trajectory, results in the GLE for the dynamics of the low-resolution system; (\emph{iii}) we provide all technical details concerning the  discretisation and minimisation of the CG action necessary to extract the optimised parameters of the GLE; (\emph{iv}) we derive the constraints on the memory kernel introduced in the minimisation workflow; and (\emph{v}) we report the numerical integrator employed in this work to solve the GLE for the CG system.

redpajama_set_name: RedPajamaArXiv


I stumbled upon this deserted hotel during a visit to some gloriously deserted coastline. Left to its own devices in the teeth of the oncoming Atlantic storms, it's certainly picked up a few scars.
There are surprisingly few abandoned buildings like this in the UK, and should you find them, they're often much harder to access than this one, too. [All the doors were blocked by heavy sleepers except one – presumably the one the people who blocked up all the other doors exited from! So I managed to squeeze in.] I spent about an hour familiarising myself with echoes of things not there. At times, certainly in the darker recesses of the north side of the building, distinctly unnerving.
The setting [not fully revealed in the mainly interior focus] overlooking Whitesands Bay in splendid isolation contained a curious symbiosis with the infamous The Overlook Hotel from The Shining. As I began processing, the appearance of a few ghostly echoes felt inevitable; the voices long gone, the series became a search for the forgotten.


text:
Australia News Brief and Action Alert
The Adani Carmichael Mega Coal Mine
February 11, 2018February 24, 2018Julian Atchison and Alex Tuai
Spotlight Issue: The Adani Carmichael Mega Coal Mine
Most loudly hailed by former Party Federal Resources Minister Matt Canavan, Deputy Prime Minister Barnaby Joyce, and Prime Minister Malcolm Turnbull, the Australian government is pushing for the construction of the Adani Energy Group's Carmichael mega coal mine despite widespread protests from groups including Aboriginal peoples, activists, and local farmers. If completed, the mine would be the largest in Australia and one of the largest in the world spanning "more than 30 kilometers. It would annually produce around 60 million tonnes of coal and consume around 12 billion liters of water".
The development of the Carmichael mega mine would take Australia off track from its pledge to "reduce emissions to 26-28% of 2005 levels by 2030". Calculations have found that if the Adani Carmichael mine proceeds, the annual emissions would exceed the annual emissions level of New York City. Allowing the mine to proceed will worsen the effects of climate change, deplete water levels in an already drought-prone region, and increase instances of human health issues like life-threatening coalworkers' pneumoconiosis or "black lung" and asthma.
The proposed mine and development of the Galilee Basin for coal mining has been condemned by several international bodies who argue that the subsequent carbon emissions will undermine Australia's pledges to the Paris Agreement. While the emissions aren't directly attributed to Australia, because the coal will be exported to India, the carbon emissions and depletion of groundwater will negatively affect Australia and its Pacific neighbors during the proposed operational timeline and for the years to come.
Personal letters, emails and phone calls have the most influence and impact on decision makers and policy makers so we encourage you to contact each individual directly. However if you don't have time, fill out your name and email and click below and the message will be sent on behalf of yourself and Climate Scorecard.
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Deputy Prime Minister and National Party Leader Barnaby Joyce
Mail: Hon. Barnaby Joyce, MP
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In 'immersoin game' about World War 1 in your municipality In Dilemma 14/18 young people are confronted with difficult and lifelike dilemmas from the First World War. The events are based on real stories from real youngsters from that time. Each group lives in a person who tried to survive during 1914-1918. The game is adaptable to local situation: you can add local character and introduce young people to local stories. pedagogical goals: Dilemma 14/18 involves young people in a playful way in the memory of the First World War. Young people experience that a war involves much more than just battles between heroic soldiers. Children and young people learn about the impact of the Great War on daily life in their own village/city. Young people feel the madness of war. Dilemma 14/18 introduces young people to different voices: a Flemish-minded farmer's son, a patriotic pastor, a refugee in France or a German family in Brussels. In this way, young people get different perspectives on the occupation. (multiperspective) Indirectly, young people learn to estimate complex dilemmas, listen to each other's opinions and work together. it is a non-formal learning, made for youth work but also suitable as an active activity for students in formal learning settings.
Dilemma 14/18 consists of three parts: the start-up, the game and the end. The start-up game divides the participants into the desired number of groups and introduces young people to the individual which which they are connected. Group 1 is Martha, group 2 lives in the role of George, group 3 is introduced to Louise, ... They come to know, for example, which hobbies their character has and how old he / she is. Through a short running game, the participants discover the characters of the other groups. The game coordinator gives the game explanation and starts the game. The young people go to five locations and each time play a game order to earn food stamps. They also solve a dilemma at each location, linked to the story of their character. In doing so, they weigh up the moral side of the dilemma, their chances of survival with each choice option and the number of food vouchers that each choice will produce. The five locations are preferably symbolic locations, which have a link with events that happened during First World War. The games are always connected to a theme of WWI (such as smuggling, poison gas, shortage of food,...) At the end of the game all participants gather back at the central spot and a groupdiscussion follows. Based on the risks that that character took in the game and the play of fate, one character dies through the lottery draw. The participants of the other characters can join a snack table and buy their snack with the collected food stamps. What choices did you make? What did you find most important: food stamps or survival chances? Or just the moral? The participants exchange their experiences through a statement game.
Actually we were somehow dissapoint by the amount the game has been bought/played. Since we believe it is a very qualitative and at the same time fun game, we hoped for more response and interest. So we definitly have some doubts whether it was an effective practice (a lot of costs in development, much material left over) But I think that the positive aspects are the interactive, playful way of approaching this topic. Very suitable for non-formal settings such as youth work. I also think that it is highly adaptable, to the local context where you are playing the game. This makes it easily recognisable, and easier to affiliate with the persons.
We gave some workshops for teachers and youth workers interested in playing the game with their target group. In this workshop we presented the game, its goals. We showed how they could adapt the game to their local situation (add a local character and write a story about this person). We also introduced them to the website, how to work with it and also showed them all the materials necessary.


To provide a structured teaching & learning program for members in building & presentation techniques.
ALUG was grown from a sense of responsibility to the wider Lego AFOL community. The idea behind ALUG is to bring together all LUGS, while still retaining individual LUGS across Australia.
The founding members of ALUG, a registered, Non-Profit Incorporation, listened to the wider community and realised there was a niche to be filled in the area of skills building and community engagement.
ALUG plans on bringing some exciting initiatives to the world of AFOL's with monthly tutorials/workshops.
The aim of ALUG is to foster a nurturing environment in which skills building and information sharing take the forefront.
What is a LUG? a LUG is a Lego User Group. But you probably already knew that.
Transparency, Skills, and Information Sharing.