eurm10 msam10

# Density jump for oblique collisionless shocks in pair plasmas: physical
solutions

Antoine Bret1,2,3,4 Colby C. Haggerty5 Ramesh Narayan3,4 Email address for
correspondence<EMAIL_ADDRESS>1ETSI Industriales, Universidad de
Castilla-La Mancha, 13071 Ciudad Real, Spain
2Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus
Universitario de Ciudad Real, 13071 Ciudad Real, Spain
3 Center for Astrophysics — Harvard & Smithsonian, Harvard University, 60
Garden St., Cambridge, MA 02138, USA
4Black Hole Initiative at Harvard University, 20 Garden St., Cambridge, MA
02138, USA
5Institute for Astronomy, University of Hawaii, Manoa, 2680 Woodlawn Dr.,
Honolulu, HI 96822, USA

(?; revised ?; accepted ?. - To be entered by editorial office)

###### Abstract

Collisionless shocks are frequently analyzed using the magnetohydrodynamics
(MHD) formalism, even though MHD assumes a small mean free path. Yet, isotropy
of pressure, fruit of binary collisions and assumed in MHD, may not apply in
collisionless shocks. This is especially true within a magnetized plasma,
where the field can stabilize an anisotropy.

In a previous article (Bret & Narayan, 2022b), a model was presented capable
of dealing with the anisotropies that may arise at the front crossing. It was
solved for any orientation of the field with respect to the shock front. Yet,
for some values of the upstream parameters, several downstream solutions were
found.

Here, we complete the work started in Bret & Narayan (2022b) by showing how to
pick the physical solution out of the ones offered by the algebra. This is
achieved by 2 means: 1) selecting the solution that has the downstream field
obliquity closest to the upstream one. This criterion is exemplified on the
parallel case and backed up by Particle-in-Cell simulations. 2) Filtering out
solutions which do not satisfy a criteria already invoked to trim multiple
solutions in MHD: the evolutionarity criterion, that we assume valid in the
collisionless case.

The end result is a model in which a given upstream configuration results in a
unique, or none (like in MHD), downstream configuration. The largest departure
from MHD is found for the case of a parallel shock.

##  1 Introduction

Shock waves are fundamental processes in fluids. They have been the subject of
numerous studies for nearly two centuries (Salas, 2007). When the frequency of
collisions between particles is high, the thickness of the shock front is of
the order of a few mean free paths, since binary collisions are ultimately the
only microscopic mechanism capable of transferring some of the kinetic energy
of the upstream medium into heat in the downstream (Zel’dovich & Raizer,
2002).

However, _in situ_ observations of the bow shock of the Earth’s magnetosphere
in the solar wind have shown that its front is about a hundred kilometers
thick, while the mean free path at the same location is of the order of the
Sun-Earth distance (Bale et al., 2003; Schwartz et al., 2011). Such a shock
cannot be mediated by collisions. It is mediated by collective plasma
electromagnetic effects (Sagdeev, 1966). This type of shock is known as
“collisionless shock”.

Strictly speaking, collisionless shocks should be studied at the kinetic
level, using the Vlasov equation, since the absence of collisions can even
make it difficult to define a local velocity, as is the case, for example, in
counter streaming systems. Due to the complexity involved in solving the
Vlasov equation, collisionless shocks, and in particular the density,
temperature or velocity jumps they present, are often interpreted via MHD.

Yet MHD relies on hydrodynamics and as such entails the same hypothesis of
small mean free path (Gurnett & Bhattacharjee, 2005; Goedbloed et al., 2010;
Thorne & Blandford, 2017). This hypothesis of small mean free path has 2
consequences that are important for the calculation of the density jump around
a shock. The first consequence is that pressure is isotropic, both before and
after the shock. In fact, even if a fluid is subjected to pressure anisotropy
during its transport from the upstream to the downstream, binary collisions
will restore isotropy on a time scale of the order of the collision frequency,
well below the macroscopic times involved. The second consequence is that all
of the upstream fluid passes into the downstream, along with the matter and
momentum it carries.

It turns out that in a collisionless shock, these 2 consequences can be
invalidated (Bret, 2020). The first, because in the absence of collisions, a
plasma can maintain a stable anisotropy in the presence of an ambient magnetic
field (Hasegawa, 1975; Gary, 1993). The second consequence, because due to the
large mean free path, plasma particles can bounce off the shock front or even
travel upstream from the downstream, as is the case with accelerated particles
(Drury, 1983; Blandford & Eichler, 1987).

This article proposes a remedy to the first consequence: how to correct the
MHD jump equations so that they can account for an anisotropy in the plasma?

Notably, in the absence of an ambient magnetic field, the Weibel instability
ensures isotropy of a collisionless plasma (Weibel, 1959; Silva et al., 2021).
Therefore, the MHD jump equations only need to account for anisotropies if a
magnetic field is present.

Several authors have derived the MHD jump equations for the non-isotropic case
(Hudson, 1970; Karimabadi et al., 1995; Erkaev et al., 2000; Gerbig &
Schlickeiser, 2011). But in all of these works, while the anisotropy of the
upstream is considered a free parameter, so is that of the downstream. These
equations are therefore unable, on their own, to derive the density jump of a
shock whose downstream is not isotropic, because they lack precisely one
parameter: the anisotropy of the downstream.

In a recent series of papers, we developed a model that precisely fills this
gap. It was first applied to the case of a parallel shock (Bret & Narayan,
2018), i.e. a shock moving parallel to the ambient magnetic field. The
assumptions made and the results obtained were confirmed by numerical
simulations (Haggerty et al., 2022).

The model was then applied to the case of a perpendicular shock (Bret &
Narayan, 2019), and finally to the case of a switch-on and of an oblique shock
(Bret & Narayan, 2022a, b). The latter case, that of an oblique shock, is
analytically much more complicated than the parallel and perpendicular cases,
due to the complexity of the MHD jump equations for an oblique field and
anisotropic temperatures.

In Bret & Narayan (2022b), hereafter referred to as Paper I, the algebra of
these equations was solved, but the solutions were left unfiltered. As a
result, several coexisted for some combinations of the upstream parameters.

Here, Paper I is completed by filtering the algebraic solutions so that for a
given combination of upstream variables, there is no more than 1 solution for
the downstream.

Figure 1: System under scrutiny. It is identical to that of Paper I. Although
the conservation equations used are valid for any upstream temperatures, we
here, like in Paper I, only treat the strong sonic case $T_{\perp
1}=T_{\parallel 1}=0$. We work in the frame of reference where
$\mathbf{v}_{1}$ is normal to the front.

##  2 Method

The system under scrutiny is represented on Figure 1. It is identical to that
of Paper I. Though the conservation equations we shall use are valid for any
upstream temperatures, we here, like in Paper I, only treat the strong sonic
case $T_{\perp 1}=T_{\parallel 1}=0$.

###  2.1 Summary of previous works

As previously said, the basic caveat of MHD is that if a collisionlessly
stable anisotropy develops at the front crossing, MHD itself cannot derive it.
The jump of quantities like the density is therefore under-determined.

For completeness, we now briefly recall the results obtained in previous
works.

In Bret & Narayan (2018) we presented a model capable of solving this issue
for a parallel shock. We reasoned that as it crosses the front, the plasma is
compressed in the direction parallel to the motion. As a consequence, its
parallel temperature increases while its perpendicular temperature remains
constant. The 3 MHD conservation equations (matter, momentum, energy111For a
parallel shock, the MHD conservation equations are identical to the fluid ones
(Kulsrud, 2005). With anisotropic temperature, they are obtained setting
$\theta_{1}=\theta_{2}=\xi_{2}=0$ in Eqs. (26-31) of Appendix B.) are
therefore completed by,

$T_{\perp 2}=T_{\perp 1},$ (1)

allowing to derive the 4 downstream unknowns ($n_{2},v_{2},T_{\perp
2},T_{\parallel 2}$), in terms of the upstream variables. Note that here, the
perpendicular direction is common to the flow and the field.

Now, the state of the downstream resulting from the conservation of
$T_{\perp}$ may be stable, or not. If it is stable, then this is the end state
of the downstream. If it is unstable, the plasma migrates to the instability
threshold222The nature of this instability will be specified shortly..
Imposing marginal stability then provides again a fourth equation allowing to
fully determine the properties of the downstream.

Bret & Narayan (2018), as well as every subsequent works of ours on the same
model, is limited to pair plasmas. The reason for this is that the equality of
the mass of the species allows to consider only one parallel and one
perpendicular temperature. In an electron/ion plasma where electrons and ions
are heated differently in the shock, a 4 temperatures model would be required
(Guo et al., 2017, 2018). Yet, since the model eventually relies on
macroscopic physics, it should also apply to electron/ion plasmas, as
preliminary Particle-in-Cell (PIC) simulations seem to indicate (Shalaby,
2024).

The model predicted, for a strong sonic parallel shock, a density jump going
from 4 to 2 in the high field regime, a prediction successfully confirmed by
PIC simulations in Haggerty et al. (2022). Such a result stands in contrast
with MHD where the density jump should always be 4, regardless of the field
strength333The jump of a strong fluid shock with adiabatic index $\gamma=5/3$
is 4. And for the parallel case, the MHD and fluid conservation equations are
identical (Kulsrud, 2005)..

The perpendicular case was treated in Bret & Narayan (2019). There, the
direction perpendicular to the flow is eventually parallel to the field, so
that the counterpart of Eq. (1) is,

$T_{\parallel 2}=T_{\parallel 1}.$ (2)

The switch-on shocks, where the field is oblique in the downstream only, was
treated in Bret & Narayan (2022a). The model has also been solved for a
parallel or a perpendicular shock, with an anisotropic _upstream_ (Bret,
2023a, b).

Finally, the general case where the field may be oblique in both the upstream
and the downstream was treated in Bret & Narayan (2022b), namely, Paper I.

In Bret & Narayan (2022a, b), the closure of the MHD jump equations was
achieved through an _ansatz_ interpolating between (1) and (2). In the limit
of a cold upstream with $T_{1}=0$, which is the regime treated in Paper I and
hereafter, the _ansatz_ reads,

$\displaystyle T_{\parallel 2}$ $\displaystyle=$ $\displaystyle
T_{e}\cos^{2}\theta_{2},$ $\displaystyle T_{\perp 2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}T_{e}\sin^{2}\theta_{2},$ (3)

where $T_{e}$ is a parameter determined when solving the equations and
$\theta_{2}$ is the angle of the downstream field with the shock normal (see
Figure 1). Eqs. (2.1) correctly reduce to (1) and (2) in their respective
limits since $\theta_{2}=0$ for the parallel case, while $\theta_{2}=\pi/2$
for the perpendicular one. Such a scheme guarantees that both perpendicular
temperatures are equal, as required by the Vlasov equation (Landau & Lifshitz
(1981), §27). Also, the total thermal energy in the 3 directions sums up to
$k_{B}T_{e}$, where $k_{B}$ is the Boltzmann constant.

In summary, and since the 2 instabilities involved are the firehose and the
mirror instabilities (see Section 4), our model can be stated as follow:

  * •

As the plasma goes through the shock front, its temperature normal to the flow
is conserved for the parallel and perpendicular cases. This translates
directly to (1) and (2) respectively. For the oblique case with cold upstream
$T_{1}=0$, Eqs. (2.1) interpolate between these 2 extremes.

  * •

The resulting state of the plasma in the downstream is called “Stage 1”.

  * •

If the downstream field is strong enough to stabilize Stage 1, then this is
the end state of the downstream.

  * •

If the downstream field is too weak to stabilize Stage 1, then

    * –

If Stage 1 is firehose unstable, it migrates to the firehose instability
threshold. This is “Stage 2 - firehose”.

    * –

If Stage 1 is mirror unstable, it migrates to the mirror instability
threshold. This is “Stage 2 - mirror”.

###  2.2 Present work

Paper I has 3 kinds of limitations:

  1. 1.

It is restricted to strong sonic shocks, namely $T_{1}=0$, and to non-
relativistic pair plasmas. These restrictions are still considered here.

  2. 2.

It considers the simplest expressions for the Alfvén velocity and the
stability criterion of the instabilities involved in our model. Yet, more
accurate expressions are required in an anisotropic plasma. The present work
accounts for one of them.

  3. 3.

It only presents the _allowed_ solutions to the conservation equations, plus
Eq. (2.1). It does not filter these solutions according to their physical
relevance. Such a filtering is the main goal of the present work.

Our purpose here is to deal with points (_b_) and (_c_) above.

Besides the variables explained on Figure 1, we shall use the following
dimensionless parameters,

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{n_{2}}{n_{1}},$
$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle\frac{T_{\perp
i}}{T_{\parallel i}},$ $\displaystyle M_{A,i}$ $\displaystyle=$
$\displaystyle\frac{v_{i}}{v_{A,i}},$ $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{B_{1}^{2}/8\pi}{\frac{1}{2}n_{1}mv_{1}^{2}}\equiv\frac{1}{M_{A,1}^{2}},$
(4)

where $v_{A,i}$ is the Alfvén velocity,

$v_{A,i}=\frac{B_{i}}{\sqrt{4\pi n_{i}m}}.$ (5)

The parameter $\sigma$ is frequently used in simulations of collisionless
shocks like Haggerty et al. (2022), while the Alfvén Mach number is common in
MHD shock literature.

In addition to the Alfvén Mach number defined above, we shall often use in the
sequel its following variant,

$M_{Aix}\equiv\frac{v_{i}\cos\xi_{i}}{v_{A,i}\cos\theta_{i}}.$ (6)

It compares the projection of the flow velocity along the shock normal ($x$
axis), to the projection of the Alfvén velocity, still along the shock normal.

Since the road map for solving our model is eventually the one already used in
MHD, we start by reminding the reader how it applies there.

##  3 Isotropic MHD results and evolutionarity

One of the criteria used here to filter out some solutions is the so-called
“evolutionarity criterion”, already present in isotropic MHD. It is therefore
convenient to present how it operates there.

The MHD conservation equations for isotropic temperatures are reported in
Appendix A. They can be used to derive an expression of the downstream field
angle $\theta_{2}$ in terms of the upstream quantities only. Namely, the
quantity,

$T_{2}\equiv\tan\theta_{2},$ (7)

is a root of the polynomial (24) in Appendix A. The MHD density jump is then
given by

$r=\frac{M_{A1}^{2}T_{2}}{M_{A1}^{2}\tan\theta_{1}+T_{2}\cos^{2}\theta_{1}-\sin\theta_{1}\cos\theta_{1}},$
(8)

in terms of the upstream Alfvén Mach number (2.2).

Although such a derivation of the MHD density jump is uncommon in the
literature, it mimics the derivation of the jump in our model. It is therefore
pedagogically valuable for the present work.

Figure 2: _Upper row_ : density jump $r$ for different angles $\theta_{1}$ in
terms of $M_{A1x}=M_{A1}/\cos\theta_{1}$, defined by Eq. (6). Sometimes there
is more than 1 solution. _Second row_ : angle $\theta_{2}$ of the field
$\textbf{B}_{2}$ with the $x$ axis (see Figure 1). _Third row_ :
evolutionarity criterion. Some branches, namely, the ones crossing the shaded
areas, are to be excluded. _Lower row_ : same as upper row, but without the
branches excluded by the evolutionarity criterion. There is now but 1 solution
for a given $M_{A1x}$, or none. The black circles for $\theta_{1}=0.5\pi/2$
are the results of our MHD simulations (see Section 3.1).

Figure 2 presents the solutions for 3 different angles $\theta_{1}$. The upper
row shows all the possible solutions. For $\theta_{1}=0$ and
$M_{A1x}\in[1,2]$, there are 2 MHD branches, the lower one pertaining to the
switch-on solutions. For $\theta_{1}=0.175\pi/2$ there are even 3 solutions
for $M_{A1x}\in[1,1.34]$.

Which one will the shock choose? This question is at the heart of this work.
Let us now see how it is solved in MHD.

The MHD answer relies on the notion of shock “evolutionarity”, which has been
discussed several times in the literature (e.g., Kennel et al. (1990); Farris
et al. (1994); Falle & Komissarov (2001); Kulsrud (2005) ). For given upstream
and downstream boundary conditions, the MHD Rankine-Hugoniot jump conditions
give a unique solution for fast shocks, where the flow speed on both sides of
the shock are super-Alfvénic, and also for slow shocks, where the fluid is
throughout sub-Alfvénic. These two types of shocks are stable and are
described as evolutionary. Four other potential shock types, each of which has
super-Alfvénic upstream fluid and sub-Alfvénic downstream fluid, have no
unique solutions to the Rankine-Hugoniot relations. In these shocks, the
transverse magnetic field switches sign across the shock, and the fluid
equations do not provide the correct number of Alfvén waves to handle the
field flip (Falle & Komissarov 1997, Kulsrud 2005). Such shocks will not arise
naturally from generic initial conditions. Even if they are artificially set
up, they will tend to deviate quickly from their initial configuration,
typically splitting into two shocks, one fast and the other slow. These
“forbidden” shocks do _not_ satisfy the evolutionarity condition. They are
called “intermediate” (Falle & Komissarov, 1997) or “extraneous” (Kulsrud,
2005) shocks.

The MHD solutions presented on the upper row of Figure 2 need therefore to be
filtered according to the aforementioned evolutionarity criterion. The
upstream Alfvén Mach number to consider for the evolutionarity analysis is not
the one given by Eq. (2.2), but its variant (6) instead. With this definition,
the downstream Alfvén Mach number reads

$M_{A2x}^{2}=1+\frac{\tan\theta_{1}\left(M_{A1}^{2}\sec^{2}\theta_{1}-1\right)}{T_{2}}.$
(9)

It is with definition (6) that the Alfvén Mach number of the MHD switch-on
shocks, see left plot of third row on Figure 2, is found equal to 1 (Goedbloed
et al. (2010), p. 853). These modes, by definition, do _not_ have
$\theta_{2}=0$ (second row on Figure 2) nor $\xi_{2}=0$. Corrections (6) to
the Alfvén Mach number (2.2) are therefore important here.

$M_{A2x}$ is plotted in terms of $M_{A1x}$ on the third row of Figure 2. The
forbidden, non-evolutionary, zones have been colored on the plots. They
feature the non-evolutionary transition just described, namely super-Alfvénic
$\rightarrow$ sub-Alfvénic, together with the reverse transition sub-Alfvénic
$\rightarrow$ super-Alfvénic. As a result, the MHD branches going through
these regions are non-evolutionary, hence not physical solutions of the MHD
problem.

The lowest row of figure 2 is the result of this evolutionary filter applied
to the upper row. There is now but 1 solution for a given $M_{A1x}$, or none,
as there are $M_{A1x}$-gaps where no solutions appear. Regardless of the
initial conditions, the shock formed is never found with a $M_{A1x}$ inside
such gaps like, for example, $M_{A1x}\in[1,1.4]$ for $\theta_{1}=0.5\pi/2$
(see Figure 2, bottom right plot).

We shall assume in the sequel that the evolutionarity criteria also applies in
the collisionless case. This will have to be checked in future works.

Figure 3: MHD simulation in which the evolutionarity criterion results in the
formation of 2 sub-shocks instead of one. The dashed blue lines show the
initial fluid configuration at $t=0$, when two fluid columns collide at $x=0$.
The solid red lines show the configuration at $t=1.5$. Each fluid column
develops two shocks, a fast shock at $|x|=1.58$ and a slow shock at
$|x|=0.34$. The little dip in density $n$ at $x=0$ for $t=1.5$ is a numerical
artifact.

###  3.1 MHD simulations

In order to illustrate these theoretical results, we ran some MHD simulations
with the code KORAL (Sa̧dowski et al., 2013, 2014). KORAL is designed for
multi-dimensional radiative MHD simulations in general relativistic
spacetimes. However, if we turn off the radiation module as well as special
and general relativity, the code reduces to a multi-dimenionsional non-
relativistic MHD code. This version of the code was used here. Some of the
results for $\theta_{2}$, $M_{A2x}$ and $r$ in terms of $M_{A1x}$ for upstream
$\theta_{1}=0.5\pi/2$, are pictured on Figure 2 by the black circles. They
perfectly line up with the theory and avoid the predicted gap in the solutions
for $M_{A1x}\in[1,1.4]$.

Figure 3 shows the result of a simulation of a shock in a non-evolutionary
case. At $t=0$, two cold fluids of identical density and opposite velocities
$\pm 1.5\mathbf{x}$ collide (blue lines). The magnetic field has modulus unity
and $\theta_{1}=0.5\pi/2$. At $t=1.5$, two shocks formed instead of one. The
non-evolutionary case gives rise to 2 sub-shocks, as predicted in Kulsrud
(2005) for example.

##  4 Instabilities involved and modified Alfvén velocity

As previously stated, the starting point for our model are the MHD
conservation equations with anisotropic temperatures. They have been derived
in Hudson (1970) and are reported in Appendix B with the notations of Bret &
Narayan (2022a, b).

For Stage 1, we solve them imposing relations (2.1). This defines Stage 1.
Then, mirror or firehose stability of Stage 1 has to be assessed. These
instabilities are discussed here.

Now, evolutionarity involves the downstream Alfvén velocity (projected onto
the shock normal). As long as the plasma is isotropic, the Alfvén velocity in
given by Eq. (5). As a result, the downstream Alfvén Mach number for the MHD
switch-on shocks is exactly 1.

Yet, in an anisotropic plasma, it was found in Abraham-Shrauner (1967) that
the Alfvén velocity reads444This expression already includes the $\cos\theta$
factor of Eq. (6). There is no need to multiply it by an additional
$\cos\theta$ when computing the Alfvén Mach number (6).,

$c_{A}=\pm v_{A}\sqrt{S_{\perp}-S_{\parallel}+1}~{}\cos\theta,$ (10)

where $v_{A}$ is still given by Eq. (5) and

$S_{\perp,\parallel}=\frac{nk_{B}T_{\perp,\parallel}}{B^{2}/4\pi}.$ (11)

In Eq. (10), the $\pm$ sign refers to waves propagating along the field, or in
the opposite direction. As shall be checked in the sequel (Sections 5.1 & 5.2
and Figure 5), with this correction to the Alfvén velocity, the switch-on
solutions of our model have $M_{A2x}=1$, exactly like in MHD.

In addition, the quantity below the square root can become negative. Such a
situation indicates that the Alfvén waves become unstable, which corresponds
to the firehose instability. For this to happen, $S_{\perp}-S_{\parallel}+1<0$
is required, which can be cast under the form,

$A\equiv\frac{T_{\perp}}{T_{\parallel}}<1-\frac{2}{\beta_{\parallel}},$ (12)

where555This parameter is just twice the $S_{\parallel}$ parameter defined by
Eq. (11). We could use only $\beta_{\parallel}$. But for clarity, we present
the result of Abraham-Shrauner (1967) with the notations of Abraham-Shrauner
(1967).,

$\beta_{\parallel}=\frac{nk_{B}T_{\parallel}}{B^{2}/8\pi}.$ (13)

The criterion (12) is therefore the one we shall use in the sequel, in order
to preserve the inner coherence with the switch-on solutions of our model
having $M_{A2x}=1$. This criterion slightly differs, by the factor 2, from the
one commonly used in the literature and in Paper I.

The Solar Wind data are a key test for the threshold of the firehose
instability. They are indeed compatible with both criteria, namely with or
without the factor 2 (Hellinger et al., 2006; Bale et al., 2009; Maruca et
al., 2011).

Figure 4: Stability diagram. The solid lines picture the criterion used here.
The dashed lines pertain to other criteria discussed in Section 4.

The other instability considered in relation to the stability of Stage 1, is
the mirror instability. While the firehose instability occurs for too low an
anisotropy $T_{\perp}/T_{\parallel}$, the mirror instability occurs for too
high an anisotropy. The standard threshold given in the literature reads
(Hasegawa, 1975; Gary, 1993),

$A>1+\frac{1}{\beta_{\parallel}}.$ (14)

Yet, a different criterion is given in Abraham-Shrauner (1967), namely

$A>1+\frac{1}{A~{}\beta_{\parallel}}.$ (15)

However, while Stage 2-firehose remains analytically tractable using (12),
Stage 2-mirror is not when using (15) instead of (14). Therefore, we adopt
criterion (12) for the firehose instability and keep (14) for mirror. Note in
this respect that 1), the Solar Wind data are compatible with both criteria
and 2), the inner coherence of the model does not impose a specific mirror
criteria, as is the case for firehose.

Figure 4 pictures the various criteria commented here for the mirror and the
firehose instabilities. Even if the corrected criterion for the mirror
instability is functionally different than the one without the correction, the
end result is qualitatively the same, and remains compatible with the Solar
Wind data.

The two stability domains are disconnected, so that the plasma cannot be
unstable to both at once. There is no possible competition between them.

Figure 5: _Left_ : All solutions offered by our model for $\theta_{1}=0$.
_Center_ : All the solutions fulfill the evolutionarity criterion. Note that
this is different from the isotropic MHD problem, where some solutions for
$\theta_{1}=0$ enter the shaded forbidden zone (Figure 2, row 3). Note also
that, with definition (10) of the Alfvén speed, $M_{A2x}=+\infty$ on the
firehose threshold. This is why the green branch on the left is sent to
$M_{A2x}=+\infty$ on the center plot. Notably, $M_{A2x}=1$ for the switch-on
shocks of our model (red segment on center plot). For this analysis, the
horizontal axis must be $\sigma_{x}=\sigma\cos^{2}\theta_{1}$, which for
$\theta_{1}=0$ makes no difference. _Right_ : End result once the
“$\theta_{2}$ closest to $\theta_{1}$” filter has been applied. This is the
result found in Bret & Narayan (2018); Haggerty et al. (2022).

##  5 Branches selection

Having specified the instabilities involved in the transition from Stage 1 to
Stage 2, together with their respective thresholds, we can proceed to the
filtering of the solutions our model offers. In this respect, it is
instructive to single out the case $\theta_{1}=0$.

###  5.1 Case $\theta_{1}=0$

Figure 5-left shows the solutions offered by our model for $\theta_{1}=0$,
without any filtering. It displays the various solutions with stable Stage 1,
plus a branch, in green, corresponding to Stage 2-firehose because Stage 1 is
firehose unstable for $\sigma\in[0,0.5]$. For $\sigma\in[0.5,1]$, there are up
to 3 possible solutions: one has $r=2$ and $\theta_{2}=0$, and the other two
pertain to the switch-on solutions in red, with $\theta_{2}\neq 0$.

Could the evolutionarity criterion help filtering them (the answer is ‘no’) ?
Figure 5-center answers the question. Here is plotted the downstream Mach
number $M_{A2x}$ of each solution, as defined by Eq. (6), where the Alfvén
speed has been corrected according to Eq. (10). Our switch-on solutions have
exactly $M_{A2x}=1$, while the others also satisfy the evolutionary
criterion666Figure 5(b) of Bret & Narayan (2022a) shows the switch-on
solutions of our model with $M_{A2x}\neq 1$. This is because correction (10)
to the Alfvén speed was not considered.. In line with the adequate definition
(6) of the Alfvén Mach number for evolutionarity analysis, we use on the
horizontal axis the parameter

$\sigma_{x}\equiv\sigma\cos^{2}\theta_{1},$ (16)

which, for the present case $\theta_{1}=0$, makes no difference.

Note that with definition (10) of the Alfvén speed, $M_{A2x}=+\infty$ on the
firehose threshold. This is the reason why the green branch on the left is
sent to $M_{A2x}=+\infty$ on the center plot. As a consequence, each time the
system eventually settles in Stage 2-firehose with $\sigma_{x}<1$ (i.e,
$M_{A1x}>1$), it is evolutionary.

As a conclusion, in the interval $\sigma\in[0.5,1]$, there are three solutions
for Stage 1, and all three satisfy the evolutionarity criterion. The
evolutionarity criterion by itself is therefore _not_ sufficient to trim the
number of solutions down to 1, or even 0.

Figure 6: _Left_ : expected value of $B_{\perp 2}/B_{1}$, in red for the
switch-on solutions with $\theta_{2}\neq 0$, and in black for the solution
with $\theta_{2}=\theta_{1}=0$. The circles show the values of $\sigma$ which
have been simulated. _Right_ : results of the PIC simulations, with the shock
density profile on top, and the ratio $B_{\perp 2}/B_{1}$ below. The $\sigma$
value for a curve is given by the circle of the same color on the left plot.
The same results are obtained for $\theta_{1}=2^{\circ}$. The flow is along
the $x$ axis and $d_{e}=c/\omega_{p}$ is the electron inertial length.

###  5.2 PIC simulations for $\theta_{1}=0$

In order to check which solution the system chooses, we ran a series a PIC
simulations for various $\sigma$’s, performed with the fully kinetic 3D PIC
code, TRISTAN-MP (Buneman, 1993; Spitkovsky, 2005)777Simulation parameters
identical to those of Haggerty et al. (2022).. The surest way to tell whether
the shock is of the switch-on type or not, is to plot the perpendicular
component $B_{\perp 2}$ of the field in the downstream. According to Eq. (27),
with $\theta_{1}=0$ it simply reads

$B_{\perp 2}=B_{1}\tan\theta_{2}.$ (17)

Figure 6-left shows the expected value of $B_{\perp 2}/B_{1}$, in red for the
switch-on solutions with $\theta_{2}\neq 0$, and in black for the solution
with $\theta_{2}=0$. The circles show the values of $\sigma$ which have been
simulated. Figure 6-right shows the results of the PIC simulations, with the
shock density profile on top, and the ratio $B_{\perp 2}/B_{1}$ below. Besides
some perturbations around the shock front at low $\sigma$, $B_{\perp 2}/B_{1}$
does not depart from 0 in the downstream for any $\sigma$. These perturbations
are due to particle acceleration and back-reaction in the precursor, and from
the instabilities triggered by the interaction of the fast upstream flow with
the front (Sironi et al., 2013; Brown et al., 2023).

Simply put, PIC simulations consistently discard the switch-on solutions. The
same results are obtained for $\theta_{1}=2^{\circ}$, so that we are not
witnessing a singular behavior fruit of a perfect, hence unrealistic,
symmetry. A similar pattern was observed in the relativistic regime in Bret et
al. (2017).

The situation for $\theta_{1}\sim 0$ is here markedly different from the MHD
case. In MHD, where isotropy is imposed, the evolutionarity criterion imposes
switch-on shocks within some $\sigma$-range, as explained in Section 3. In our
model, where an anisotropy is driven by the field, the evolutionarity
criterion does _not_ impose switch-on solutions, while PIC simulations
consistently choose the non switch-on ones.

Arguably this explains why so few detections of switch-on shocks have been
made in the Solar System. Indeed, among the thousands of shocks observed
(Farris et al. (1994); Russell & Farris (1995), Balogh & Treumann (2013)
§2.3.6.) only one “possible” detection of an interplanetary switch-on shock
was reported in Feng et al. (2009). Also, Russell & Farris (1995) reported the
detection of only one switch-on shock among the International Sun-Earth
Explorer data.888Switch-on shocks have been produced in the laboratory (Craig
& Paul, 1973), but within a collisional environment, where MHD rules. We here
deal with collision _less_ plasmas.

Still, what about these exceptions, since our collisionless scenario should
rule them out? Several explanations are possible,

  * •

The detections were faulty. Hence the adjective “possible” associated with one
of them.

  * •

We only solve our model for a cold upstream, namely $T_{1}=0$. Maybe a finite
$T_{1}$ would have our model allowing for some switch-on shocks.

  * •

The _ansatz_ (2.1) is not accurate enough, and a better version would allow
for some switch-on shocks.

At any rate, PIC simulations and observations tell switch-on shocks in
collisionless plasma are rare. We therefore propose the following criterion
allowing to choose between various solutions:_the system chooses the solution
with $\theta_{2}$ closest to $\theta_{1}$._

Note that this “$\theta_{2}$ closest to $\theta_{1}$” criterion stems from our
PIC simulations, as well as others of parallel shocks in pair (Bret et al.,
2017; Haggerty et al., 2022) or electron/ion (Niemiec et al., 2012; Zeković,
2019) plasmas, where the same, non switch-on branch, is consistently chosen.
This is why we used the verb “propose”. Its robustness on longer time scales,
or other shock geometries, is beyond the scope of this work and should be
assessed in further works.

Figure 5-right eventually shows the end result once the “$\theta_{2}$ closest
to $\theta_{1}$” filter has been applied to the $\theta_{1}=0$ case. There is
now but one solution for any $\sigma$, which indeed is the one found in Bret &
Narayan (2018); Haggerty et al. (2022).

Figure 7: Flowchart of the resolution of our model. In case Stage 1 is mirror
unstable for a pair $(\sigma,\theta_{1})$, there can be a degeneracy in the
choice of Stage 2-mirror for the same pair $(\sigma,\theta_{1})$. In such a
case, we choose the Stage 2-mirror with the $\theta_{2}$ closest to the
$\theta_{2}$ of the Stage 1 it comes from, as was already done in Bret &
Narayan (2022b). Such a situation never happens with Stage 2-firehose.

###  5.3 General algorithm for branches selection

We may now lay out the general algorithm for branches selection. The criteria
used to eliminate some are, applied in this order,

  1. 1.

Exclude Stage 1 solutions with $r<1$. Then select the one with the
$\theta_{2}$ closest to $\theta_{1}$.

  2. 2.

In case Stage 1 is unstable, is the anisotropy $A_{2}$ of a given Stage 2
solution, negative? This was already implemented in Paper I and allows to
eliminate some Stage 2 -firehose and -mirror solutions. This never happens
with Stage 1 since it has, by design from Eqs. (2.1),
$A_{2}=\frac{1}{2}\tan^{2}\theta_{2}$.

  3. 3.

Does the resulting solution fulfill the evolutionarity criterion ?

The evolutionarity criterion is applied last because it operates on time
scales related to the propagation of the shock999See Section 6.3 in Falle &
Komissarov (2001)., whereas the other criteria operate on much shorter time
scales, related to plasma instabilities.

The algorithm is eventually represented by the flowchart on Figure 7. In case
Stage 1 is mirror unstable for a given pair $(\sigma,\theta_{1})$, there can
be a degeneracy in the choice of the Stage 2-mirror states for the same pair
$(\sigma,\theta_{1})$. In such a case, we choose the Stage 2-mirror with the
$\theta_{2}$ closest to the $\theta_{2}$ of the Stage 1 it comes from, as was
already done in Bret & Narayan (2022b). Such a situation never happens with
Stage 2-firehose.

According to the flowchart on Figure 7, there is necessarily only 1 solution
left before applying the evolutionarity criterion. There is therefore no need
to apply the “$\theta_{2}$ closest to $\theta_{1}$” filter in (_c_), since
only 1 branch can make it to this stage. Yet, this does not mean the
evolutionarity filter is useless, as it can simply forbid the existence of a
solution in some $\sigma$ range, as is the case in MHD (see bottom row of
Figure 2).

Figure 8: Detailed progression of the application of the filtering process
described by the flow chart on Figure 7, for $\theta_{1}=0.3\pi/2$. On Figure
(d), the green color indicates that these Stage 1 solutions are firehose
unstable. Then on Figure (g), the green color shows the density jump of Stage
2-firehose. The red branch on Figure (f) with $A_{2}\sim-1$ discards Stage 2
firehose solutions with $r>4$, not shown on Figure (e).

##  6 Results

While the fruit of our algorithm has already been explained for
$\theta_{1}=0$, it is interesting to detail how it unfolds for an oblique
field, like for example $\theta_{1}=0.3\pi/2$.

Figure 8(a) shows all possible density jumps for Stage 1. Notably, $r<1$ for
$\sigma\in[1,1.3]$. The corresponding branches are then eliminated on Figure
8(b).

For these Stage 1 solutions with $r>1$, Figure 8(c) then shows the
corresponding values of $\theta_{2}$. The horizontal red line is at
$\theta_{2}=\theta_{1}$. The result of the “$\theta_{2}$ closest to
$\theta_{1}$” selection rule is displayed on Figure 8(d).

On Figure 8(d), the green color indicates that some Stage 1 solutions at low
$\sigma$ are firehose unstable. We therefore need to examine Stage 2-firehose
solutions. All of them are displayed on Figure 8(e). Yet, Figure 8(f) shows
that the lower branch has $A_{2}<0$ and needs to be eliminated. As a
consequence, the only physical Stage 2-firehose solution available when Stage
1 is firehose unstable, is the upper branch.

Replacing then the firehose unstable Stage 1 solutions of Figure 8(d), by the
corresponding Stage 2-firehose solutions, gives Figure 8(g).

We finally need to apply the evolutionarity test to the solutions of Figure
8(g). This is done on Figure 8(h) where the downstream Alfvén Mach number
$M_{A2x}$ has been computed for the solution plotted on Figure 8(g), with the
forbidden zones shaded. Note that for such an analysis, the horizontal scale
has to be $\sigma_{x}=\sigma\cos^{2}\theta_{1}$. Then and only then has the
evolutionary analysis some branches passing exactly through the point $(1,1)$,
like on Figure 8(h).

The Stage 2-firehose branch visible in green on Figure 8(g) at small $\sigma$
passes the evolutionarity test since it has $\sigma_{x}<1$ and
$M_{A2x}=+\infty$. The analysis shows that some stable Stage 1 solutions do
not pass the evolutionarity test. As a result, Figure 9 for
$\theta_{1}=0.3\pi/2$ displays a gap without solution for $\sigma\in[1,1.17]$,
that does not show on Figure 8(g).

Figure 9: Result of the filtering process elaborated in this work for various
angles $\theta_{1}\in[0,\pi/2]$. All of them but $\theta_{1}=0.3\pi/2$, are
the ones considered in Figure 8 of Bret & Narayan (2022b). The gray dashed
lines picture the MHD result, evolutionary-filtered. The green solutions
pertain to Stage 2-firehose, and the red ones to Stage 2-mirror.

The end result of such a filtering process is eventually shown on Figure 9 for
several angles $\theta_{1}$ between 0 and $\pi/2$. For comparison, all of them
but $\theta_{1}=0.3\pi/2$, are the ones considered in Figure 8 of Bret &
Narayan (2022b). The gray dashed line pictures the MHD result, evolutionary-
filtered.

Figure 9 eventually interpolates between the parallel case treated in Bret &
Narayan (2018), and the perpendicular one treated in Bret & Narayan (2019).
While Section 5 showed that some filtering is needed to get to the end result
for the parallel case, no filtering at all is required for the perpendicular
case. Figure 9 for $\theta_{1}=0.9\pi/2$ gives the result already found in
Bret & Narayan (2019), without any filtering but the $A_{2}<0$ for the mirror
solutions. The main reason for this is that for the perpendicular case, there
is but one Stage 1 solution, which, when stable, fulfills the evolutionarity
criterion.

##  7 Conclusion

Applying the MHD formalism to analyse collisionless shocks may be problematic
in the presence of an ambient magnetic field, capable of stabilizing pressure
anisotropies. In a series of recent papers, we developed a method allowing to
determine the downstream anisotropy of a collisionless shock (Bret & Narayan,
2018, 2019, 2022a, 2022b; Bret, 2023a, b). The anisotropy can then be included
in the MHD conservation equations for anisotropic pressures, and the modified
density jump derived. The case of a parallel shock has been successfully
tested by PIC simulations in Haggerty et al. (2022), confirming that for a
strong enough field, the density jump can go from 4, the expected MHD value,
to only 2, the anisotropy corrected one.

Once the density jump is found, all the other jumps like pressure,
temperature, entropy, etc., can be straightforwardly derived (Bret, 2021).

As can be seen on Figure 9, our results differ from the isotropic MHD ones in
2 ways,

  * •

The ranges of $\sigma$ without solutions differ.

  * •

For $\sigma$ values with a solution, our density jump is usually, though not
always, lower than the MHD one.

Overall, our results and the MHD ones bear a “family resemblance”, the largest
discrepancy being found for the parallel case $\theta_{1}=0$. The predicted
large reduction of the density jump could have important consequences for
particle acceleration, since their index scales like $(r-1)^{-1}$ (Blandford &
Ostriker, 1978; Axford et al., 1977; Bell, 1978a, b; Haggerty & Caprioli,
2020; Caprioli et al., 2020).

Besides the strong sonic shock assumption of the present work, namely
$P_{1}=0$, its main limitation may lie in the composition of the plasma, here
a pair plasma. As stated in the introduction, such an assumption allows to
consider only one parallel and one perpendicular temperature. Yet, our
formalism being eventually macroscopic, we expect our conclusions to hold for
electron/ion plasmas as well. Such a conjecture is currently being tested
through PIC simulations of such plasmas (Shalaby, 2024).

## Acknowledgments

Thanks are due to Alexander Velikovich and Federico Fraschetti for valuable
inputs.

## Funding

A.B. acknowledges support by the Ministerio de Economía y Competitividad of
Spain (Grant No. PID2021-125550OB-I00). R.N. acknowledges support from the NSF
Grant No. AST-1816420. C.C.H. was partially supported by NSF FDSS grant
AGS-1936393 as well as NASA grants 80NSSC20K1273 and 80NSSC23K0099.

Simulations were performed on computational resources provided by TACC’s
Stampede2 and Purdue’s ANVIL through ACCESS (formally XSEDE) allocation TG-
AST180008. Some of the work was supported by the Geospace Environment Modeling
Focus Group “Particle Heating and Thermalization in Collisionless Shocks in
the Magnetospheric MultiScale Mission (MMS) Era”.

R.N. and A.B. thank the Black Hole Initiative at Harvard University for
support. The BHI is funded by grants from the John Templeton Foundation and
the Gordon and Betty Moore Foundation.

## Declaration of Interests

The authors report no conflict of interest.

##  Appendix A Isotropic MHD conservation equations for an oblique shock

The MHD conservation equations for an oblique shock and a fluid of adiabatic
index $\gamma=5/3$ read (Kulsrud, 2005),

$\displaystyle n_{2}v_{2}\cos\xi_{2}$ $\displaystyle=$ $\displaystyle
n_{1}v_{1},$ (18) $\displaystyle B_{2}\cos\theta_{2}$ $\displaystyle=$
$\displaystyle B_{1}\cos\theta_{1},$ (19) $\displaystyle
B_{2}v_{2}\sin\theta_{2}\cos\xi_{2}-B_{2}v_{2}\cos\theta_{2}\sin\xi_{2}$
$\displaystyle=$ $\displaystyle B_{1}v_{1}\sin\theta_{1},$ (20)
$\displaystyle\frac{B_{2}^{2}\sin^{2}\theta_{2}}{8\pi}+n_{2}k_{B}T_{2}+mn_{2}v_{2}^{2}\cos^{2}\xi_{2}$
$\displaystyle=$
$\displaystyle\frac{B_{1}^{2}\sin^{2}\theta_{1}}{8\pi}+n_{2}k_{B}T_{1}+mn_{1}v_{1}^{2},$
(21)
$\displaystyle-\frac{B_{2}^{2}}{4\pi}\sin\theta_{2}\cos\theta_{2}+mn_{2}v_{2}^{2}\sin\xi_{2}\cos\xi_{2}$
$\displaystyle=$
$\displaystyle-\frac{B_{1}^{2}\sin\theta_{1}\cos\theta_{1}}{4\pi},$ (22)
$\displaystyle\mathcal{A}v_{2}\sin\xi_{2}+\mathcal{C}$ $\displaystyle=$
$\displaystyle
mn_{1}v_{1}\left(\frac{5k_{B}T_{1}}{2m}+\frac{B_{1}^{2}\sin^{2}\theta_{1}}{4\pi
mn_{1}}+\frac{v_{1}^{2}}{2}\right),$ (23)

where $m$ is the mass of the particles, $k_{B}$ the Boltzmann constant, and

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-\frac{B_{2}^{2}}{4\pi}\sin\theta_{2}\cos\theta_{2},$
$\displaystyle\mathcal{C}$ $\displaystyle=$ $\displaystyle
mn_{2}v_{2}\cos\xi_{2}\left(5\frac{k_{B}T_{2}}{2m}+\frac{B_{2}^{2}\sin^{2}\theta_{2}}{4\pi
mn_{2}}+\frac{v_{2}^{2}}{2}\right).$

After some algebra, $\overline{T}_{2}\equiv\tan\theta_{2}$ is found solution
of101010This quantity is written here with a bar to avoid confusion with the
downstream temperature $T_{2}$ in Eqs. (21,23). Such a confusion is excluded
in the rest of the paper since the downstream temperature splits into 2
different quantities, namely $T_{\parallel 2}$ and $T_{\perp 2}$.

$\sum_{k=0}^{4}a_{k}\overline{T}_{2}^{k}=0,$ (24)

with,

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 2M_{A1}^{2}\left(\sin
2\theta_{1}-2M_{A1}^{2}\tan\theta_{1}\right)^{2},$ $\displaystyle a_{1}$
$\displaystyle=$
$\displaystyle-\frac{1}{8}M_{A1}^{2}\tan\theta_{1}\left[-4M_{A1}^{2}\left(31\cos
2\theta_{1}+21\right)+80M_{A1}^{4}+4\cos^{2}\theta_{1}\left(21\cos
2\theta_{1}+11\right)\right],$ $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M_{A1}^{2}\cos^{2}\theta_{1}\left(15\cos
2\theta_{1}+1\right)-M_{A1}^{4}\left(7\cos 2\theta_{1}+3\right)+2M_{A1}^{6},$
$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}\sin
2\theta_{1}M_{A1}^{2}\left(-2M_{A1}^{2}+\cos 2\theta_{1}+1\right),$
$\displaystyle a_{4}$ $\displaystyle=$ $\displaystyle
3\cos^{4}\theta_{1}M_{A1}^{2}.$ (25)

##  Appendix B Anisotropic MHD conservation equations for an oblique shock

The conservation equations for anisotropic temperatures were derived in Hudson
(1970); Erkaev et al. (2000). They have been re-derived in Bret & Narayan
(2022a) with the present notations. They are formally valid even for
anisotropic upstream temperatures, with $T_{\parallel 1}\neq T_{\perp 1}$.
Writing them for $T_{\parallel 1}=T_{\perp 1}\equiv T_{1}$, they read,

$\displaystyle n_{2}v_{2}\cos\xi_{2}$ $\displaystyle=$ $\displaystyle
n_{1}v_{1},$ (26) $\displaystyle B_{2}\cos\theta_{2}$ $\displaystyle=$
$\displaystyle B_{1}\cos\theta_{1},$ (27) $\displaystyle
B_{2}v_{2}\sin\theta_{2}\cos\xi_{2}-B_{2}v_{2}\cos\theta_{2}\sin\xi_{2}$
$\displaystyle=$ $\displaystyle B_{1}v_{1}\sin\theta_{1},$ (28)
$\displaystyle\frac{B_{2}^{2}\sin^{2}\theta_{2}}{8\pi}+n_{2}k_{B}(T_{\parallel
2}\cos^{2}\theta_{2}+T_{\perp
2}\sin^{2}\theta_{2})+mn_{2}v_{2}^{2}\cos^{2}\xi_{2}$ $\displaystyle=$
$\displaystyle\frac{B_{1}^{2}\sin^{2}\theta_{1}}{8\pi}+n_{2}k_{B}T_{1}+mn_{1}v_{1}^{2},$
(29) $\displaystyle\mathcal{A}+mn_{2}v_{2}^{2}\sin\xi_{2}\cos\xi_{2}$
$\displaystyle=$
$\displaystyle-\frac{B_{1}^{2}\sin\theta_{1}\cos\theta_{1}}{4\pi},$ (30)
$\displaystyle\mathcal{A}v_{2}\sin\xi_{2}+\mathcal{B}+\mathcal{C}$
$\displaystyle=$ $\displaystyle
mn_{1}v_{1}\left(\frac{5k_{B}T_{1}}{2m}+\frac{B_{1}^{2}\sin^{2}\theta_{1}}{4\pi
mn_{1}}+\frac{v_{1}^{2}}{2}\right),$ (31)

where

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\sin\theta_{2}\cos\theta_{2}n_{2}k_{B}\left(T_{\parallel
2}-T_{\perp 2}\right)-\frac{B_{2}^{2}}{4\pi}\sin\theta_{2}\cos\theta_{2},$
$\displaystyle\mathcal{B}$ $\displaystyle=$ $\displaystyle
v_{2}\cos^{2}\theta_{2}\cos\xi_{2}n_{2}k_{B}(T_{\parallel 2}-T_{\perp 2}),$
$\displaystyle\mathcal{C}$ $\displaystyle=$ $\displaystyle
mn_{2}v_{2}\cos\xi_{2}\left(\frac{k_{B}}{2m}(T_{\parallel 2}+4T_{\perp
2})+\frac{B_{2}^{2}\sin^{2}\theta_{2}}{4\pi
mn_{2}}+\frac{v_{2}^{2}}{2}\right).$

It can be checked that setting $T_{\parallel 2}=T_{\perp 2}=T_{2}$ gives back
the MHD Eqs. (18-23).

##  Appendix C Main quantities for Stages 1 & 2

Implementing the algorithm described in Figure 7 requires computing the
density ratio $r$, the anisotropy $A_{2}$ and the downstream Alfvén Mach
number $M_{A2x}$, for Stage 1, Stage 2-firehose and Stage 2-mirror. The
results are presented below.

###  C.1 Stage 1

Solving the system of equations (26-31) with prescription (2.1) allows to
derive a polynomial for the quantify $T_{2}=\tan\theta_{2}$ defined in (7). It
has been derived in Paper I111111See Eqs. (4.2-4.5) in Paper I.. Using now the
expression (10) for the Alfvén velocity, the Alfvén Mach number for Stage 1
reads,

$M_{A2x}^{2}=\frac{4\left(T_{2}^{4}+2\right)\sec^{2}\theta_{1}M_{A1}^{2}}{\left(T_{2}^{2}-2\right)\sec^{2}\theta_{1}\left(4M_{A1}^{2}(r-1)-r\cos
2\theta_{1}+r\right)+2r\left(T_{2}^{4}+2T_{2}^{2}+4\right)}.$ (32)

From Eqs. (2.1), the anisotropy of Stage 1 is simply,

$A_{2}=\frac{1}{2}\tan^{2}\theta_{2}.$ (33)

The density ratio $r$ is the same as in Paper I, namely,

$r=\frac{4\mathcal{M}_{A1}^{2}T_{2}^{3}(1+T_{2}^{2})}{\sum_{k=0}^{5}b_{k}T2^{k}},$
(34)

where,

$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle
8\mathcal{M}_{A1}^{2}\tan\theta_{1}-4\sin 2\theta_{1},$ $\displaystyle b_{1}$
$\displaystyle=$ $\displaystyle-8\mathcal{M}_{A1}^{2}+6\cos 2\theta_{1}+2,$
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
b_{3}$ $\displaystyle=$ $\displaystyle 4\mathcal{M}_{A1}^{2}+\cos
2\theta_{1}+3,$ $\displaystyle b_{4}$ $\displaystyle=$ $\displaystyle
4\mathcal{M}_{A1}^{2}\tan\theta_{1}-2\sin 2\theta_{1},$ $\displaystyle b_{5}$
$\displaystyle=$ $\displaystyle 2\cos^{2}\theta_{1}.$ (35)

###  C.2 Stage 2 firehose

If Stage 1 has $A_{2}<1-2/\beta_{\parallel 2}$, then Stage 2-firehose is the
end state. Since the stability criterion differs from that used in Paper I by
the factor 2, the properties of Stage 2 firehose change with respect to Paper
I. A polynomial equation can still be derived for $T_{2}=\tan\theta_{2}$ as,

$\sum_{k=0}^{3}a_{k}T_{2}^{k}=0,$ (36)

with,

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle-8\left(\sin
2\theta_{1}-2M_{A1}^{2}\tan\theta_{1}\right)^{2}$ (37) $\displaystyle a_{1}$
$\displaystyle=$ $\displaystyle\left(-2M_{A1}^{2}+\cos
2\theta_{1}+1\right)\left(-20M_{A1}^{2}+9\cos
2\theta_{1}-1\right)\tan\theta_{1}$ (38) $\displaystyle a_{2}$
$\displaystyle=$ $\displaystyle 8\cos
2\theta_{1}M_{A1}^{2}-8\left(M_{A1}^{4}+M_{A1}^{2}\right)-\cos 4\theta_{1}+1$
(39) $\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\left(2M_{A1}^{2}-\cos 2\theta_{1}-1\right)\sin 2\theta_{1}.$
(40)

The density jump is then given by

$r=\frac{M_{A1}^{2}T_{2}}{M_{A1}^{2}\tan\theta_{1}-\sin\theta_{1}\cos\theta_{1}}.$
(41)

As for the Alfvén Mach number, we found in Section 4 that the Alfvén speed
vanishes on the firehose instability threshold. Therefore, Stage 2-firehose
has,

$M_{A2x}=+\infty.$ (42)

As a consequence, when the system eventually settles in Stage 2-firehose with
$\sigma_{x}<1$ (i.e, $M_{A1x}>1$), it is evolutionary.

The anisotropy of Stage 2-firehose is also modified with respect to Paper I,
due to the modified stability criterion. It now reads,

$A_{2}=1-\frac{2(T_{2}^{3}+T_{2})\cos^{2}\theta_{1}}{2M_{A1}^{2}(T_{2}-\tan\theta_{1})+T_{2}^{3}\cos^{2}\theta_{1}+T_{2}\sin^{2}\theta_{1}+\sin
2\theta_{1}}.$ (43)

###  C.3 Stage 2 mirror

Since the stability threshold for the mirror instability is the same as in
Paper I, the polynomial for $T_{2}$, the anisotropy $A_{2}$ and the density
ratio are also the same. The Alfvén Mach number reads here

$M_{A2x}^{2}=\frac{2}{3}\frac{\sec^{2}\theta_{1}}{r}M_{A1}^{2}.$ (44)

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$\displaystyle\varphi(\textbf{t})(J)=$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{t_{i}(J):i\in
S_{v}\cup S_{w}\\}\cup\\{\omega_{J,S_{v}\cup
S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{t})}(\textbf{t}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{z)}})\\}$
$\displaystyle\leq$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{t_{i}(J):i\in
S_{v}\cup
S_{w}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$
$\displaystyle\leq$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}\min\\{v_{i}(J):i\in
S_{v}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$
$\displaystyle\varphi(\textbf{t})(J)\leq$
$\displaystyle\max_{S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{w_{i}(J):i\in
S_{w}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$

Therefore $\varphi(\textbf{t})(J)=\varphi(\textbf{v})(J)$. ∎

###  D.11. Proof for OC proposition 5.20

###### Proof.

$\Rightarrow:$ Let us consider a partition
$\mathcal{N}_{1}\cup\mathcal{N}_{2}=\mathcal{N}$ of the set of voters. For any
voting profile t, let $\textbf{v}=\textbf{t}_{-(\mathcal{N}_{2})}$ and
$\textbf{w}=\textbf{t}_{-(\mathcal{N}_{1})}$. We define
$k_{1}=\\#(\textbf{v}(J)\cup\mathcal{F}(\textbf{v}))$ and
$k_{2}=\\#(\textbf{w}(J)\cup\mathcal{F}(\textbf{w}))$.

We know that the $g_{J}(k_{1})$-th and $g_{J}(k_{2})$-th smallest members the
voting pools $(\textbf{v}(J)\cup\mathcal{F}(\textbf{v}))$ and
$(\textbf{w}(J)\cup\mathcal{F}(\textbf{w}))$ respectively have the same value.
$g_{J}$ has no way on ensuring that any other element of these voting pools
have the same value. As such when we merge the voting pools we must select one
of those 2 elements. It follows that
$g_{J}(k_{1}+k_{2})\in\\{g_{J}(k_{1})+g_{J}(k_{2})-1,g_{J}(k_{1})+g_{J}(k_{2})\\}$.

$\Leftarrow:$

Suppose that for all $p_{1},p_{2}$ we have
$g_{J}(p_{1}+p_{2})\in\\{g_{J}(p_{1})+g_{J}(p_{2})-1,g_{J}(p_{1})+g_{J}(p_{2})\\}$.
Then it is immediate that when we merge two voter pools that provided the same
outcome, we obtain said outcome.

∎

###  D.12. Proof for IC theorem 5.22

###### Proof.

$\Rightarrow:$

  1. (1)

If we have
$\omega_{J,S}^{T}(\textbf{v}_{-T})=\omega_{J,S}^{T^{\prime}}(\textbf{w}_{-T^{\prime}})$
then for an appropriate choice of $\textbf{z}(J)$ values we get
$\alpha=\varphi(\textbf{v})(J)=\omega_{J,S}^{T}(\textbf{v}_{-T})=\omega_{J,S}^{T^{\prime}}(\textbf{w}_{-T^{\prime}})=\varphi(\textbf{w})(J)$
that does not belong to $\textbf{z}(J)$ unless
$\alpha\in\\{\inf\mathcal{A},\sup\mathcal{A}\\}$. Therefore by IC we get
$\varphi(\textbf{z})(J)=\alpha$, it follows that $\omega_{J,S\cup
S^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})=\alpha$.

  2. (2)

Suppose that we have
$\max(\omega_{J,\emptyset}^{T}(\textbf{v}_{-T}),\omega_{J,\emptyset}^{T^{\prime}}(\textbf{w}_{-T^{\prime}}))\leq\alpha\leq\min(\omega_{J,S}^{T}(\textbf{v}_{-T}),\omega_{J,S}^{T^{\prime}}(\textbf{w}_{-T^{\prime}}))$.
Any changes to the grades for $J$ does not affect the phantom-mappings. For
all $i\in S\cup S^{\prime}$ we take $t_{i}(J)=\alpha$ and for all others we
take $t_{i}(J)=\beta\leq\alpha$. We have
$\varphi(\textbf{v})(J)=\varphi(\textbf{w})(J)=\alpha$, therefore by (IC) we
have that $\varphi(\textbf{z})(J)=\alpha$ Suppose we could take
$\beta<\alpha$. For any $U\not\subseteq S\cup S^{\prime}$ we have that
$\min\\{z_{i}(J):i\in U\\}=\beta$, as such:

$\displaystyle\alpha=\varphi(\textbf{z})(J)=$ $\displaystyle\max_{U\subseteq
T\cup T^{\prime}}\min\\{z_{i}(J):i\in U\\}\cup\\{\omega_{J,U}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\\}$ $\displaystyle=$
$\displaystyle\max\\{\min\\{\alpha,\omega_{S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\\},\omega_{J,\emptyset}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\\}$

Since $\omega_{J,\emptyset}^{T\cup T^{\prime}}$ was not selected we therefore
have that $\omega_{J,\emptyset}^{T\cup T^{\prime}}\leq\alpha$. As such:

$\varphi(\textbf{v})(J)=\min\\{\alpha,\omega_{S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\\}$

Therefore we have $\alpha\leq\omega_{S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})$.

If it was impossible to take $\beta<\alpha$, then we have
$\alpha=\inf\mathcal{A}$. Since we take the $\max$ is the characterization we
have $\omega_{J,\emptyset}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup
T^{\prime})})\leq\alpha$. Then by the lemma A.1, we have
$\alpha\leq\omega_{S\cup S^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup
T^{\prime})})$.

  3. (3)

Suppose that we have
$\max(\omega_{J,S}^{T}(\textbf{v}_{-T}),\omega_{J,S^{\prime}}^{T^{\prime}}(\textbf{w}_{-T^{\prime}}))\leq\alpha\leq\min(\omega_{J,T}^{T}(\textbf{v}_{-T}),\omega_{J,T}^{T^{\prime}}(\textbf{w}_{-T^{\prime}}))$.
Any changes to the grades for $J$ does not affect the phantom-mappings. For
all $i\not\in S\cup S^{\prime}$ we take $z_{i}(J)=\alpha$ and for all $i\in
S\cup S^{\prime}$ we take $z_{i}(J)=\beta\geq\alpha$. We have
$\varphi(\textbf{v})(J)=\varphi(\textbf{w})(J)=\alpha$. Therefore by IC we get
that $\varphi(\textbf{z})(J)=\alpha$. Suppose that we can find $\beta>\alpha$.
If we had $\omega_{S\cup S^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup
T^{\prime})})>\alpha$ then we would have $\min\\{z_{i}(J):i\in S\cup
S^{\prime}\\}\cup\\{\omega_{J,S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\\}>\varphi(\textbf{z})(J)$. This
is absurd, as such $\omega_{J,S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\leq\alpha$. If $\omega_{J,T\cup
T^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})<\alpha$ then
for all $U$ we have $\omega_{J,U}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup
T^{\prime})})<\alpha$ therefore we would have $\varphi(\textbf{v})(J)<\alpha$.
This is absurd, as such $\alpha\leq\omega_{J,S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})$.

If we could not find $\beta>\alpha$ then we have that
$\alpha=\sup\mathcal{A}$. According to the lemma A.1, if $\omega_{J,S\cup
S^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})>\alpha$ then
$\omega_{J,\emptyset}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup
T^{\prime})})>\alpha$. We therefore contradict
$\varphi(\textbf{z})(J)=\alpha$. Similarly $\omega_{J,T\cup T^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})<\alpha$ implies that
$\varphi(\textbf{z})(J)<\alpha$. As such $\omega_{J,S\cup S^{\prime}}^{T\cup
T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})\leq\alpha\leq\omega_{J,T\cup
T^{\prime}}^{T\cup T^{\prime}}(\textbf{z}_{-(T\cup T^{\prime})})$.

$\Leftarrow:$ Let us suppose that we have $\varphi$ that verifies the
characterization. Suppose that
$\varphi(\textbf{v})(J)=\varphi(\textbf{w})(J)$:

$\displaystyle\varphi(\textbf{z})(J)=$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{z_{i}(J):i\in
S_{v}\cup S_{w}\\}\cup\\{\omega_{J,S_{v}\cup
S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{z})}(\textbf{z}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{z)}})\\}$
$\displaystyle=$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{v_{i}(J):i\in
S_{v}\\}\cup\\{w_{i}(J):i\cup S_{w}\\}\cup\\{\omega_{J,S_{v}\cup
S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{z})}(\textbf{z}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{z)}})\\}$
$\displaystyle\geq$ $\displaystyle\min\\{v_{i}(J):i\in
S\\}\cup\\{w_{i}(J):i\cup S^{\prime}\\}\cup\\{\omega_{J,S\cup
S}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{z})}(\textbf{z}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{z)}})\\}$
$\displaystyle\geq$ $\displaystyle\min\\{v_{i}(J):i\in
S\\}\cup\\{w_{i}(J):i\cup
S^{\prime}\\}\cup\\{\min(\omega_{J,S}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S^{\prime}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$
$\displaystyle\geq$ $\displaystyle\varphi(\textbf{v}))(J)$

$\displaystyle\varphi(\textbf{z})(J)=$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{z_{i}(J):i\in
S_{v}\cup S_{w}\\}\cup\\{\omega_{J,S_{v}\cup
S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{z})}(\textbf{z}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{z)}})\\}$
$\displaystyle\leq$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v}),S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{z_{i}(J):i\in
S_{v}\cup
S_{w}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$
$\displaystyle\leq$
$\displaystyle\max_{S_{v}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}\min\\{v_{i}(J):i\in
S_{v}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$
$\displaystyle\varphi(\textbf{z})(J)\leq$
$\displaystyle\max_{S_{w}\subseteq\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}\min\\{w_{i}(J):i\in
S_{w}\\}\cup\\{\max(\omega_{J,S_{v}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{v})}),\omega_{J,S_{w}}^{\mathcal{D^{\prime\prime}}_{J}(\textbf{w})}(\textbf{w}_{-\mathcal{D^{\prime\prime}}_{J}(\textbf{w)}}))\\}$

Therefore $\varphi(\textbf{z})(J)=\varphi(\textbf{v})(J)$. ∎

##  Appendix E Section Ranking

###  E.1. Proof for determinism property 6.1

###### Proof.

Let $P_{J}^{\psi}:\mathcal{B}^{n}\rightarrow\mathcal{B}^{n}$ be function that
turns a multi-set $\mathcal{V}$ in an ordered-set defined as:

  1. (1)

$P_{J}^{\psi}:=[]$; $S=\mathcal{V}$;

  2. (2)

If $S=\emptyset$; Return $P_{J}^{\psi}$, END

  3. (3)

Else $\alpha:=\mu_{g(\\#S)}(S)$ $P_{J}^{\psi}:=(R_{J}^{\psi},\alpha)$;
$S=S-\alpha$;

  4. (4)

Go back to (2)

By definition of a multi-set we have that $P_{J}^{\psi}$ is well defined.

Let us now show that for we have:

$\forall\textbf{v},R_{J}^{\psi}(\textbf{v})=P_{J}^{\psi}(\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v})).$

Recall that $\psi$ is defined as
$\mu_{g(\\#\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v}))}(\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v}))$.

It is therefore sufficient to show that for any $i$ we have that if
$\textbf{w}=\textbf{v}[\forall J,v_{i}(J)=\emptyset]$ then if
$v_{i}(J)\in\mathcal{A}$ then
$\textbf{w}(J)\cup\mathcal{F}_{J}(\textbf{w})=(\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v}))-\\{v_{i}(J)\\}$
and if $v_{i}(J)\not\in\mathcal{A}$ we have
$\textbf{w}(J)\cup\mathcal{F}_{J}(\textbf{w})=(\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v}))-\\{f_{i,J}(J)\\}$.

  * •

If $v_{i}(J)\in\mathcal{A}$ then since $f_{i,J}(v_{i})[\forall
J,v_{i}(J)=\emptyset]=\emptyset)$ (by definition of the proxy functions) and
since all the other proxy functions do not depend on $v_{i}$ we have
$\mathcal{F}_{J}(\textbf{v})=\mathcal{F}_{J}(\textbf{w})$.
$\textbf{v}(J)=\textbf{w}(J)\cup\\{v_{i}(J)\\}$. Therefore we have the
equality.

  * •

If $f_{i,J}(v_{i}(J))=\emptyset$ then
$f_{i,J}(v_{i}(J))=f_{i,J}(v_{i}(J)[\forall J,v_{i}(J)=\emptyset])$. Since
none of the other proxy functions depend on $v_{i}$ we have
$\mathcal{F}_{J}(\textbf{v})=\mathcal{F}_{J}(\textbf{w})-\\{f_{i,J}(v_{i})\\}$.
We also have $\textbf{v}(J)=\textbf{w}(J)$.

  * •

If $f_{i,J}(v_{i}(J))\in\mathcal{B}$ then since none of the other proxy
function depend on $v_{i}$ we have
$\mathcal{F}_{J}(\textbf{v})=\mathcal{F}_{J}(\textbf{w})-\\{f_{i,J}(v_{i})\\}$.
We also have $\textbf{v}(J)=\textbf{w}(J)$.

We therefore have the desired equality. This ends our proof. ∎

###  E.2. Proof for total ranking 6.3

###### Proof.

Because $\psi$ is fair (F) we have that any 2 candidates that provide the same
voting pool obtain the same ranking of their elements. If 2 voting pools then
an ordering of their elements must also be different. The fact that the
lexicographic order is a total order provides the rest. ∎

###  E.3. Proof for SP in the new space 6.8

###### Proof.

Suppose that we have $(v_{i}(J))<_{\psi}R_{J}^{\psi}(\textbf{v})$ (with
$v_{i}(J)\in\mathcal{A}$). Let $w_{i}$ be such $w_{i}(J)\in\mathcal{A}$. Let
$\textbf{w}=\textbf{v}[v_{i}:=w_{i}]$.

We have $\textbf{w}(J)=(\textbf{v}(J)\cup\\{w_{i}(J)\\})-\\{v_{i}(J))$ and
$\mathcal{F}_{J}(\textbf{v})=\mathcal{F}_{J}(\textbf{w})$.

Let $k$ be the first value such that $R_{J}^{(}\textbf{v})(k)\neq
R_{J}(\textbf{w})(J)$.

This is therefore the first time the loop could not choose the same element in
the multi-set. As such since for all other voters the vote that represents
them in the two voting pools is the same this is also the first time we cannot
remove the same voter in both instances.

Since $R_{J}^{\psi}$ is defined iterative we can therefore assume $k=1$
without loss of generality. It follows that we are comparing
$\psi(\textbf{v})$ and $\psi(\textbf{w})$. Since $\psi$ is SP
$\psi(\textbf{v})\leq\psi(\textbf{w})$ therefore we have
$R_{J}^{\psi}(\textbf{v})<R_{J}^{\psi}(\textbf{v})$.

This concludes the proof. (The case $v_{i}(J)>R_{J}^{\psi}(\textbf{v})$ is
symmetrical).

∎

###  E.4. Proof for the duplicate proposition 6.5

###### Proof.

Let $\psi$ be a phantom-proxy mechanism that verifies (OC). For any v and w
such that w is obtained by duplicating v $k$ times ($k\geq 1$). Suppose that
our proposition is false. That is to say that there are two voters $i$ and $j$
such that for v the vote (or proxy-vote) of $i$ was selected by $R$ before the
vote of $j$ and for w the first time we selected a proxy for $j$ was before
the first time we selected a proxy for $i$. Wlog we will assume that $j$ is
the first voter whose duplicate got selected early and $i$ the first voter
whose first duplicate got selected late.

Wlog we will assume that all votes are different. (If not we are careful in
our order of selection of duplicate when we have the choice).

Let $\tilde{v_{i}(J)}$ (resp $\tilde{v_{j}(J)})$) be the vote that represents
$i$ (resp $j$) in v. Let $n$ be the number of elements left in the voting pool
when $g$ selects $\tilde{v_{i}(J)}$. Let $g(n)=r$.

  * •

If $\tilde{v_{i}(J)}$ was selected as an element that is lower than
$\tilde{v_{j}(J)}$. Let $T$ be the remaining voting pool when
$\tilde{v_{i}(J)}$ was selected by v. Let $S\subset T$ be the remaining votes
in the voting pool for v when $\tilde{v_{i}(J)}$ was selected that were
considered lower than $\tilde{v_{i}(J)}$. Let us now consider the moment a
duplicate of $v_{j}(J)$ was selected. Any duplicates of an element of $S$ was
never selected. There are $rk$ such duplicates. All of these elements are
considered smaller than $\tilde{v_{j}(J)}$. There are also $k^{\prime}\geq 0$
elements that are not a duplicate of an element of $T$ remaining in the voting
pool of w. These elements are also considered smaller than $\tilde{v_{j}(J)}$.
We therefore have $g(nk+k^{\prime})=rk+k^{\prime}+1$. By (OC) we have
$g(nk+k^{\prime})\leq kg(n)+k^{\prime}=kr+k^{\prime}$. We have reached a
contradiction.

  * •

If $\tilde{v_{i}(J)}$ was selected as an element that is greater than the
$\tilde{v_{j}(J)}$. We have that $g(n)=g(n-1)+1=r$. Let $T$ be the remaining
voting pool when $\tilde{v_{i}(J)}$ was selected by v. Let $S\subset T$ be the
remaining votes in the voting pool for v when $\tilde{v_{i}(J)}$ was selected
that were considered greater than $\tilde{v_{i}(J)}$. Let us now consider the
moment a duplicate of $v_{j}(J)$ was selected. Any duplicates of an element of
$S$ was never selected. There are $(n-r+1)k$ such duplicates. All of these
elements are considered greater than $\tilde{v_{j}(J)}$. There are also
$k^{\prime}\geq 0$ elements that are not a duplicate of an element of $T$
remaining in the voting pool of w. These elements are also considered greater
than $\tilde{v_{j}(J)}$. We therefore have $g(nk+k^{\prime})=k(r-1)$. By (OC)
we have $g(nk+k^{\prime})\geq g(kn)+g(k^{\prime})\geq
kg(n)-(k-1)+g(k^{\prime})\geq rk-(k-1)=r(k-1)+1$. We have reached a
contradiction.

This concludes our proof. ∎

###  E.5. Proof for the corollary 6.6

###### Proof.

Since $\psi$ is fair we have the same $g$ for $J$ and $I$. As such is we order
our votes so the $k$-th smallest vote for $I$ correspond to the same voter as
the $k$-th smallest vote for $J$. For all voters $i$, we have that each time a
duplicate of $i$ is chosen in the voting pool for $I$, a duplicate for $i$ is
selected for $J$. It follows that the order in which we discover a duplicate
belonging to a new voter for the first time determines the ranking of $I$
compared to $J$. The property 6.5 for (OC) phantom-proxy mechanisms therefore
concludes the proof. ∎

##  Appendix F Pareto optimality

###### Definition F.1 (Pareto Optimal :OP).

A voting method
$\varphi:\mathcal{O}^{\mathcal{M}\times\mathcal{N}}\rightarrow{(\mathcal{B}\cup\\{\emptyset\\})}^{\mathcal{M}}$
is Pareto Efficient if there does not exist an outcome $\alpha\in\mathcal{B}$
such that for at least one voter $i$ that gave a grade $v_{i}(J)$ we have
$\alpha$ is closer to $v_{i}(J)$ than $\varphi(\textbf{v})(J)$ and for no
voters $j\in\mathcal{D^{\prime\prime}}_{J}$, $\alpha$ is further away from
$v_{j}(J)$ than $\varphi(\textbf{v})(J)$.

###### Theorem F.2 (Pareto Optimal characterization).

A voting method
$\varphi:\mathcal{O}^{\mathcal{M}\times\mathcal{N}}\rightarrow{(\mathcal{B}\cup\\{\emptyset\\})}^{\mathcal{M}}$
is Pareto Optimal in regards to candidates iff it is unanimous.

###### Proof.

$\Rightarrow:$Suppose that we are Pareto Optimal in regards to candidates than
if all voters gave the same grade $\alpha$ by Pareto Optimality the outcome
must be $\alpha$. The method is therefore unanimous.

$\Leftarrow:$Suppose that we are unanimous then let $i$ be the voter that gave
$J$ the smallest grade $v_{i}(J)=\alpha$ and $j$ the voter that gave $J$ the
largest grade $v_{j}(J)=\beta$. Let $S$ be such that
$\omega_{J,S}^{\mathcal{D^{\prime\prime}}_{J}}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}})$
is greater or equal to $v_{i}(J)$.

$\min\\{v_{k}(J):k\in
S\\}\cup\\{\omega_{J,S}^{\mathcal{D^{\prime\prime}}_{J}}(\textbf{v}_{-\mathcal{D^{\prime\prime}}_{J}})\\}\geq
v_{i}(J)$

Therefore we have that $\varphi(\textbf{v})(J)\geq v_{i}(J)$. When
$S\neq\emptyset$, we have that

$\min\\{v_{k}(J):k\in S\\}\leq v_{j}(J)$

and $\omega_{J,\emptyset}^{\mathcal{D^{\prime\prime}}_{J}}\leq v_{j}(J)$
therefore $\varphi(\textbf{v})(J)\leq v_{i}(J)$. As such any alternative
outcome will result in $i$ or $j$ getting further away from the outcome. We
are Pareto optimal ∎

##  Appendix G Reinforcing absentees

Let us now suggest a small modification that reinforces our concept of what
absentee votes represent.

Ranking by using $\psi$ is considered fair because all voters are represented
at most once in the voting pool and $\mu_{g(\\#\mathcal{V}(J))}$ does not
distinguish between elements of the voting pool. As such the mechanism is
still considered fair if an element is added to the voting pool that
represents a voter that was not yet represented in the voting pool.

The spirit of the SC rule is that if voter $i\in\mathcal{N}$ abstained then he
might as well have voted for the outcome $\alpha$. When considering SI,
phantom-proxy methods $\psi$ functions, if a voter abstained then that voter
does not have a proxy vote. As such we remain fair in our model if we add an
element worth $\alpha$ representing $i$ to the voting pool. This late addition
to the voting pool represents voter $i$ consenting with the outcome. We call
these votes the absentee votes. When $\mathcal{A}=\mathcal{B}$, we can
represent this when ordering our candidates by using the S-grading range
$P_{J}^{\psi}$ instead of the grading range to $R_{J}^{\psi}$ where
$P_{J}^{\psi}$ is defined as:

  1. (1)

$P_{J}^{\psi}:=[]$; $S=\emptyset$;

  2. (2)

$\textbf{w}:=\textbf{v}[v_{i}(J):=\psi(\textbf{v}):\forall i,v_{i}(J)=\circ]$;

  3. (3)

If $\psi(\textbf{w}[\forall J,\forall i\in
S,w_{i}(J)=\emptyset])(J)=\emptyset$; Return $P_{J}^{\psi}$, END

  4. (4)

Else $\alpha:=\psi(\textbf{w}[\forall J,\forall i\in
S,w_{i}(J)=\emptyset])(J)$ $P_{J}^{\psi}:=(P_{J}^{\psi},\alpha)$;

  5. (5)

Find $i\in\mathcal{N}-S$ such that $w_{i}(J)=\alpha$ or
$f_{i,J}(w_{i})=\alpha$; $S:=S\cup\\{i\\}.$

  6. (6)

Go back to (3)

###### Proposition G.1.

If $\forall i,v_{i}(J)\neq\circ$ then
$P_{J}^{\psi}(\textbf{v})=R_{J}^{\psi}(\textbf{v})$.

As such the S-grading range $P_{J}^{\psi}$ is an extension of the grading
range $R_{J}^{\psi}$ that strengthens the meaning of an absentee vote.

###### Proposition G.2.

If $R_{J}^{\psi}(\textbf{v})<\psi(\textbf{v})(J)$ then we have
$R_{J}^{\psi}(\textbf{v})<P_{J}^{\psi}(\textbf{v})<\psi(\textbf{v})(J)$ and if
$R_{J}^{\psi}(\textbf{v})>\psi(\textbf{v})(J)$ then we have
$R_{J}^{\psi}(\textbf{v})>P_{J}^{\psi}(\textbf{v})>\psi(\textbf{v})(J)$

As such we have that the absentee voter is strengthening the outcome. The
previous method has be summarized for a finite number of candidates in the
algorithm 1.

Data: v and $\psi$

Result: A ranking of the candidates

for  _$J\in\mathcal{M}$_ do

$\mathcal{V}(J)\leftarrow\textbf{v}(J)\cup\mathcal{F}_{J}(\textbf{v})$;

if  _We wish to verify the reinforced absentee property_ then

for  _$i\in\\{i:v_{i}(J)=\circ\\}$_ do

$\mathcal{V}(J)\leftarrow\mathcal{V}(J)\cup\psi(\textbf{v})(J)$;

end for

end if

if  _$\mathcal{V}(J)=\emptyset$_ then

REMOVE $J$ from the ranking process.

end if

end for

$k\leftarrow\Pi_{J\in\mathcal{M}}\\#\mathcal{V}(J)$;

for  _$J\in\mathcal{M}$_ do

$\mathcal{V}(J)\leftarrow$ $\dfrac{k}{\\#\mathcal{V}(J)}\mathcal{V}(J)$;

$R_{J}\leftarrow[]$;

while  _$\mathcal{V}(J)\neq\emptyset$_ do

$R_{J}\leftarrow(R_{J},\mu_{g(\\#\mathcal{V}(J))}(\mathcal{V}(J))$;

$\mathcal{V}(J)\leftarrow\mathcal{V}(J)-\mu_{g(\\#\mathcal{V}(J))(\mathcal{V}(J))}$;

end while

end for

Rank the candidates according to the $R_{J}$ ranking.

ALGORITHM 1 The Ranking algorithm

##  Appendix H end