text:
\section{Choice of an analytic representation of non-Maxwellian distribution functions}
\label{ToC.NMDFs}
This section deals with the velocity phase space representation of non-Maxwellian distribution functions (NMDFs) using specific analytic forms instead of the usual numerical discretization. In fact, the numerical approximation is not efficient for the description of some NMDFs because of the requirement to use a high number of terms. For example, it is not efficient to use a numerical approximation of a Maxwellian distribution function (MDF) in comparison to its analytic form which uses only 3 variables (density, fluid velocity and temperature). Indeed, this work argues the advantage of using a mesh-free representation of the distribution function in the velocity phase space by using fluid parameters (called here the hidden variables) instead of the discretization in the velocity phase space. We adopt the notation of ${\rm v}$ for the one dimension velocity phase space coordinate, $\vpartb{x}$ for the three dimension position and $t$ for the time. The generalization to three dimensional velocity phase space is ongoing for isotropic distribution functions but some complexities appear for anisotropic ones~\cite{Izacard_2016_Paper2}. Further work will include anisotropic NMDFs. For simplicity of notations the division of the temperature by the mass is omitted in the distribution function. \\
In this section we show notations for existing distributions (the KDF and the sum of two MDFs) and we introduce a new class of NMDFs (i.e., the INMDF) which can be viewed as a non-orthogonal generalization of the sum of Maxwellians and Hermite polynomials.

\subsection{The Kappa distributions}
The KDFs are commonly used in astrophysics~\cite{Scudder:1992,Leubner:2008} to describe different tail power laws than the Maxwellian. The definition of the KDF $f_{\kappa}(\vpartb{x},{\rm v},t)$ is
\begin{eqnarray}
\label{eq.nM.fKappa}
\displaystyle
f_{\kappa} &=& \Delta_{\kappa}  \left(1 + \frac{{\rm v}^2}{W_{\kappa}}\right)^{-(\kappa+1)},
\end{eqnarray}
where
\begin{eqnarray}
\displaystyle
\Delta_{\kappa} &=& \frac{n}{\sqrt{\pi W_{\kappa} }} \left( \frac{\Gamma (\kappa+1)}{\Gamma \left(\kappa-\frac{1}{2}\right)} \right)^{1/3}, \\
\displaystyle
W_{\kappa} &=& (2 \kappa-3) T,
\end{eqnarray}
and $\Gamma(z)$ is the Euler Gamma function. The fluid quantities, which are evolving in time and are functions of the position, are $n(\vpartb{x},t)$, $W_{\kappa}(\vpartb{x},t)$ (or $T(\vpartb{x},t)$) and $\kappa(\vpartb{x},t)$. For a shifted distribution function the fluid velocity $v(\vpartb{x},t)$ appears by the change of variable ${\rm v} \rightarrow {\rm v}-v$ in Eqs.~(\ref{eq.nM.fKappa}), (\ref{eq.nM.KqkappaT}) and~(\ref{eq.nM.KqkappaTa}). The KDF recovers the MDF for $\kappa \rightarrow \infty$. Many properties of the KDF have been published in the literature and are not reproduced here. Integrals of KDFs over the velocity phase space introduce functions detailed in App.~\ref{ToC.App.KqkappaTa}
\begin{eqnarray}
\label{eq.nM.KqkappaT}
\displaystyle
K_{q}(\kappa ,T) &=& \int_{-\infty}^{\infty} {\rm v}^{q} \left( 1 + \frac{{\rm v}^2}{W_{\kappa}} \right)^{-(\kappa +1)} d{\rm v}, \\
\label{eq.nM.KqkappaTa}
\displaystyle
K_{q}(\kappa ,T,a) &=& \int_{a}^{\infty} {\rm v}^{q} \left( 1 + \frac{{\rm v}^2}{W_{\kappa}} \right)^{-(\kappa +1)} d{\rm v}.
\end{eqnarray}
By using these integrals, it is trivial to obtain all fluid moments of any shifted KDF centered at the fluid velocity $v$ as function of a finite sum of the function $K_q(\kappa,T)$. The KDF is successfully used to describe different power laws in velocity phase space. However, because an over population of fast particles localized around a specific velocity is often observed, the sum of two Maxwellian can be a better approximation for these cases.

\subsection{The sum of Maxwellians}
The approximation of localized super-thermal tails is commonly used~\cite{Popov:2009,Jaworski:2013,Jaworski_2015_Private}. It is usually called the bi-modal distribution function where two different temperatures dominate as function of the velocity coordinate ${\rm v}$. The sum of two shifted MDFs $f_{2M}(\vpartb{x},{\rm v},t)$ reads
\begin{eqnarray}
\label{eq.nM.f2Maxw}
\displaystyle
f_{2M} &=& \Delta \exp\left( -\frac{1}{2 T} {\rm v}^2 + \frac{v}{T} {\rm v} \right) \nonumber\\&+& \Delta_f \exp\left( -\frac{1}{2 T_f} {\rm v}^2 + \frac{v_f}{T_f} {\rm v} \right),
\end{eqnarray}
with
\begin{eqnarray}
\label{eq.nM.f2Maxw.del}
\displaystyle
\Delta &=& \frac{n}{\sqrt{2\pi T}} \exp\left( -\frac{v^2}{2 T} \right), \\
\label{eq.nM.f2Maxw.delf}
\displaystyle
\Delta_f &=& \frac{n_f}{\sqrt{2\pi T_f}} \exp\left( -\frac{v_f^2}{2 T_f} \right),
\end{eqnarray}
where the density, fluid velocity and temperature are respectively for the bulk and the super-thermal (i.e., fast particles) populations $n(\vpartb{x},t)$, $v(\vpartb{x},t)$, $T(\vpartb{x},t)$ and $n_f(\vpartb{x},t)$, $v_f(\vpartb{x},t)$, $T_f(\vpartb{x},t)$.
The velocity integrals are directly given by the function $J_k(a,b)$ detailed in App.~\ref{ToC.App.Jkab} and defined by
\begin{eqnarray}
\label{eq.Gen.Jkab}
\displaystyle
J_k(a,b) &=& \int_{-\infty}^{\infty} {\rm v}^k \exp\left(-a {\rm v}^2 + b {\rm v}\right) d{\rm v}.
\end{eqnarray}
Another useful function is $J_k(a,b,c)$ detailed in App.~\ref{ToC.App.Jkab0} and App.~\ref{ToC.App.Jkabc} and defined by
\begin{eqnarray}
\label{eq.Gen.Jkabc}
\displaystyle
J_k(a,b,c) &=& \int_{c}^{\infty} {\rm v}^k \exp\left(-a {\rm v}^2 + b {\rm v}\right) d{\rm v}.
\end{eqnarray}
The generalization with a sum of more than two MDFs is possible with the Radial Gaussian Basis Function (RGBF) which has successfully been used for artificial neural networks. Many properties of the bi-modal distribution function and the RGBF have been published in the literature and are not reproduced here. \\
This representation is accurate for example when there is an external production of fast particles and when there is no interaction between the bulk plasma (of collisionality $\nu_{\rm th-th}$) and these fast particles (of collisionality $\nu_{\rm th-f}$). This means that the RGBF is valid only at two collisional regimes: the limit of no collision between thermalized and fast particles (i.e., $\nu_{\rm th-f}=0$) or the limit of infinite collisionality (i.e., $\nu_{\rm th-th}/\nu_{\rm th-f} \rightarrow \infty$). In other words, the RGBF is not consistent with interactions between the bulk plasma and the fast particles at finite collisionality because each of them become non-Maxwellian. The novel representation introduced below is a possible solution of this inconsistency.

\subsection{The interpreted non-Maxwellian}
\label{ToC.NMDFs.INMDF}
The definition of the INMDF $f_I(\vpartb{x},\vpartb{v},t)$ was introduced for the first time in one dimension by Refs.~\cite{Izacard_2013_GA_Talk,Izacard_2013_Emails_Candy,Izacard_2016_FischSymposium}
\begin{eqnarray}
\label{eq.nM.fINMDF}
\displaystyle
f_I &=& f_0 + \delta f,
\end{eqnarray}
with
\begin{eqnarray}
\label{eq.nM.fINMDF.f0}
\displaystyle
f_0 &=&  \Delta \exp\left( -\frac{1}{2 T} {\rm v} + \frac{v}{T} {\rm v} \right), \\
\displaystyle
\label{eq.nM.fINMDF.delf}
\delta f &=& \Delta_I (\vpart{v}-c) \exp\left( -\frac{1}{2 W} {\rm v} + \frac{c}{W} {\rm v} \right),
\end{eqnarray}
and with $\Delta$ given by \refeq{eq.nM.f2Maxw.del} and
\begin{eqnarray}
\displaystyle
\Delta_I &=& \frac{\Gamma}{\sqrt{2\pi W^3}} \exp\left(-\frac{c^2}{2W}\right),
\end{eqnarray}
where the fluid hidden variables $\{n,v,T,\Gamma,c,W\}$ are function of $({\bf x},t)$. The first part $f_0$ is a MDF and due to the second part $\delta f$, the INMDF $f_I$ cannot be described with a finite number of terms using any of the existing analytic representations. The second part is related to a physical interpretation and could be valid for all collisionality regimes because $\delta f$ represents an enhancement of the energy of a population of particles. This new formula represents a real asset for the analytic modeling of NMDFs. More details of the difference between the INMDF and all existing analytic NMDFs are given in Sec.~\ref{ToC.NMDFs.Gen}. The moments $M_k = \int f {\rm v}^k d{\rm v}$ of the non-Maxwellian $f=f_I$ are
\begin{eqnarray}
\label{eq.fI.M0}
\displaystyle
M_0 &=& n, \\
\displaystyle
M_1 &=& n v + \Gamma, \\
\displaystyle
M_2 &=& n \left( T + v^2 \right) + 2 \Gamma c,  \\
\displaystyle
M_3 &=& n v \left( 3 T + v^2 \right) + 3 \Gamma \left( W + c^2 \right), \\
\displaystyle
M_4 &=& n \left( 3 T^2 + 6 T v^2 + v^4 \right) + 4 \Gamma c \left( 3 W + c^2 \right), \\
\displaystyle
M_5 &=& n v \left(15 T^2 + 10 T v^2 + v^4 \right) \nonumber\\&&+ 5 \Gamma \left( 3 W^2 + 6 W c^2 + c^4 \right), \\
\label{eq.fI.M6}
\displaystyle
M_6 &=& n \left(15 T^3 + 45 T^2 v^2 + 15 T v^4 + v^6 \right) \nonumber\\&&+ 6 \Gamma c \left( 15 W^2 + 10 W c^2 + c^4 \right),
\end{eqnarray}
and generally given as function of $J_k(a,b)$. The usual moments $P_k = 1/M_0 \int f (\vpart{v}-M_1/M_0)^k d\vpart{v}$ for $k \geq 2$ of the non-Maxwellian $f_I$ are also function of the hidden variables. We do not write these moments since it is not used here, but if needed, readers can easily obtain them from the given $M_k$ moments.
Nevertheless, we remark as expected for a MDF that when $\Gamma = 0$ the odd moments $P_{2k+1}$ are equal to 0 and the even moments are $P_{2k} = (2k-1)!!\ T^k$.
The coefficient $\Gamma$ has the dimension of a momentum (a density times a velocity) then $\Gamma/n$ has the dimension of a velocity, $c$ of a velocity, and $W$ of a temperature (square of a velocity, including the division of the temperature by the mass which is omitted by simplicity of notations). \\
We remark that, by using the method of hidden variables, there is no reason to resolve the inversion of the Eqs.~(\ref{eq.fI.M0})-(\ref{eq.fI.M6}) in order to extract the hidden variables as function of the fluid moments. To  date, the inversion of these formulas to extract the hidden variables $\{n,v,T\}$ as function of the moments $\{M_0,M_1,M_2\}=\{n,nv,nT+nv^2\}$ is possible only for a MDF and it seems too constraining to use this criteria for the description of non-Maxwellian steady-state distribution functions observed experimentally. \\
Fig.~\ref{fig.Izacardian_examples1}.(a) represents an INMDF (solid blue curve) with a correction (dashed red curve) centered at the mean velocity with respect to the MDF (dashed black curve). This example corresponds to an asymmetric distribution function observed at least in presence of ion orbit losses~\cite{Battaglia:2013} and will be investigated in future work. Fig.~\ref{fig.Izacardian_examples1}.(b) represents the same correction $\delta f$ but centered at a higher velocity coordinate (i.e., higher value of $c$).
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{}
\centering
\includegraphics[height=30mm,natwidth=585,natheight=441]{Fig1a.pdf}
\label{fig.Izacardian_examples1.1}
\end{subfigure}%
\begin{subfigure}[]{}
\centering
\includegraphics[height=30mm,natwidth=585,natheight=441]{Fig1b.pdf}
\label{fig.Izacardian_examples1.2}
\end{subfigure}%
\caption{Schema of the Maxwellian (dashed black curve) and the INMDF (solid blue curve) distribution functions and the difference (dashed red curve). These curves are obtained with the density $n=10^{19}$ m$^{-3}$, the temperature $T=10^4$ eV, the velocity space ${\rm v} \in \left[ 0 , 10^5 \right]$, the number of points in the velocity space $N_{\rm v}=1001$, the fluid velocity $v = ({\rm v}_{max}-{\rm v}_{min})/2$, the kinetic flux $\Gamma=0.1n\sqrt{T}$, the central flow (a) $c=0$ (b) $c=1.25 v$ and the width of the heat spread $W=T/2$. The physical interpretation of these three additional variables is given below.}
\label{fig.Izacardian_examples1}
\end{figure}
The tail at high energy (centered at $c>v$) is unstable when the distribution function increases and is not monotonous. In another words, the distribution function is unstable when its first derivative is positive around ${\rm v}=c$. Indeed, the stability criteria of the INMDF $f_I=f_0+\delta f$ is obtained when the derivative of the Maxwellian part $f_0$ is bigger than the derivative of the additional part $\delta f$ (i.e. $\partial f_I/\partial {\rm v}<0$). We found the stability criteria
\begin{eqnarray}
\displaystyle
&& \left( 1 - \frac{({\rm v}-c)^2}{W} \right) \frac{\Gamma}{\left(2\pi W^3 \right)^{1/2}} \exp\left( -\frac{\left({\rm v}-c\right)^2}{2 W} \right) \nonumber\\
\displaystyle
&&\quad < \frac{{\rm v}-v}{T} \frac{n}{\left(2\pi T \right)^{1/2}} \exp\left( -\frac{\left({\rm v}-v\right)^2}{2 T} \right),
\end{eqnarray}
then, for $W < T$ the highest value at ${\rm v}=c$ gives for $c>v$ the stability criteria
\begin{eqnarray}
\label{eq.nM.Izacardian.StabilityCriteria}
\displaystyle
\frac{\Gamma}{n (c-v)} \left(\frac{T}{W}\right)^{3/2} \exp\left( \frac{\left(c-v\right)^2}{2 T} \right) < 1.
\end{eqnarray}
For $W > T$, similar stability criterion can be obtained for a range of velocity but are not detailed here.
Moreover, the constraint to consider a strictly positive distribution function is given by the relation $f_I(\vpart{v})>0$ for all ${\rm v}$, particularly when, for $\vpart{v}<c$, the additional non-Maxwellian part $\delta f$ is minimum (for $\Gamma>0$). The minimum value of the additional non-Maxwellian part $\delta f$ is obtained at
\begin{eqnarray}
\displaystyle
\frac{\partial \delta f}{\partial {\rm v}} = 0 &\Leftrightarrow & \frac{({\rm v}-c)^2}{W} = 1, \nonumber\\
\displaystyle
&\Leftrightarrow & {\rm v} = c - \sqrt{W}.
\end{eqnarray}
Then the constraint of a strictly positive distribution function is given by $f_I\left({\rm v}=c - \sqrt{W}\right) > 0$, so
\begin{eqnarray}
\displaystyle
\frac{\Gamma \sqrt{T}}{n W} \exp\left( \frac{\left(c-v-\sqrt{W}\right)^2}{2 T} - \frac{1}{2} \right) < 1.
\end{eqnarray}
We remark that for $\Gamma<0$, similar relations can be obtained using the minimum value of $\delta f$ obtained for ${\rm v}>c$ (i.e., at ${\rm v} = c+\sqrt{W}$). \\
Before investigating corrections of some existing theories due to non-thermal population of particles, we have to understand how the previously given formulas could accurately describe deviations from MDFs. For that, the physical interpretation of the INMDF is given here.
\begin{figure}[!htbp]
\centering
\includegraphics[height=65mm,natwidth=585,natheight=441]{Fig2.pdf}
\caption{Schema of the physical interpretation of the INMDF. The Maxwellian $f_0$ (black curve), the INMDF $f_I$ (blue curve) and the difference $\delta f = f_I-f_0$ (red curve) are shown. These curves are obtained with the density $n=10^{19}$ m$^{-3}$, the temperature $T=10^4$ eV, the velocity space ${\rm v} \in \left[ 0 , 10^5 \right]$, the number of points in the velocity space $N_{\rm v}=1001$, the fluid velocity $v = ({\rm v}_{max}-{\rm v}_{min})/2$, the kinetic flux $\Gamma=0.1n\sqrt{T}$, the central flow $c=v+1.25\sqrt{T}$ and the width of the heat spread $W=T/2$.}
\label{fig.nM.interpretation}
\end{figure}
The fluid quantity $\Gamma({\rm x},t)$ is called here the kinetic flux, $c({\rm x},t)$ the central flow and $W({\rm x},t)$ the width of the heat spread. Similar to the graphical interpretation of $n$, $v$ and $T$ for the MDF, the graphical interpretation of the non-Maxwellian part $\Gamma$, $c$ and $W$ is shown in Fig.~\ref{fig.nM.interpretation}. \\
Fig.~\ref{fig.nM.interpretation} represents the displacement of a population of particles ($\sim 5\%$) from the dark-green area to the light-green area. The hidden variable $\Gamma$ is called the kinetic flux since it happens in the velocity phase space and has a dimension of a particle flux (i.e., a density times a velocity). The central flow $c$ is the velocity where the INMDF equals the MDF. Finally, the width of the heat spread $W$ characterizes the width in velocity phase space of the super-thermal population modified by an external source of energy (e.g., the current drive). In the following figures, the coefficients $(q,r,s)$ are used to parameterize the super-thermal tail such that $\Gamma = 2 q n \sqrt{T}/100$ where $q$ represents approximately the percentage of super-thermal particles over the total number of particles, $r$ represents the position of the central flow by the relation $c=v+r\sqrt{T}$, and $s$ represents the ratio of the width of the heat spread over the temperature $W=s^2 T$ (i.e., $s = \sqrt{W/T}$ is the ratio of the widths).

\subsection{Generalization with other INMDFs}
\label{ToC.NMDFs.Gen}
It is possible to use other formulas for the description of non-Maxwellians such as shown in
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{}
\includegraphics[width=40mm,natwidth=585,natheight=441]{Fig3a.pdf}
\label{fig.Izacardian_examples2.1}
\end{subfigure}%
\begin{subfigure}[]{}
\includegraphics[height=30mm,natwidth=585,natheight=441]{Fig3b.pdf}
\label{fig.Izacardian_examples2.4}
\end{subfigure}%
\caption{Example of two INMDFs obtained from \refeq{eq.fnM.Generalization} with different values of the fluid hidden variables.}
\label{fig.Izacardian_examples2}
\end{figure}
Fig.~\ref{fig.Izacardian_examples2} where different NMDFs are obtained from
\begin{eqnarray}
\label{eq.fnM.Generalization}
\displaystyle
f &=& \sum_{k=0}^{N_k} \frac{ a_k \left( {\rm v} - b_k \right)^{n_k} }{\left(2\pi d_k^{m_k}\right)^{1/2}}  \exp\left( -\frac { \left( {\rm v}-c_k\right)^2 }{ 2e_k }  \right),
\end{eqnarray}
with $m_k = 2 {\rm E}[(n_k+1)/2]+1$, $n_k \in \mathbb{N}$ where ${\rm E}[x]$ is the floor function and $a_k$, $b_k$, $c_k$, $d_k$, $e_k$ are fluid coefficients for all integer $k$. The set of coefficients $\{ a_k, b_k, c_k, d_k, e_k \}$ are the hidden variables. We notice that $a_k$ has a dimension of a density if $n_k$ is even and of a particle flux if $n_k$ is odd, $b_k$ and $c_k$ have the dimension of velocities and $d_k$ and $e_k$ of temperatures. This general form can describe asymmetric distribution functions $f$ which deviate from MDFs $f_0$ and the effective density can be different to $n$ (e.g., when we consider odd values of $n_k$). Moreover, because the hidden variables $b_k$ and $c_k$ as well as $d_k$ and $e_k$ can be different, in opposite to the Fig.~\ref{fig.Izacardian_examples1}, we can consider asymmetric corrections $\delta f$ as shown by red curves in Fig.~\ref{fig.fnM.Generalization.AsymTail}.
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{}
\centering
\includegraphics[height=30mm,natwidth=585,natheight=441]{Fig4a.pdf}
\label{fig.fnM.Generalization.AsymTail.1}
\end{subfigure}%
\begin{subfigure}[]{}
\centering
\includegraphics[height=30mm,natwidth=585,natheight=441]{Fig4b.pdf}
\label{fig.fnM.Generalization.AsymTail2}
\end{subfigure}%
\caption{A possible asymmetric correction of INMDFs given by \refeq{eq.fnM.Generalization}.}
\label{fig.fnM.Generalization.AsymTail}
\end{figure}
Further investigations using this generalized INMDFs given by \refeq{eq.fnM.Generalization} are possible. In summary, this formulation (i.e., the generalized INMDFs) can be related to a generalization of the Hermite polynomial representation~\cite{Grad:1949a,Grad:1949b,Grad:1963} (due to the presence of $({\rm v}-b_k)^{n_k}$ term) but each term is associated with a different MDF. The representation given by \refeq{eq.fnM.Generalization} forms a non-orthogonal basis in opposition to usual work found in the literature where orthogonal basis are used in order to project the distribution function on the basis set, but here this non-orthogonal basis is relevant to a significant reduction of the number of required hidden variables. Moreover, the completeness property is intrinsically inherited from the usual Hermite polynomial since \refeq{eq.fnM.Generalization} can be reduced to an Hermite polynomial with specific relation between all hidden variables (similar argument with a possible reduction to the RGBF, see below). The main advantage of this representation against the Hermite polynomial is that the required number of terms can be drastically reduced for the description of non-Maxwellians with high flows.
Some researchers argue that one needs to keep at least hundreds or thousands of terms with the usual Hermite polynomial representation (so at least hundreds or thousands independent fluid moments, see Ref.~\cite{Izacard_2016_Paper2}) for an accurate description of non-Maxwellian plasmas. With our generalization given by \refeq{eq.fnM.Generalization}, it is conceivable to keep much less hidden variables (related to much less independent fluid moments). Having less hidden variables is an argument of the advantage of the INMDF. Another argument is the physical interpretation (see the end of Sec.~\ref{ToC.NMDFs.INMDF}) of the hidden variables. We remark that our general representation can also reproduce the sum of shifted MDFs (i.e., the RGBF) by imposing other constraints such as $n_k=0$ and $m_k=1$.\\
The moments $M_q$ of the generalized distribution function given by \refeq{eq.fnM.Generalization} become
\begin{eqnarray}
\label{eq.fnM.Generalization.Mq}
\displaystyle
M_q &=& \sum_{k=0}^{N_k} \Bigg[ \Delta_k \sum_{p=0}^{n_k} \left(\begin{array}{c}n_k\\p\end{array}\right) \left(-b_k\right)^{(n_k-p)} \Bigg. \nonumber\\
\Bigg. && \qquad\qquad\qquad
\times
J_{p+q}\left( \frac{1}{2e_k}, \frac{c_k}{e_k} \right) \Bigg],
\end{eqnarray}
where $\Delta_k = a_k \left(2\pi d_k^{m_k}\right)^{-1/2} \exp\left( - c_k^2 / ( 2e_k ) \right)$, $\left(\begin{array}{c}m\\n\end{array}\right)$ is the combination and the function $J_k(a,b)$ is introduced above. The introduced INMDF given by \refeq{eq.nM.fINMDF} or \refeq{eq.fnM.Generalization} is a new minimal formulation for a local non-Maxwellian deviation. This distribution function cannot be exactly obtained with a finite number of terms by any of the existing basis functions. In comparison to the sum of MDFs (i.e., the RGBF~\cite{Hirvijoki:2015}), valid only at two collisionality limits, the INMDF is the first distribution function which can be self-consistent with MDFs steady states at finite collisionality. More properties of the INMDF will be published later.

\subsection{Physical reality of the INMDFs}
\label{ToC.NMDFs.Reality}
The readers may be concerned about the physical reality of the newly introduced INMDFs with respect to the well observed KDF in astrophysics or the natural expansion with a sum of MDFs. In order to argue about the physical picture introduced by the displacement of population of particles in the velocity phase space of the INMDFs, we describe below a qualitative fitting of an indirect experimental measurement of the distribution function. In fact, to date there is no universal direct measurement of the velocity phase space variations of the distribution function but it is common to indirectly compute the distribution function from different diagnostics. Moreover, even if we choose to perform the qualitative fitting (without the necessity to use the experimental data) of only one published article on the electron temperature measurement discrepancy, it is at least possible (not shown here, see Ref.~\cite{Izacard_2016_Paper2}) to fit NMDFs observed by Fokker-Planck numerical codes in presence of lower hybrid current drive or by particle in cell codes in presence of ion orbit losses.
\begin{figure}[!htbp]
\centering
\includegraphics[height=50mm,natwidth=545,natheight=339]{Fig5.pdf}
\caption{Over plot of our qualitative fitting of a MDF $f_0$ (blue dashed cruve) and two INMDFs $f_{I,1}$ (green dashed curve) and $f_{I,2}$ (cyan dashed curve). Our fitting is done with respect to the numerical model NMDF (red dashed curve) of Ref.~\cite{Beausang_2011_RSI} which explains the inconsistency between the electron cyclotron emission and the Thomson scattering measurements. The background figure is reprinted from [Beausang, \textit{et al.}, \textit{Rev. Sci. Instrum.} \textbf{82}, 033514 (2011)] with the permission of AIP Publishing. See Fig.~2.a in Ref.~\cite{Beausang_2011_RSI}.}
\label{fig.Real.ECE-TS}
\end{figure}
Fig.~\ref{fig.Real.ECE-TS} proves the capability of INMDFs to describe experimentally observed physical processes. With some density and temperature constraints in presence of NBI and ICRF in JET~\cite{Luna_2003_RSI,Beausang_2011_RSI} and TFTR~\cite{Taylor_1996_PPCF,Fidone_1996_PoP,White_2012_NF}, it has been observed a $20\%$ discrepancy of the electron temperature between the interpretation from the electron cyclotron emission (ECE) and Thomson scattering (TS) due to kinetic effects~\cite{Beausang_2011_RSI}. De La Luna, \textit{et al.} and Beausang, \textit{et al.} numerically found a model NMDF (dashed red curve) constructed from the spectrum of the ECE. {In fact, the measurement of the ECE spectrum was not consistent with a MDF. De La Luna, \textit{et al.} originally found the way to better recover this ECE spectrum by numerically modifying the distribution function and obtaining the called model NMDF. It turned out that} this model NMDF resolves the discrepancy of the interpreted electron temperature by TS. The physical processes at the origin of this model NMDF are not understood. Our three over plots (blue, green and cyan dashed curves) in Fig.~\ref{fig.Real.ECE-TS} can help to understand the origin of the detected NMDF. The blue curve $f_0$ corresponds to a MDF with a specific set of parameters. The green curve $f_{I,1}=f_0+\delta f_1$ is obtained by adding $\delta f_1$ with the parameters $(q,r,s)=(1.7,0.52,0.48)$. Then, the cyan dashed curve $f_{I,2}=f_{I,1}+\delta f_2$ is obtained by adding $\delta f_2$ with $(q,r,s)=(-0.2,1.3,0.2)$. This means that we detect both, an enhancement (i.e., a heating $\delta f_1$) and a reduction (i.e., a cooling $\delta f_2$) of the energy of two different populations of particles. {We notice that because we use interpreted parameters, we do not need to access the data (i.e., the density, fluid velocity and temperature of the background MDF). Our additional parameters ($\Gamma,c,W$) are the only way to modify the shape of the distribution function and the use of the terms ($q,r,s$) make our figure independent of the data.} We prove here that the $20\%$ TS-ECE discrepancy is due to approximately $1.7\%$ of heated particles and $0.2\%$ of cooled particles, and both are attracted by a resonant-like process around $1.0\ u_{\rm th}$. {We highlight the fact that this result may not be the detection of a new physical effect but, at least, it helps to describe and understand the unsolved problem of the TS-ECE discrepancy. Future work could try to link our understandings with known heating and current drive resonant theories or may lead to the discovery of a new physical effect. Fokker-Planck codes may help to fully understand the details of the energy transfer. In summary, we do not attempt to describe here what is resonant with what in this complex unsolved problem where an electron NMDF, associated here with a kind of current drive resonance, is experimentally observed in presence of ion heating and current drive (NBI and ICRH).
However, even if w}e do not know yet the exact origin of these detected physical processes (heating and cooling), the analytic formula of the INMDF $f_{I,2}$ can be directly used to understand the effects of non-Maxwellian bulks on other diagnostics and on the transport and turbulence. Indeed, even without theoretically or numerically describing the physical processes at the origin of the heating and cooling down we observed here, we can still have a good description of the deviations of the NMDF with respect to a MDF and analytically predict and validate the kinetic effects from this observed NMDF on different diagnostics. {Our capability to calibrate different experimental diagnostics is indispensable since we prove here that less than $2\%$ of super-thermal particles can lead to a $20\%$ discrepancy of the electron temperature interpretation between TS and ECE when a MDF is assumed. It is specially indispensable for the prediction of the effects of super-thermal tails on different diagnostics in ITER where a $90\%$ of self-heating by fusion reactions is planned in addition to some external heating.} Future work will focus on our detected INMDF $f_{I,2}$ and will help to initiate standard {experimental} procedures to measure NMDFs and to resolve inconsistencies between diagnostics more or less sensitives to kinetic effects. \\

In the next section, the advantage of using one of these NMDFs is shown for the analytic computation of kinetic corrections on the secondary electron emission, the Langmuir probe characteristic curve and the entropy.

\section{Kinetic corrections due to Non-Maxwellians}
\label{ToC.KinCorrections}
This section applies the NMDFs shown in the previous section on some existing theories relevant to current and future plasma devices. Analytic corrections of these theories are given as function of the fluid hidden variables. These corrections can be implemented in existing numerical simulations in order to better describe some kinetic effects when they need to be taken into account.

\subsection{Corrections for the secondary electron emission}
In tokamaks scrape-off-layer (SOL), incident primary particles (i.e., charged or neutral particles) release secondary electron from a solid surface. The secondary electron emission (SEE) reduces the sheath potential and can modify many phenomena in the plasma-material interface~\cite{Bruining:1954}. The computation of the SEE is obtained by formulating the SEE yield for incoming particles of energy $E$ related to the material under investigation (Sternglass formula~\cite{Bruining:1954,
Sternglass_1957_PR,
Dionne:1975}) multiplied by the proportion of particles at this energy (i.e., the distribution function $f(E)$) and has been under investigation in the literature~\cite{Meyer-Vernet_1982_AA,
Mayer_1988_PRB,
Bacharis_2010_PoP_17,
Raitses:2011,
Shotorban_2015_PRE}. By integrating over the possible energy of incoming particles, the secondary electron emission $\delta_{see}$ with respect to the incoming particles distribution function $f(E)$ becomes
\begin{eqnarray}
\label{eq.SEE}
\displaystyle
\delta_{see} = \frac{\int_0^{\infty} \sqrt{\frac{2E}{m}} f(E) \delta_{s}(E) dE}{\int_0^{\infty} \sqrt{\frac{2E}{m}} f(E) dE},
\end{eqnarray}
with $\delta_{s}(E) = \Delta_{s}\ E \exp\left( -2 \sqrt{E} / \sqrt{E_{\rm max}} \right)$, the constant coefficient $\Delta_{s} = (2.72)^2 \delta_{\rm max} / E_{\rm max}$, and $E=m{\rm v}^2/2$ is the kinetic energy. Some values of $E_{\rm max}$ and $\delta_{\rm max}$ have been obtained experimentally and are dependent of the material under consideration (e.g., see Refs.~\cite{Bacharis_2010_PoP_17,Dionne:1975}). The value of $E_{\rm max}$ corresponds to the energy associated to the value of $\delta_{\rm max}$, the maximum SEE measured and used to obtain a normalized $\delta_{see}$. The empirical formula for a MDF obtained in Ref.~\cite{Bacharis_2010_PoP_17} is
\begin{eqnarray}
\displaystyle
\log_{10}\left[ \delta_{see}(T) \right] \approx C_3 x^3 + C_2 x^2 + C_1 x + C_0,
\end{eqnarray}
with $x = \log_{10}(T)$, the coefficient $C_k$ are given ad-hoc for each species, and the dependence with respect to the fluid velocity $v$ disappears by using the assumption of a Maxwellian plasma with no mean flow. However, we can reformulate \refeq{eq.SEE} to
\begin{eqnarray}
\label{eq.SEE2}
\displaystyle
\delta_{see} = \frac{\int_0^{\infty} {\rm v}^2 f({\rm v}) \delta_{s}({\rm v}) d{\rm v}}{\int_0^{\infty} {\rm v}^2 f({\rm v}) d{\rm v}},
\end{eqnarray}
with $\delta_{s}({\rm v}) = \Delta_{s}\ m {\rm v}^2/2  \exp\left( -2{\rm v}/ u \right)$, and $u=\sqrt{2E_{\rm max}/m}$. \begin{figure}[!ht]
\centering
\begin{subfigure}[]{}
\centering
\includegraphics[height=65mm,natwidth=576,natheight=432]{Fig6a.pdf}
\label{fig.fnM.SEE_DF}
\end{subfigure}%
\begin{subfigure}[]{}
\centering
\includegraphics[height=65mm,natwidth=576,natheight=432]{Fig6b.pdf}
\label{fig.fnM.SEE_Modif}
\end{subfigure}%
\caption{Distribution functions (a) and effects of the tails on the SEE (b) from \refeq{eq.SEE.dsec0} for a Maxwellian and from Eqs.~(\ref{eq.SEE.dsec2M}) and (\ref{eq.SEE.dsecI}) for different INMDFs.}
\label{fig.fnM.SEE}
\end{figure}
Here we found the following exact analytic formula from \refeq{eq.SEE2} for a MDF given by \refeq{eq.nM.fINMDF.f0}
\begin{eqnarray}
\label{eq.SEE.dsec0}
\displaystyle
\delta_{see,0}(v,T) = \delta \frac{J_{4}\left(\frac{1}{2T},\frac{v}{T}-\frac{2}{u},0\right)}{J_{2}\left(\frac{1}{2T},\frac{v}{T},0\right)},
\end{eqnarray}
with $\delta = (2.72)^2 / (8 \sqrt{\pi}) \delta_{\rm max} / u^2$ and where the general form and some functions $J_k(a,b,0)$ are detailed in App.~\ref{ToC.App.Jkab0} where
\begin{eqnarray}
\label{eq.SEE.Jkab0}
\displaystyle
J_k(a,b,0) = \int_0^{\infty} {\rm v}^k \exp\left(-a{\rm v}^2+b{\rm v}\right) d{\rm v}.
\end{eqnarray}
{Following the same analytic method, it has been possible to obtain the corrections $\delta_{see,\kappa}$ from a KDF. However, we do not focus on this result here because }
it involves the hypergeometric PFQ function instead of $K_q(\kappa,T)$ due to of the presence of $\exp(-2 {\rm v} / u)$ in the term $\delta_{s}({\rm v})$ and it may not be possible to obtain a simple analytic form in comparison to the following results. This may mean that the KDF is less natural than the sum of two MDFs or the INMDF (see results below) due to the complexity of the solution of the SEE $\delta_{see,\kappa}$. We found for the sum of two Maxwellians the following corrections
\begin{eqnarray}
\label{eq.SEE.dsec2M}
\displaystyle
\delta_{see,2M} &=& \delta \Bigg[ \Delta J_{4}\left(\frac{1}{2T},\frac{v}{T}-\frac{2}{u},0\right) \Bigg. \nonumber\\
\displaystyle
\Bigg. &&\quad
+ \Delta_f J_{4}\left(\frac{1}{2T_f},\frac{v_f}{T_f}-\frac{2}{u},0\right)  \Bigg] \nonumber\\
\displaystyle
&&
\times \Bigg[ \Delta J_{2}\left(\frac{1}{2T},\frac{v}{T},0\right) \Bigg. \nonumber\\
\displaystyle
\Bigg. &&\quad
+ \Delta_f J_{2}\left(\frac{1}{2T_f},\frac{v_f}{T_f},0\right) \Bigg]^{-1},
\end{eqnarray}
and for the INMDF given by \refeq{eq.nM.fINMDF} the following corrections
\begin{eqnarray}
\label{eq.SEE.dsecI}
\displaystyle
\delta_{see,I} &=& \delta \Bigg[ \Delta J_{4}\left(\frac{1}{2T},\frac{v}{T}-\frac{2}{u},0\right) \Bigg. \nonumber\\
\displaystyle
\Bigg. &&\quad
+ \Delta_I J_{5}\left(\frac{1}{2W},\frac{c}{W}-\frac{2}{u},0\right) \Bigg. \nonumber\\
\displaystyle
\Bigg. &&\quad
- c \Delta_I J_{4}\left(\frac{1}{2W},\frac{c}{W}-\frac{2}{u},0\right)   \Bigg] \nonumber\\
\displaystyle
&&
\times \Bigg[ \Delta J_{2}\left(\frac{1}{2T},\frac{v}{T},0\right) + \Delta_I J_{3}\left(\frac{1}{2W},\frac{c}{W},0\right) \Bigg. \nonumber\\
\displaystyle
\Bigg. &&\quad
- c \Delta_I J_{2}\left(\frac{1}{2W},\frac{c}{W},0\right)   \Bigg]^{-1},
\end{eqnarray}
where the secondary electron emissions $\delta_{see,0}(v,T)$, $\delta_{see,2M}(n,v,T,n_f,v_f,T_f)$ and $\delta_{see,I}(n,v,T,\Gamma,c,W)$ are functions of all fluid hidden variables and the quantities $\delta$, $\Delta$, $\Delta_f$ and $\Delta_I$ are previously defined.\\
Fig.~\ref{fig.fnM.SEE}.(a) shows different NMDFs computed from the bi-modal NMDF given by \refeq{eq.nM.f2Maxw} and from the INMDF given by \refeq{eq.nM.fINMDF} for a temperature $T=1keV$. The corrections of the secondary electron emission as function of the temperature are shown in Fig.~\ref{fig.fnM.SEE}.(b). \\
An important result is that the black curve obtained for a MDF with our exact formula does not match exactly the red curve of the empirical formula given in Ref.~\cite{Bacharis_2010_PoP_17}. The reason is due to an undetected presence of super-thermal particles in Ref.~\cite{Bacharis_2010_PoP_17} because our analytic formula $\delta_{see,I}$ with the INMDF $f_I(q=3,r=1.25,s=0.5)$ (cyan dashed curve) matches better the empirical formula (red curve) better in opposition to our analytic formula $\delta_{see,0}$ with the MDF $f_0(n_0=10^{18},v=0,T=10^3)$ (black curve). Moreover, the empirical formula is better matched for $T<10^2eV$ with $f_I(q=2,r=1.2,s=0.45)$ (blue dashed curve) and for $T>3.10^2eV$ with $f_I(q=4,r=1.3,s=0.55)$ (green dashed curve). We remark that only for the cyan curve in Fig.~\ref{fig.fnM.SEE}.(b) we used $\delta_{\rm max}=0.95$ instead of $1$, but it is possible to find other parameters which match the red empirical curve (e.g., keeping $\delta_{\rm max}=1$ and multiplying $n$ and $\Gamma$ by 0.95 because $\delta_{see}$ is linear with respect to these two quantities). Finally, it seems that the super-thermal particles of the empirical formula are clearly better described by the INMDF rather than by a sum of two MDFs (yellow solid curve is one of the best fit). Nevertheless, the goal is to predict the modifications of the secondary electron emission in presence of different super-thermal particles. The effects of a tail on the secondary electron emission are significant specially for a low temperature (i.e., $T \leq 10eV$) of the background plasma since a factor around $5$ can be observed (i.e., $(q,r,s)=(2,2.5,1)$ not shown here). Then, it is indispensable to take into account super-thermal particles even if this population represents less than $4\%$ because, at least, the dynamics of dusts, neutrals and impurities are highly impacted by the secondary electron emission. It looks like our analytic prediction (with $s>1$) is able to observe two peaks in the SEE like it has been experimentally observed (see Ref.~\cite{Raitses:2015}). Future work will investigate this two peaks observation, particularly the addition of other terms in \refeq{eq.nM.fINMDF}, in order to avoid negative distribution function for all ${\rm v}<c$.

\subsection{Corrections for the Langmuir probes interpretations}
A commonly used diagnostic of edge (cold) plasma is the Langmuir probe. By applying an electric potential scan on the Langmuir probe, we can measure the current of the probe as function of the applied potential. This give us the well known characteristic curve. The theory of the Langmuir probe has successfully been developed in the past century~\cite{Stangeby_2000,Hutchinson_2002} and properties of the plasma can be extracted from the characteristic curves such as the electron temperature, or the floating and plasma potentials by interpreting the measurement for a background Maxwellian plasma. However, a recurrent discrepancy of the interpretation of the plasma properties with other diagnostics (e.g., with Thomson scattering measurements~\cite{Luna_2003_RSI,
Beausang_2011_RSI,
Jaworski_2015_Private} in attached plasma) has been observed. This discrepancy has been related to NMDFs as detailed in Refs.~\cite{Popov:2009,Jaworski_2012_FED,Jaworski:2013} where bi-modal distribution functions (distribution function of $2$ populations of particles at different temperatures) have been used. Other results are published in the literature describing effects of non-Maxwellians on the Langmuir probe measurements~\cite{Arslanbekov,Sudit_1994_JAP,
Stangeby_1995_PPCF,
Batishchev_1997_PoP,
Demidov_2002_RSI,
Tskhahaya_2011_JNM,
Sheehan_2011_PoP,
Godyak_2011_JPD,
Jaworski_2012_FED,
Popov_2012_PSST,
Popov_2014_CPP,
Godyak_2015_JAP}. The bi-modal approximation is the first efficient way to describe NMDFs, but it assumes the superposition of two populations of particles at the Maxwellian equilibrium and the self-consistency is very restricted since in this bi-modal description the plasma is enough collisional to assure a MDF for each species but do not allow collisions between these two populations. In order to resolve this inconsistency, we propose to use the INMDF of the plasma (only one population of particles with $1$ effective temperature) which contains super-thermal particles. In our case, all particles (viewed as one species) collide together and external sources or sinks of energy create non-Maxwellian steady states. \\
The general formula of the electron current~\cite{Arslanbekov,Demidov_2002_RSI,Jaworski_2012_FED,Popov_2014_CPP} can be written as a function of any distribution function $f(E)$ of the kinetic energy $E$ of particles
\begin{eqnarray}
\label{eq.LP.Ie.E}
\displaystyle
I_e(U) = - \frac{8\pi e S}{3 m^2} \int_{eU}^{\infty} \frac{(E-eU) f(E)}{\gamma(E) \left[ 1 + \frac{E-eU}{E} \psi(E) \right]} dE,
\end{eqnarray}
where $\psi(E)$ is a diffusion parameter, $\gamma(E)$ a geometric parameter of the probe, $U=U_p-U_{pl}$ is the difference between the applied potential at the probe and the plasma potential, $S$ the surface of the probe, and $m$ is the mass of electron. The classical regime is obtained by assuming a diffusionless limit ($\psi(E) \ll 1$), and with $\gamma=4/3$ for a spherical probes. This formula computes the flux of electron which have higher relative energy than the potential $eU$ because electrons of lower energy cannot contribute to the current due to the Coulomb barrier generated by other electrons at the probe interface. With the change of coordinate $E=m{\rm v}^2/2$ and $u=\sqrt{2eU/m}$, the electron current becomes
\begin{eqnarray}
\label{eq.LP.Ie.v}
\displaystyle
I_e(U) = - \frac{2\pi e S}{m} \int_{u}^{\infty} \left(\frac{m{\rm v}^2}{2}-eU\right) {\rm v} f({\rm v}) d{\rm v}.
\end{eqnarray}
If we assume a background MDF $f_0$ given by \refeq{eq.nM.fINMDF.f0}, the electron current becomes
\begin{eqnarray}
\label{eq.LP.Ie0}
\displaystyle
I_{e,0}(U) &=& - \Delta \frac{2\pi e S}{m} \bigg( \frac{m}{2} J_3\left(\frac{1}{2T},\frac{v}{T},u\right) \bigg. \nonumber\\
\displaystyle
\bigg. && \qquad\qquad\quad
- eU J_1\left(\frac{1}{2T},\frac{v}{T},u\right) \bigg),
\end{eqnarray}
with the definition of the function
\begin{eqnarray}
\label{eq.LP.Jkabc}
\displaystyle
J_k(a,b,c) = \int_{c}^{\infty} {\rm v}^k \exp\left(-a{\rm v}^2+b{\rm v}\right) d{\rm v},
\end{eqnarray}
detailed in App.~\ref{ToC.App.Jkabc}.
Then, from the KDF given by \refeq{eq.nM.fKappa} the electron current becomes
\begin{eqnarray}
\label{eq.LP.IeKappa}
\displaystyle
I_{e,\kappa}(U) &=& - \Delta_{\kappa} \frac{2\pi e S}{m} \bigg( \frac{m}{2} K_3\left(\kappa,T,u\right) \bigg. \nonumber\\
\displaystyle
\bigg. && \qquad\qquad\quad
- eU K_1\left(\kappa,T,u\right) \bigg),
\end{eqnarray}
from the sum of two MDFs given by \refeq{eq.nM.f2Maxw} the electron current becomes
\begin{eqnarray}
\label{eq.LP.Ie2M}
\displaystyle
I_{e,2M}(U) &=& - \frac{2\pi e S}{m} \bigg( \frac{m}{2} \Delta J_3\left(\frac{1}{2T},\frac{v}{T},u\right) \bigg. \nonumber\\
\displaystyle
&& \qquad\quad
- eU \Delta J_1\left(\frac{1}{2T},\frac{v}{T},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
+ \frac{m}{2} \Delta_f J_3\left(\frac{1}{2T_f},\frac{v_f}{T_f},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
- eU \Delta_f J_1\left(\frac{1}{2T_f},\frac{v_f}{T_f},u\right) \bigg),
\end{eqnarray}
and from the INMDF $f_I$ given by \refeq{eq.nM.fINMDF} the electron current becomes
\begin{eqnarray}
\label{eq.LP.IeI}
\displaystyle
I_{e,I}(U) &=& - \frac{2\pi e S}{m} \bigg( \frac{m}{2} \Delta J_3\left(\frac{1}{2T},\frac{v}{T},u\right) \bigg. \nonumber\\
\displaystyle
&& \qquad\quad
- eU \Delta J_1\left(\frac{1}{2T},\frac{v}{T},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
+ \frac{m}{2} \Delta_I J_4\left(\frac{1}{2W},\frac{c}{W},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
- \frac{m}{2} c \Delta_I J_3\left(\frac{1}{2W},\frac{c}{W},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
- eU \Delta_I J_2\left(\frac{1}{2W},\frac{c}{W},u\right) \nonumber\\
\displaystyle
&&\qquad\quad
+ eUc \Delta_I J_1\left(\frac{1}{2W},\frac{c}{W},u\right) \bigg).
\end{eqnarray}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{}
\centering
\includegraphics[height=65mm,natwidth=576,natheight=432]{Fig7a.pdf}
\label{fig.fnM.LangmuirProves_DF}
\end{subfigure}%
\begin{subfigure}[]{}
\centering
\includegraphics[height=65mm,natwidth=576,natheight=432]{Fig7b.pdf}
\label{fig.fnM.LangmuirProbe_CharacteristicCurve}
\end{subfigure}%
\caption{Logarithms of distribution functions (a) and effects of the tails on the Langmuir probe current (b) from \refeq{eq.LP.Ie0} for a Maxwellian and \refeq{eq.LP.IeI} for different INMDFs.}
\label{fig.fnM.LangmuirProbe}
\end{figure}
As shown in Fig.~\ref{fig.fnM.LangmuirProbe}, different NMDFs (see Fig.~\ref{fig.fnM.LangmuirProbe}.(a)) with the same Maxwellian background (black curve) can significantly modify (see Fig.~\ref{fig.fnM.LangmuirProbe}.(b)) the electric current measured by a Langmuir probe. This figure is obtained for parameters relevant to the SOL plasmas close to the divertor plates (i.e., $T=15eV$, $n=10^{20} m^{-3}$). However, we remark that the amplitudes are not consistent with the experimental measurements by the Langmuir probes since the formula given by \refeq{eq.LP.Ie.E} (see Refs.~\cite{Arslanbekov,Demidov_2002_RSI,Jaworski_2012_FED,Popov_2014_CPP}) is not dimensionally correct. Usually, results are given in arbitrary units in the literature~\cite{Jaworski_2015_Private} in order to avoid this dimensional inconsistency. This dimensional issue can be investigated later since it is not the focus here. \\
Moreover, \refeq{eq.LP.Ie.v} (directly linked to the Druyvesteyn formula~\cite{Arslanbekov,Demidov_2002_RSI,Popov_2012_PSST,Popov_2014_CPP}) is the diffusionless limit of \refeq{eq.LP.Ie.E}. In comparison to other studies~\cite{Jaworski_2015_Private}, the INMDF used in the diffusionless limit \refeq{eq.LP.Ie.v} can reproduce similar curves than the use of a MDF in the diffusional \refeq{eq.LP.Ie.E}. This observation is not surprising because the particles diffusion describes an enhancement of the displacement of particles as well as the population of super-thermal particles. This is a very important observation because instead of using ad-hoc diffusion parameter in order to reproduce experimental observations, the existence of NMDFs could replace those.

Finally, all common interpretations of the plasma parameters are modified by the NMDF. For example, the floating potential $U_f$ is defined by $I_e(U_f)=I_i(U_f)$ where $I_i(U)$ is the current of the Langmuir probe generated by ions and commonly assumed to be constant due to ion mass and temperature involved. From the result given by \refeq{eq.LP.IeI} and its version for the ion current, it is possible to better evaluate the floating potential from the distribution function. Of course, all other quantities such as the plasma potential $U_{pl}$, the electron temperature $T_e$ and the radial electric field $E_r$ are interpreted differently due to NMDFs. Additional investigations will be reported later.

\subsection{Entropy decrease}
In the thermodynamics theory, the entropy is interpreted as the degree of disorder of a system. A generalization of the entropy is required for the description of non thermodynamics equilibrium. By sharing similar perspectives than Refs.~\cite{Kranys_1970_ARMA,Livadiotis_2014_Entropy}, corrections of the entropy can be analytically computed from a NMDF. In an isolated plasma, the entropy can only increase due to collisions in order to reach its maximum value when the distribution function is Maxwellian. The definition of the statistical entropy developed by Boltzmann is
\begin{eqnarray}
\label{eq.B.s}
\displaystyle
s &=& - k_B \int_{-\infty}^{\infty} \frac{f}{n} \log\left( \frac{f}{n} \right) d{\rm v},
\end{eqnarray}
where $f$ is the distribution function of the particles of density $n=\int f d{\rm v}$ and $k_B$ is the Boltzmann constant. The maximal value of the entropy is obtained with a MDF $f_0$ (see App.~\ref{ToC.App.Entropy})
\begin{eqnarray}
\displaystyle
s_0 &=& - k_B \int_{-\infty}^{\infty} \frac{f_0}{n} \log\left( \frac{f_0}{n} \right) d{\rm v}, \\
\displaystyle
&=& \frac{k_B}{2} \big[ 1 + \log(2\pi T) \big],
\end{eqnarray}
such that $\partial_t s_0 = - k_B \partial_t \int f_0/n \log\left( f_0/n \right) d{\rm v} = k_B/(2T) \partial_t T= 0$ for isolated equilibrium plasmas. For other distribution functions than the MDF, the entropy can only be smaller to this value and increase ($\partial_t s \geq 0$). However, in order to make a link with observed NMDFs for non-isolated plasmas, it makes sense to interpret the entropy by a level of sharing energy or information. The maximal entropy obtained with a MDF of an isolated plasma is reached when the interaction between all particles of the plasmas have shared the total energy of the system after a thermalization characteristic time $t_{\rm th}$. This level of sharing can decrease when an external source or sink of energy of a non-isolated plasma appears locally in the velocity phase space at a finite collisionality. The presence of a steady-state super-thermal population of particles due to external sources introduces corrections of the entropy because at finite collisionality some particles can be sensitive to the external source of energy in time range $t < t_{\rm th}$, shorter than the one associated to the sharing of energy. The analytic computation of the corrections becomes possible using the INMDF written in the form $f_I = f_0 +\delta f = f_0 ( 1 + \delta f/f_0)$ with the MDF $f_0$ and the corrections $\delta f$ previously defined (respectively by Eqs.~(\ref{eq.nM.fINMDF.f0}) and~(\ref{eq.nM.fINMDF.delf})). The modified entropy due to the presence of a super-thermal population reads
\begin{eqnarray}
\displaystyle
s_I &=& - k_B \int_{-\infty}^{\infty} \frac{f_I}{n_I} \log\left( \frac{f_I}{n_I} \right) d{\rm v},
\end{eqnarray}
where $n_I=\int f_I d{\rm v} = n$. Using the Taylor expansion of the logarithm of $(1+x)$ around $0$ (i.e., $\log(1+x)=- \sum_{k=1}^{\infty} (-1)^k x^k / k$) if we assume relatively small deviations from the Maxwellian (i.e., $\vert \delta f \vert \ll f_0$), then, following the details of App.~\ref{ToC.App.Entropy}, the entropy becomes $s_I = s_0 + \delta s$ with
\begin{eqnarray}
\displaystyle
&&\delta s = - k_B \frac{1}{n} \int_{-\infty}^{\infty} \delta f \log\left( \frac{f_0}{n} \right) d{\rm v} \nonumber\\
\displaystyle
&&+ k_B \frac{\Delta}{n} \sum_{k=1}^{\infty} \frac{(-1)^k}{k(k+1)} \left( \frac{\Delta_I}{\Delta} \right)^{k+1} \sum_{m=0}^{k+1} (-c)^{k+1-m} \nonumber\\
\displaystyle
&&\times \left(\begin{array}{c}k+1\\ m\end{array}\right) J_m\left( \frac{k+1}{2W}-\frac{k}{2T}, \frac{(k+1)c}{W}-\frac{kv}{T} \right). \qquad
\end{eqnarray}
At the first order in the expansion of $\vert \delta f \vert \ll f_0$ (i.e., the sum is of higher order, see App.~\ref{ToC.App.Entropy}) and with the use of the following relations
\begin{eqnarray}
\displaystyle
J_1\left(A,B\right) - c J_0\left(A,B\right) &=& 0, \\
\displaystyle
J_2\left(A,B\right) - c J_1\left(A,B\right) &=& W J_0\left(A,B\right), \\
\displaystyle
J_3\left(A,B\right) - c J_2\left(A,B\right) &=& c (W-c^2) J_0\left(A,B\right), \\
\displaystyle
\Delta_I J_0\left(A,B\right) &=& \frac{\Gamma}{W},
\end{eqnarray}
when $A=1/(2W)$, $B=c/W$ and with the functions $J_k(a,b)$ given in App.~\ref{ToC.App.Jkab}, the correction of the entropy is
\begin{eqnarray}
\displaystyle
\delta s &\approx & - k_B \int_{-\infty}^{\infty} \frac{\delta f}{n} \log\left(\frac{f_0}{n}\right) d{\rm v}, \\
\displaystyle
&\approx & - k_B \frac{\Gamma}{n T} \left[ v - \frac{c}{2} + \frac{c^3}{2W} \right],
\end{eqnarray}
In comparison to Refs.~\cite{Bizarro_2011_PRE,Bizarro_2015_PoP} where the entropy can decrease in presence of frictions, here we assume an INMDF steady-state without friction. We found that the first order correction of the entropy can either be positive or negative as function of the hidden variables describing the INMDF. We remark that our INMDF is not related to an isolated system, so the inclusion of the collision and the source of energy (e.g., radio-frequency waves, neutral beam, runaway electron) would recover known thermodynamic results of isolated systems (e.g., similar to Ref.~\cite{Bizarro_2015_PoP}). This means that when an external source or sink of energy is turned on, the corrected entropy of the plasma ($\delta s$) computed from an INMDF can increase or decrease with respect to the entropy $s_0$ obtained from a MDF. Both cases are possible without violating the second law of the thermodynamics since here the plasma is non-isolated and the total entropy increase due to the external source is not taken into account. In fact, instead of describing a complete theory of the source and the plasma, we rather focus on the statistical description of the plasma in presence of sources. Then, for a non-isolated system, the maximum value of the entropy is a higher value than $s_0$ as function of the plasma parameters. Moreover, since both variations of $s_I$ are possible, the intermediate case $\delta s=0$ is the solution where the super-thermal particles do not modify globally the entropy of the Maxwellian at the first order. There are $2$ solutions of $\delta s=0$. The first, for finite values of the density and temperature when $\Gamma=0$, recovers the MDF. The second solution is obtained for $c \neq 0$ and $c \neq v$ when
\begin{eqnarray}
\label{eq.B.Wcv}
\displaystyle
W &=& \frac{c^2}{1-2\frac{v}{c}}.
\end{eqnarray}
The physical interpretation of this specific value of the width of the heat spread still needs to be understood and is still under investigation. However, with this specific INMDF profile, the statistical entropy of the plasma, i.e., without including the entropy increase due to the external sources, is locally constant. This means that the increase of the energy (which increases the entropy) is compensated exactly with the departure of the INMDF with respect to the MDF (which decreases the entropy). The time derivative of the entropy is found by using the Boltzmann equation and omitting the collision operator. It reads $\partial_t s_{(0)} = - {\nabla}  { s}_{(1)}$ where ${ s}_{(k)} = - k_B \int (f/n) \log(f/n) {\rm v}^{k} d{\rm v}$. For the INMDF, $\partial_t s_{I,(0)} = - {\nabla}  { s}_{I,(1)}$ with ${ s}_{I,(0)} = s_I$, and ${ s}_{I,(1)} = { s}_{0,(1)} + \delta { s}_{(1)}$. Using the same expansion of the logarithm, we found that the time evolution of the entropy is modified at the first order by the spatial derivatives of
\begin{eqnarray}
\displaystyle
{ s}_{0,(1)} &=& k_B \frac{v}{2} \bigg[ \log(2\pi T) - \frac{v^2}{T} \bigg], \\
\displaystyle
\delta { s}_{(1)} &\approx & k_B \frac{\Gamma}{2 n} \bigg[ \log\left(2\pi T\right) + \frac{v^2}{T} -\frac{2vc}{T} \left( 1-\frac{c^2}{W} \right) \bigg. \nonumber\\
\displaystyle \bigg.
&& \qquad\qquad\qquad\qquad
+ \frac{3 (W+c^2)}{T} - 2 \bigg].
\end{eqnarray}
These relations can be used in order to compute the localized statistical entropy evolution of the plasma as a fluid quantity. As a numerical error criteria, for a MDF and without sources, we have to verify that $\partial_t s_{0,(0)} = - {\nabla}  { s}_{0,(1)} = 0$ where $s_{0,(0)} = s_{0}$.\\
The increase or decrease of the entropy in time is given here as function of the spatial gradients of the the hidden variables. A very important interpretation of this result is following:\\
In many experiments (even for other areas than plasmas) non-Maxwellian steady-state distribution functions are observed. This means that in order for us to approximate this steady-state with the proposed INMDF or by creating new analytic NMDFs with as few hidden variables as possible, we have to verify that the local entropy of a non-isolated plasma (including sources) is constant in time because the entropy increase due to the sources is compensated by the entropy reduction of the NMDF steady-state with respect to the maximum entropy of a heated MDF. Then, there cannot be evolving local entropy at NMDF steady-state. From our result of $\partial_t s_{I,(0)} = - {\nabla}  { s}_{I,(1)}$ written as function of the $6$ hidden variables, there are an infinite number of solutions which locally conserve the entropy $\partial_t s_{I,(0)} = 0$ at least at the first order in the expansion of small non-Maxwellian deviations. For example, we obtain one class of these solutions by trivially extracting the density $n$ as function of the other fluid hidden variables $(v,T,\Gamma,c,W)$ since the equation ${ s}_{I,(1)} = { s}_{0,(1)} + \delta { s}_{(1)} = 0$ contains only one occurrence of $n$. We can write this equation as $A+B/n = 0$ with $A={ s}_{0,(1)}$ and $B=n \delta { s}_{(1)}$ which are both functions of $(v,T,\Gamma,c,W)$. This class of solutions of locally constant entropy is obtained when $n = - B/A$, if $A \neq 0$ (i.e., if $v \neq 0$ and $v^2 \neq T \log(2\pi T)$). We have an infinite number of solutions because we have $6$ parameters to resolve the equation $\partial_t s_{I,(0)} = - {\nabla}  { s}_{I,(1)} = 0$. Another class of solution can be found by canceling the derivative of ${ s}_{I,(1)}$. However, even if a decrease of the local entropy can be found in this case, it is an artifact decrease because we intentionally omit the entropy of the external sources which generate the tail (i.e., via the collision operator). This means that by including the collision operator for the self-collisions and the collisions with the sources, the entropy have to be able to increase only following the second law of thermodynamics of isolated systems.\\
As a summary, this result highly suggest the existence of an infinite number of analytic solutions of the Boltzmann equation for non-isolated systems. All these solutions can be obtained as function of the collision frequencies between the plasma and the sources.

The result of the allowed decreasing entropy demystifies the Maxwell's demon since the statistical description of the plasma is still possible even for a non-isolated system. There is no violation of the second law of the thermodynamics which postulates only increasing entropy until the Maxwellian equilibrium of an isolated system. This work could enhance a large range of present technologies since the isolation of a system is very rare. This new approach offers many perspectives since instead of developing more accurate ``interaction'' theories to include very complex external phenomena in a much larger isolated system, we are statistically describing non-isolated systems without dealing with the exact description of the external ``interaction'' theories. Ongoing investigations directly motivated by this result could be on (i) the expansion of the thermodynamics theory for non-isolated systems, (ii) the inclusion of higher orders in $(\delta f/f_0)^k$, (iii) the inclusion of the collision operator in the time evolution of the entropy as function of the hidden variables or, (iv) the numerical observation of NMDF steady states in presence of sources by evolving in time the dynamic equation of the local entropy.

A very large number of other theories usually assume a MDF and can be modified following the three examples shown above. The first goal could be to describe self-consistently the dynamics of dusts, neutrals and impurities from the corrections of measured quantities in presence of non-Maxwellian plasma background. The second goal could be to develop new diagnostics and interpretations of data to systematically measure in experiments non-Maxwellian distribution functions.

\section{Conclusion}
\label{ToC.CCL}
We describe in Sec.~\ref{ToC.NMDFs} the KDF and the sum of MDFs which are successfully used by researchers for numerical post-analyses. The KDF observed in astrophysics is one example of NMDF, even if to date there is no physical interpretation of the parameter $\kappa$. Another choice of NMDF commonly used is the bi-modal distribution function (the sum of two MDFs) but this choice can be consistent only at a very specific collisionality regime (the self-collisionality of the bulk and the fast population are both much larger than the collisionality of the interaction between them). Moreover, a new function called the INMDF is introduced here in order to offer more choices (consistent with a finite collisionality) for the representation of NMDFs. Readers can create as new analytic distribution functions as needed to represent the phenomenon under investigation in such a way that the velocity phase space integrals are analytically manageable as function of as few hidden variables as possible. In comparison to the bi-modal distribution function, the INMDF is a possible solution for other collisionality regimes since it describes a displacement of a population of particles from one energy to another. As a summary of this work, INMDFs are new proposed functions that seem much more efficient in describing non-thermalized plasmas than any existing formula thanks to the low number of required parameters (i.e., hidden variables). Moreover, INMDFs help to understand unsolved problems such as the ones at the origin of non-Maxwellian bulks observed in JET and TFTR, thanks to the physical interpretation of the hidden variables. With the distribution functions described in Sec.~\ref{ToC.NMDFs}, it is shown in Sec.~\ref{ToC.KinCorrections} that analytic predictions of kinetic corrections are possible and examples on the SEE, the Langmuir probe characteristic curve and the entropy are shown. Because these examples are not directly related to a specific plasma, {this proves the universal property of the INMDFs and the physical interest to consider non-orthogonal basis sets.} More applications will be found later. Moreover, other details of NMDFs will be published elsewhere such as the fluid reduction (see Ref.~\cite{Izacard_2016_Paper2}), which opens the access to the next generation of fluid codes by including non-collisional and collisional kinetic effects (see Ref.~\cite{Izacard_2016_FischSymposium}), or the description of the transport. Indeed, some results suggest that fluid models can be similar to kinetic codes such as the observation of asymmetric heat flux inside an island by using a fluid model with some finite Larmor radius terms~\cite{Izacard_2016_PoP}, using profiles of transport coefficients~\cite{Chankin_2007_NF_47a,
Chankin_2007_NF_47b,
Groth_2013_NF_53} to reduce the radiation shortfall or using nonlocal fluid closures~\cite{Held_2004_PoP,Dimits_2013_PoP}. A clarification of these examples will be detailed later. \\
For the physical interpretation of the kinetic effects shown in Sec.~\ref{ToC.NMDFs.Reality} and Sec.~\ref{ToC.KinCorrections}, we found four groundbreaking results
\begin{itemize}
\item[(i)] The physical reality of the INMDFs introduced here is proved by the over plot fitting of the numerical NMDF model obtained in Ref.~\cite{Beausang_2011_RSI} for the explanation of the TS-ECE discrepancy of the electron temperature interpretation in JET. We found that some particles at low energy are heated and others at higher energy are cooled down. This interpretation would help to understand the physical origin of the NMDF observed in JET and to resolve discrepancies between diagnostics due to the presence of NMDFs.
\item[(ii)] The presence of super-thermal particles in the Langmuir probe characteristic curve induces diffusion effects which have been commonly observed by using a diffusion term in the formula when a MDF was assumed. This result highly suggests more investigations to replace the usual ad-hoc dissipative coefficients and the diffusion terms by NMDFs.
\item[(iii)] The use of the analytic computation of the SEE in presence of NMDFs allows us the unexpected observation that the empirical formula of the SEE published in Ref.~\cite{Bacharis_2010_PoP_17} is not consistent with a MDF but is consistent with a presence of $\sim 3\%$ of super-thermal particles. Moreover, because the sum of $2$ MDFs {does not} reproduce much better the empirical formula {(in contrary to the INMDF)}, it suggests that the collisionality {in Ref.~\cite{Bacharis_2010_PoP_17}} was {not} negligible or infinite, but finite as explained above.
\item[(iv)] Finally, the physical motivation of the INMDF given by \refeq{eq.nM.fINMDF} is shown by the explicit decrease of the entropy for a non-isolated system without violating the second law of thermodynamics. This is the first simple analytic function which can be consistent with NMDFs at finite collisionality in presence of external source of energy. With this work, the entropy is not viewed as a degree of disorder but is interpreted as a level of sharing information (or energy) between particles of the plasma.
\end{itemize}
From these results obtained thanks to our new INMDFs, many perspectives are possibles such as the description of ion orbit losses~\cite{Stacey_2013,Stacey_2016} and the fast ions~\cite{Fasoli,Fasoli_NaturePhys}, or the development of experimental measurements of NMDFs using for example Thomson scattering~\cite{Luna_2003_RSI,Beausang_2011_RSI} or Langmuir probes. In summary, because different measurements can be more or less sensitive to the presence of super-thermal particles, better interpretations of experimental data are possible and more accurate measurement techniques of the distribution function can be developed.

\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
This work was supported by the LLNL Postdoctoral independent research funding. The author would like to acknowledge M.~Jaworski (PPPL) for our discussions on Langmuir probes measurements, and B.~Cohen (LLNL) and D.P.~Brennan (Princeton University) for their very helpful comments on this manuscrip
. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

redpajama_set_name: RedPajamaArXiv


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During preparations for this book, I contacted many remnants of the political movement from the late 60s. Most had become respectable members of the community. Some remained outsiders, iconoclasts. I left my phone number with a lot of people. When the phone rang on June 22, 2011, I thought it might be another clue to the story of Gideon Holley.
The caller addressed me by name, and I asked if I knew him.
"No," he said, "we've never met. But I heard you were doing your homework on Gideon Holley."
"Yes, I am," I said. "Would you mind telling me who you are?"
"My name is Anthony Colletti," he said. "I think we should talk."
The speech Gideon gave at Flagstaff is an excerpt from a real speech written by a friend, A.V. Archer, myself, and others aboard something we called the ELF Bus. The speech was delivered convincingly by an earnest looking member of our group to a filled hall of revolutionary leaders at the National Students Association National Congress at Macalester College in St. Paul, Minnesota, the summer of 1970.
The speaker, whose name was Peter, spoke with passion. The rest of us, who had been on the road for many months watching the riots and demonstrations and wanted these campus leaders to understand the difference between rhetoric and reality, planted ourselves in the audience. We cheered Peter's speech and applauded at every intake of breath. Soon, our cheers were drowned out by the loud support of the crowd.
The magpie was real, too. It stayed with me for a long time. Go figure.


text:
Henri Paul Arsène Bléhaut (22 November 1889 – 8 December 1962) was French politician and counter admiral. He was Secretary of State for the Navy and the Colonies from 26 March 1943 to 10 September 1944 in Vichy France.

Life

Early career
In 1908, he entered the École navale. On the outbreak of the First World War he was attached to the torpedo-boat Commandant-Rivière on operations against the Austro-Hungarian fleet in the Adriatic, particularly during the evacuation of the Serbian Army. He was mentioned in dispatches for his bravery in battle on the night of 22 December 1916. He commanded four submarines (Cigogne, Clorindre, Joëssel and Requin) between 1918 and 1927. He commanded the surface ships Panthère and Lamotte-Picquet in 1927. In May 1929 he was put in command of the 11th Torpedo Boat Division in the Mediterranean on board the Alcyonand reveals himself as "an exceptionally talented officer and a brilliant maneuverer". From 1932 to 1934 he was chief of staff of French naval staff in the Far East on board the Primauguet. From 1936 to 1938 he commanded Maillé-Brézé and a destroyer flotilla. In 1938 he became chief of staff of the maritime prefecture of Toulon.

World War II
From 1939 to 1941 he became chief of staff to the French naval forces in the Mediterranean fleet then chief of staff of the French naval forces of the south. During this period he was also promoted to counter admiral in 1940. His next post came in 1941, as commander of the 3rd Cruiser Division, flying his flag on La Marseillaise at Toulon. It lasted until 1942. On the Allied landings in North Africa on 13 November 1942, he tried in vain to convince Jean de Laborde, commander-in-chief of Vichy France's high seas fleet, to sail his force to Africa. At dawn on 27 November that year, when German forces arrived in the Toulon arsenal, he ordered its cruisers to scuttle themselves.

In 1943, he was approached to become Vichy's Secretary of State for the Navy and the Colonies. He hesitated but decided to take it up to avoid the post falling into pro-German hands. From then onwards "he never stopped struggling at all times against the German and Italian armistice commissions, sabotaging any work in the arsenals that would assist the enemy and saving personnel in the naval and colonial ministry from being sent to Germany. Thanks to him, naval security worked to inform the Allies and naval infrastructure was preserved".

On 20 August 1944, he and Philippe Pétain were arrested at Vichy by the Gestapo commander in Vichy and the German Military Police. He was taken to Sigmaringen. The new French authorities dismissed him from the navy in September 1944. Bléhaut was part of Pétain's inner circle when Pétain was evacuated from Sigmaringen at the end of April 1945 ahead of the Allied advance. After crossing into Switzerland, Pétain and his circle voluntarily surrendered at the border post at Valborde a few days later. Bléhaut was thus imprisoned at Fresnes Prison in the suburbs of Paris. He was provisionally freed in March 1946, but he decided to flee to Switzerland rather than appear before the Haute Cour de Justice. He was found guilty of crimes against state security and of 'indignité nationale' by that court and condemned in his absence to 10 years' imprisonment and national degradation. He voluntarily returned to France in 1955. The decree revoking him was annulled by the Conseil d'État in May 1956.

In 1946, Lieutenant-Commander André Storelli, decorated with the Resistance Medal and who would later be Admiral and Chief of Staff of the French Navy (1970–1972), gave a testimony on the actions of Admiral Bléhaut:
"Admiral, first of all, just as we have seen our efforts officially rewarded for having executed your orders, I affirm that you are entitled to the gratitude of the Resistance, . [...] All those who knew you and had the honor of serving you, in the difficult circumstances in which you were their leader, know that your action was only inspired by the desire to defend, within your sphere of command, what remained of our Navy capacity. None of them had to be ashamed of his actions by complying with your orders. [...] You gave them [these officers] direction and advice, either verbally or through your staff, to fight in all areas against the occupier. It was with your full consent that the Navy spontaneously and very largely made available to the resistance the means at its disposal. [...] I can assure you that you have done the maximum. [...] In addition, many adversaries, particularly suspicious of the sailors, surrounded us, and it was obvious that a more direct action would have unmasked you immediately, resulting in the immediate control by the enemy of the means valuable and effective you still had.".

Awards
 Legion of Honor – Commander (France)
 War Cross 1914–1918 (France)
 Medal of Military Values – Silver (Italy)
 Order of the Savior – Officer (Greece)
 Order of the Star of Romania – Officer (Romania)
 Order of the Rising Sun – 4th class (Japan)
 Order of Leopold – Commander (Belgium)

References

French Naval Ministers
People of Vichy France
Recipients of the Croix de Guerre 1914–1918 (France)
Commandeurs of the Légion d'honneur
French military personnel of World War I
French Navy admirals of World War II
1889 births
1962 deaths
French politicians convicted of crimes

redpajama_set_name: RedPajamaWikipedia


text:
Copyright (C) 2023 by Patrick's People all rights reserved Updated 24 January 2023
STEWART, Walter 6th High Steward of Scotland
(About 1292-1327)
BRUCE, Marjorie Princess of Scots
MURE, Adam, Sir
(About 1295-)
MURE, Janet heiress of Polkellie
STEWART, Robert II King of Scots
MURE, Elizabeth 'of Rowallan'
STEWART, Isabella
(About 1351-About 1410)
1. DOUGLAS, James 2nd Earl of Douglas, Sir
2. EDMONSTONE, John ancestor of Duntreath family, Sir
STEWART, Isabella 2
Born: About 1351
Marriage (1): DOUGLAS, James 2nd Earl of Douglas, Sir 24 September 1371(papal dispensation) 1 2
Marriage (2): EDMONSTONE, John ancestor of Duntreath family, Sir by 1390 1 2
Died: About 1410
Another name for Isabella was STEWART, Isabel.1
User ID: K67.
Isabella married Sir James DOUGLAS 2nd Earl of Douglas, son of William DOUGLAS Earl of Douglas, 'Earl of Mar, jure uxoris' and Margaret Countess of Mar, 24 September 1371(papal dispensation).1 2 (Sir James DOUGLAS 2nd Earl of Douglas was born about 1350, died before 18 August 1388 in Battle of Otterburn, Northumberland, England 1 and was buried in Melrose, Roxburghshire, Scotland 1.)
Marriage Notes:
"Isabella, was married first (by dispensation 24 September 1371), to James, Earl of Douglas ..."
from Scots Peerage (vol 1)
"The Earl's wife was Isabel Stewart, daughter of King Robert II. The dispensation for their union is dated 24 September 1371, but it is not certain whether the marriage took place at that time or two years later, when 500 was paid on account of the marriage-contract .... By her the Earl had, according to Godscroft, one son, but he died in infancy, and his name has not been recorded."
from Scots Peerage (vol 3) 1 2
Isabella next married Sir John EDMONSTONE ancestor of Duntreath family by 1390.1 2 (Sir John EDMONSTONE ancestor of Duntreath family was born about 1340.)
"Isabella, was married .... second, between 1388 and 1390, to Sir John Edmonstone, ancestor of the family of Duntreath."
"The Earl's wife was Isabel Stewart, daughter of King Robert II.....She survived the Earl, and married, secondly, before 1390, Sir John Edmonstone, ancestor of the Edmonstones of Duntreath. She died about 1410. By her the Earl had, according to Godscroft, one son, but he died in infancy, and his name has not been recorded."
1 e-books, The Scots Peerage ed. Sir James Balfour Paul vol. 3 (1906).

redpajama_set_name: RedPajamaCommonCrawl


A workshop designed to help us ground and center and bring what lies in our roots to consciousness.
We have seven energetic centers or chakras that run along our spine that govern different segments of the body and different behavior. The Muladhara Chakra governs our survival and the way we manage our daily lives – our family, occupation, money, etc. When balanced we are grounded, relaxed and able to manage the demands of daily life. When imbalanced this chakra can be either tamasic, (underactive) which manifests as being fearful, anxious, lacking in focus and disconnected from the body or rajasic (overactive) lethargic, depressed and unable to bring about change. The practice will begin with gentle grounding asana, followed by delicious restorative poses with gentle hands-on adjustments, and will end with a thirty to forty-minute long yoga nidra. The asana, pranayama, restorative poses, visualizations and Yoga Nidra are all designed to ground, center and relax us and bring what lies in our roots to consciousness. The Yoga Nidra practice will help release memories stored in our subconscious and unconscious and in our DNA that is carried through the generations.


text:
World invited to watch museum restore Rembrandt's Night Watch
Rijksmuseum in Amsterdam has begun the biggest ever restoration of Rembrandt's The Night Watch, building a giant glass case around the famed painting so the world can see the work carried out live.
art and culture Updated: Jul 09, 2019 13:05 IST
Rijksmuseum in Amsterdam has begun the biggest ever restoration of Rembrandt's The Night Watch, building a giant glass case around the famed painting so the world can see the work carried out live. (Rijksmuseum/Instagram)
An Amsterdam museum on Monday began the biggest ever restoration of Rembrandt's The Night Watch, building a giant glass case around the famed painting so the world can see the work carried out live.
In what has been compared to a military operation, experts at the Rijksmuseum in the Dutch capital will spend a year studying the 1642 masterpiece before embarking on a huge makeover that could take several years more.
Operation Night Watch has started! It's the biggest and most wide-ranging research and restoration project in the history of Rembrandt's masterpiece. Join the exclusive Q&A with experts Katrien Keune & Susan Smelt, 5 pm CET live on Instagram. Questions? Leave them in the comments. Follow our special account @operation.nightwatch for more updates! #rijksmuseum #nightwatch #amsterdam
A post shared by Rijksmuseum (@rijksmuseum) on Jul 8, 2019 at 1:55am PDT
The multi-million-euro revamp of the tableau -- the survivor of a difficult history including several acts of vandalism and a period in hiding from the Nazis -- will also be livestreamed online.
"The Night Watch belongs to everyone," Rijksmuseum director Taco Dibbits told a media conference in front of the glass restoration case, alongside the 25-strong research team.
Met deze speciale scanner gaan onderzoekers de Nachtwacht doorlichten. En daarna opknappen. De laatste grote renovatie is bijna 45 jaar geleden. #nachtwacht #rijksmuseum #opknapbeurt #nosop3
A post shared by NOS op 3 (@nosop3) on Jul 8, 2019 at 9:38am PDT
"More than two and a half million people come and see it each year. It belongs to everybody who lives in the Netherlands, and the world. And we felt that the public has the right to see what happens to that painting."
- White haze -
Rembrandt van Rijn, the greatest artist from of the Dutch "Golden Age" of painting, created his masterpiece after a commission by the mayor and leader of the civic guard of Amsterdam, Frans Banninck Cocq, to depict the officers and other members of the so-called "Night Watch" militia.
Now one of the world's famous paintings, it remains under constant scrutiny by experts, who have recently noticed some changes.
In particular, a white haze has appeared on the surface of some parts, especially in the area around damage caused in 1975 when a mentally disturbed man slashed the painting 12 times. The haze is now bleaching out the figure of a small dog.
While most restoration work happens behind closed doors, the Rijksmuseum has decided to let daylight in on the magic.
Experts will work on the Night Watch inside a glass case designed by French architect Jean-Michel Wilmotte, who was behind revamps of both the Rijksmuseum and the Louvre gallery in Paris.
Where did the Night Watch go? Don't panic! Rembrandt's masterpiece will be on view in one of the cabinets in the Rijksmuseum's Gallery of Honour. Operation Night Watch is the most comprehensive and multi-faceted research and conservation project of the Night Watch ever conducted @rijksmuseum 🔍 Or experience our own interpretation of the Night Watch in our dedicated Art Suite by Rembrandt @conservatoriumhotel #TheSetExperience #Rijksmuseum #Amsterdam #OperationNightwatch
A post shared by Conservatorium Hotel (@conservatoriumhotel) on Jun 27, 2019 at 8:11am PDT
"The Night Watch was restored in the 1970s... while the restorers were at work, we closed the curtains at that time. And now it will all be on view for the public to see so in a sense it's the first time," Dibbits told AFP.
"There will a team of 25 scientists, researchers, conservators, curators, restorers.... So it's a great team of people who have a lot of experience with Rembrandt because the Rijksmuseum has 22 Rembrandts and we've treated them over the past generation."
Experts will examine the painting using high-resolution photography and computer analysis of every layer including varnish, paint and canvas before deciding on the best restoration techniques.
Whatever happened to the pieces that were cut off from the Night Watch? This was asked by @dudodrums following our call for questions about Rembrandt's most famous painting. In 1715, four pieces were cut off from the Night Watch in order to fit the painting on the wall at its new location, the Town Hall (now Royal Palace) on Dam Square. Unfortunately, the pieces have never been found... Read more in our stories. #operationnightwatch Image: Copy of The Night Watch, attributed to Gerrit Lundens, c. 1642 - c. 1655. On loan from the National Gallery, London. Bequeathed by the Revd Thomas Halford, 1857. #Rijksmuseum #Amsterdam #Rembrandt #NightWatch
A post shared by Rijksmuseum (@rijksmuseum) on Jun 18, 2019 at 6:54am PDT
- Historic survivor -
But the museum warned that anyone expecting fast-paced excitement might be better off watching paint dry.
"The research phase will take about one year," said Petria Noble, head of paintings conservation at Rijksmuseum.
"Then we will be able to make a treatment plan and that will be the next step in the restoration. At this moment, we don't really know how long the actual treatment of the painting will take."
Noble said the sheer scale of the three-metre-by-four-metre (nine-foot-by-13-foot) painting -- which is housed in its own special room at the museum -- had also contributed to the decision to restore it in public view.
"Given the size of the painting, it's really not possible to take it out of the galleries. That also imposes a certain risk for the painting," she said.
In many ways it is a miracle Rembrandt's brooding painting has survived the last three and a half centuries to be restored at all.
Large chunks were cut from each side of the Night Watch during a move in the 1700s, followed by several bouts of work on the varnish that darkened the picture and helped give it its name.
In 1911 a man stabbed it with a knife. Then in September 1939 the painting was evacuated from the Rijksmuseum and hidden in the cave, just months before Nazi Germany invaded the Netherlands. After the war in 1945 it needed major restoration.
Then came the 1975 knife attack, but its sufferings were still not over, as a man sprayed it with acid in 1990.
(This story has been published from a wire agency feed without modifications to the text.)
more from art and culture


text:
Victor Spirescu: Who Is Really Taking The Piss
Spiv &The Daily Chimpanzee.
Oh the dirty lying, no good, underhand cunts!
"Couldn't agree more Spiv. What have the NSPCC been up to now"?
Hi Voicey, fuck knows. Probably something that stinks, but I wasn't talking about the paedophile fronted 'charity'.
"Woops! My bad. Who are you on about then my bald headed, tattooed, Jahoo loving friend"?
I'm on about the Chimp and their propaganda bollocks in regard to this Romanian stooge Victor Spaghetti Spirescu.
You see, Spaghetti, or whatever the stooges name is, was/is the fella who had supposedly taken advantage of the relaxing of the UK immigration laws and was splashed all over the front pages of the National Press after being greeted by the slimy, slippery toad MP, Keith Vaz-Aline, who – if you think back – was also inexplicably ever present at the side of the Saldanha family following the *aherm aherm* suicide of the Royal Nurse Jacintha, back in December 2012.
PHOTO: Slippy Vazaline with the Saldanha family. The Toad MP spent far too much time cuddling Jacintha's 14 year old daughter, than is normal for even a family member to do – let alone a stranger… Just accusin'.
Now quite why sinister Vaz the Spaz would go to the airport to meet 'n' greet the Chimps anticipated massive influx of Romanian 'free loaders' is anyone's guess.
But in any case, the mass exodus from Eastern to Western Europe never materialised – just like I had repeatedly stated that it wouldn't.
In fact according to the Romanian Ambassador, Dr Ion Jinga, the total number of immigrants as of the 14th of January 2014 stood at twenty four.
Never the less,Vaz-Aline supposedly bought Spaghetti a cup of coffee on his arrival from Romania – at least he did according to the MSM – but like most things you read about where our Sniffenpissin MP's play a part, this simply wasn't true. (See HERE and photo below)
So why all the MSM interest in the Spaghetti Western script – get it? … Go me.
I mean, just 3 days after getting here on New Years Day, Spaghetti was again the focus of a Chimp article:
Victor Spirescu, 30, insisted he was in Britain 'to work' and yesterday he was true to his word, arriving at an Aldi in Biggleswade, Bedfordshire, at 8am to clean cars until 6pm.
And then, on the 8th of January 2014 the Chimp ran the following headline:
After one day washing cars, migrant quits: Workers say Romanian met by Keith Vaz at Luton airport has vanished
Strange that the Chimpanzee's at the Chimp knew that, don't cha think?
I mean, what are his fellow 'cut throat gypsies', tell tale tits?
You see, today the Chimp has published the article below that carrys the headline:
Now Romanian migrant treated to Costa Coffee by Keith Vaz lands a £60-a-day building job in London after quitting car wash following just one shift
He didn't vanish very fucking well then did he you diabolic excuses for journalists.
In fact, if you type into the Chimps search bar: Victor Kajagoogoo – or whatever the fraud cunts name is – a total of Twenty articles where he is mentioned pop up … Got that ? TWENTY, 20 Chimp articles about the fella since he arrived here on the 1st of January!
And since it is now the 20th of January, that figure equates to an article a day, don't cha know!
But why? As in: why so many articles about a Romanian Immigrant?
Well, I would imagine that since our intrepid immigrant is portrayed as a hard working, benefit scrounging, drug addicted, wife beating mush, who is only here so as he can save up enough money for a Mercededs Benz, the Chimp's message – and therefore the government's message – is something along the lines of:
"we told you, you lovely, honest, upstanding, extremely intelligent middle class people that the dangerous scroungers would come here in force to steal our benefits and our well paid jobs – of which there are plenty, plenty, and very well paid at that, not that the benefit scrounging, work shy British wasters are interested in them".
Isn't it a shame that the Monkey Cunts don't show the same commitment to investigation when it comes to the subject of our wholly corrupt, perverted Sniffenpissin MP's and scum cunt royal paedophile family?
Oh, and just one more thing before I go.
When talking about Vic the Dick and his bird, Catlaliar in regard to their hard life back in Romania, the Chimp had this to say:
For the time being Catalina is fending for herself by chopping enough wood to burn so she can stay warm in temperatures as low as minus 25c.
She then has to clean out a sty where a pig is being reared. It will be killed to eat next Christmas.
There is no running water in the crumbling one-bedroom house she bought with Mr Spirescu for £2,000 two years ago in the village of Pelisor, which is 200 miles north of the capital Bucharest.
Miss Curcean has to repeatedly lower a bucket on a winch down a well on the mud lane outside her home for cooking, cleaning and drinking. There is an outside toilet which consists of a hole in the ground covered by a wooden hut…
"poor cow. You can see why they are so keen to come over here can't you Spiv mate"?
No, not really Voicey… Look at the car, the telegraph pole and the fucking SATELLITE DISH.
The MSM will be the death of me, I'm fucking telling you… Just sayin.
Victor Spirescu quit his first job washing cars after just one day
Now he is earning £60-a-day on a building site in east London
Says his fiancee plans to move to the capital and get work as a cleaner
Couple are saving to buy a Mercedes and say they will go home in a year
Spirescu has also acknowledged that he assualted his ex-girlfriend
By CHRIS PLEASANCE
PUBLISHED: 17:21, 18 January 2014 | UPDATED: 17:21, 18 January 2014
Three weeks ago Romanian migrant Victor Spirescu arrived in the UK with the promise that he wanted to work and then go home.
But after just one day washing cars with a friend in Biggleswade, he suddenly quit and disappeared.
Now it seems he has landed a new job earning £60-a-day working on a building site somewhere in east London, and is planning on moving his 19-year-old fiancee Catalina Curcean to the UK as well.
Victor Spirescu, the first Romanian to arrive in the UK on New Years Day, has got a new job on a building site in east London
He quit his firs job washing cars after just two days but blames media attention following claims he assaulted his ex-girlfriend and tried to drown her in a lake
Spirescu says Catalina is hoping to move from their pig and sheep farm in Transylvania and get a job as a cleaner.
He says her money will be spent paying for all their bills and food, while his money will be saved to buy a Mercedes when they move back to Romania in a year's time.
Romanian tourist who went missing for four days after Tube train doors shut on his son survived on just £20…and still had £3 left!
Police swoop and detain six Romanian prostitutes in raids on brothels run in the suburbs
British diplomat who fell to his death from Romanian hotel 'left a note saying he was depressed about not starting a family'
Spirescu has refused to talk about the crime, though insists he is now a changed man
Miss Curcean has to repeatedly lower a bucket on a winch down a well on the mud lane outside her home for cooking, cleaning and drinking. There is an outside toilet which consists of a hole in the ground covered by a wooden hut.
After quitting his first job Spirescu says he has just £5 left until payday, and is on the lookout for a better paying job such as working in the North Sea crab fishing, or on an oil platform.
Speaking to the Guardian, the 30-year-old says he quit his first job after media pressure surrounding his private life forced him out.
It was revealed shortly after he arrived in the UK that Siprescu had a past conviction for assault against his ex-girlfriend, which he has refused to talk about in detail, insisting that he is a changed man.
It was claimed last week the Spirescu beat ex-girlfriend Ana, 26, to the ground in a park before threatening to drown her in a lake.
After that Spirescu says he received more than 50 calls a day from journalists and had photographers pretending to need a car wash in order to take pictures of him.
People even stopped him in the street to ask for an autograph.
Little more is known about Spirescu's past, except that he was born in Bucharest before moving to the small village of Pelisor. He says he left school at 18, learned English from the television, and then got on a plane to London.
Victor is now planning to move his girlfriend Catalina over to London so she can get a job as a cleaner and the couple can save money for a Mercedes
Speaking about his time in England he says he expected the capital to be cleaner, and criticised British food for being 'tasteless' compared to the chicken which he kills and eats himself at home.
Speaking about his welcome party waiting at the airport for his arrival, he said: 'There were television cameras, journalists and this guy Keith Vas – what was that about? I had no idea.'
Read more: http://www.dailymail.co.uk/news/article-2541789/Now-Romanian-migrant-treated-Costa-Coffee-Keith-Vaz-lands-60-day-building-job-London-quitting-car-wash-following-just-one-shift.html#ixzz2qy3OX15V
By Needlz • UK News •
Barry & The Drag Queen: A lovers tiff or much more to it than that?	The Who's who of Satanic Child Abuse
On the subject of Immigrant Benefits, I had my attention drawn to something a day or so ago in which TPB were planning on restricting Housing Benefit for 6 months or something. I have to say that most 'Folks' will applaud this and support it, but cast your mind back to when the UK Insurance industry were taken to the Euro courts regarding Female drivers policies, the judges ruled that it was Sexist and as such women had to now pay what their male counterparts pay (Oh no, it couldn't possibly have gone the other way could it, where Men's policies were brought in line with womens……just sayin').
Now, do you not think at some point in the future someone will take the same action, and as such EVERYONE that finds themselves in the unfortunate situation where they have to claim benefit will have it restricted to six months?
It's always the same, they claim to represent the national interest and the people, but in reality they want everything for themselves and make sure that those less fortunate, or anyone wishing to be a little bit more choosy about what employment they gain get treated like fu**ing dirt……and what's worse, people continually fall for this bollocks (and it's their money too).
Absolutely right Buster. Demonise and place multiple restrictions on EU immigrants and then when the REAL government in Brussels tells our PRETEND government that they can't treat immigrants differently than the indigenous plebs , what's going to happen? Right! We'll all be treated the same. Result! Expect to be interrogated if you're unfortunate enough to require A &E treatment and woe betide you if your spoken English isn't up to CaMORON'S Eton and Oxford standard.
Pissed off bastard
Hehe is that the Eton & Oxford standard where a million pound education gets you an O' level and a couple of CSE grades?..oh and the ability to think elitist while being puppeteered by bankers?
These figureheads for the Coprorations have the dumbness shining out of their eyes, can you imagine Levitt on The Chase lol
Strange how the photograph of Catalina Curcean has been cut to exclude the car on the MAIL Online site. Propaganda at work.
Named after a flying boat I believe 😉
No to Victimisation of Harmless People
To be fair, the car could be one used by the journalist/photographer, of course.
Yes the car may be 'somebody' else's, but the telegraph pole isn't and neither is the satellite dish! I'm unemployed and cannot afford a landline telephone and have NEVER had any need nor the desire to own a satellite dish! As to your article Chris, so according to the Romanian Ambassador as of the 14th of January only 24 'immigrants' have come here to become 'benefit scroungers' and/ or take our jobs! Oh and apparently old spag balls is now working in London for £60 a day as a builder. £60 a day in London! How the fuck does he or anybody survive on that in London? Over a week that wouldn't be enough to pay for a roof over his or anybody's head and they'll find themselves homeless and sleeping on the streets………oh sorry I forgot they are already, as the Chimp has already seen fit to report that! http://www.dailymail.co.uk/news/article-2532613/Beggars-sleeping-rough-Marble-Arch-arrested-UK-Border-police-just-day-British-immigration-laws-relaxed.html As I've said before Chris: Why the fuck would any Romanian, who whoever from wherever want to come to this now dump of a country to earn £60 a day and end up sleeping rough on the streets is beyond me, but clearly it's not beyond being reported by the eminent and highly qualified reporters of the Daily Chimp!
mrsWorried
"For the time being Catalina is fending for herself by chopping enough wood to burn so she can stay warm in temperatures as low as minus 25c"
Yup… but at least she has a way to keep warm, not constantly feeding her lekky meter and going cold when she has no money for a few days to top it up!
"There is no running water in the crumbling one-bedroom house she bought with Mr Spirescu for £2,000 two years ago"
I could cope with water from a well (no water bills) if I could buy a house+land for £2,000! (I wish) And the main part of the house certainly isn`t crumbling… looks quite sound.
Fair enough… at least the pig isnt pumped full of anti-biotics and growth hormones, and if you eat meat, why not rear the pig with respect, using household food scraps ? And who could eat a WHOLE pig over xmas?! Jointed it could last ages.
I dont know who these people are, but this is a propaganda piece.
Mr spaghetti might be a wife beating piece of shit (?) Who knows.
But I feel sorry for the people coming here thinking it is a "better" way of life.


text:
More demand for permanent workers and rising pay in Scotland for second month in a row
September 9, 2017 johnrobertson834
From the BBC website (and Reporting Scotland or Good Morning Scotland?):
'The jobs market in Scotland showed continued signs of strength and rising pay last month, according to a regular survey of recruiters. The IHT Markit report showed a further fall in the number of people available for permanent jobs, along with a rise in the number starting new roles. Low unemployment levels helped push up starting pay, though the rate of increase slowed in August.'
This looks like further evidence that the Scottish economy is growing and, of course, is more bad news for the Scottish Tories and Labour. The survey repeats the findings of only a month ago when I wrote:
'Hopefully, you've already seen that the Scottish unemployment rate was down again last month for the third time in a row and that the gap with the rest of UK is getting wider at 3.8% as opposed to 4.5%. In May, the Scottish rate was 4.4% compared to the UK rate of 4.6%. In June, it was 4.0% while the UK rate was still 4.6%. There's always the worry that many of the new jobs are temporary or casual in some way. However, research by the Recruitment & Employment Confederation, reported in insider.co.uk, indicated a 'steep' increase in permanent jobs as well as temporary appointments.'
http://www.insider.co.uk/news/steep-increases-permanent-jobs-starting-10949189
These increases clearly relate to growth in the overall economy. The Scottish economy grew by four times as much as the UK as-a-whole in the last quarter and an earlier survey reported Scottish businesses to be more confident than those in England.
Scottish businesses showing signs of greater health than those in the rest of the UK
Good news for the Scottish economy again! Big rise in permanent jobs and starting salaries climb in Scotland
http://www.bbc.co.uk/news/uk-scotland-scotland-business-41196397
Well done the BBC website for reporting the news Good morning Scotland and Reporting Scotland are clearly too busy for.
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