11institutetext: Universite Claude Bernard Lyon 1, CRAL UMR5574, ENS de Lyon,
CNRS, Villeurbanne, F-69622, France
11email<EMAIL_ADDRESS>22institutetext: Monash Centre for
Astrophysics (MoCA) and School of Physics and Astronomy, Monash University,
Vic. 3800, Australia

# Aeolian erosion in protoplanetary discs:
How impactful it is on dust evolution?

Stéphane Michoulier 11 Jean-François Gonzalez 11 Evgeni Grishin 22 Clément
Petetin 11

(Received 13 November 2023; accepted 26 February 2024)

###### Abstract

Context.  Many barriers prevent dust to form planetesimals via coagulation in
protoplanetary discs, such as bouncing, collisional fragmentation or aeolian
erosion. Modelling dust and the different phenomena that can alter its
evolution is therefore needed. Multiple solutions have been proposed, but
still need to be confirmed.

Aims.  In this paper, we explore the role aeolian erosion plays in the
evolution of dust.

Methods.  We use a monodisperse model to account for dust growth and
fragmentation, implemented in a 1D model to compute the evolution of single
grains and a 3D SPH code to compute the global evolution of dust and gas. We
test the erosion model in our code and ensured it matches previous results.

Results.  With a model of disc reproducing observations, we show with both 1D
and 3D studies that erosion is not significant during the evolution of dust
when we take fragmentation into consideration. With a low-viscosity disc,
fragmentation is less of a problem, but grain growth is also less important,
preventing the formation of large objects anyway. In dust traps, close to the
star, erosion is also not impactful, even when fragmentation is turned off.

Conclusions.  We show in this paper that aeolian erosion is negligible when
radial drift, fragmentation and dust traps are taken into account and does not
alter the dust evolution in the disc. However, it can have an impact on later
stages, i.e. when the streaming instability forms large clumps close to the
star, or when planetesimals are captured.

###### Key Words.:

methods: numerical – planets and satellites: formation – protoplanetary discs

##  1 Introduction

Protoplanetary discs consist of gas and dust that orbit young stars and
provide the necessary material for the agglomeration and growth of
planetesimals, the building blocks of planets. The dynamical and physical
processes occurring within these discs play a crucial role in shaping the
characteristics and composition of planetary systems. Among these processes,
aeolian erosion of large dust particles might influence the dynamics and
evolution of the dust in the inner regions of protoplanetary discs (Blum &
Wurm, 2000; Wurm et al., 2001).

Aeolian erosion is a process where dust particles are ejected from a larger
object due to the combined action of gas drag and turbulent motions within the
disc. It has been studied by Blum & Wurm (2000); Wurm et al. (2001); Paraskov
et al. (2006), and more recently by Rozner et al. (2020) and Grishin et al.
(2020) (hereafter R20 and G20). They showed that large aggregates can be
eroded in a short time, typically ranging from a few years to a few thousand
years, with sizes going from several hundred metres down to a couple of
centimetres (; ). This process is therefore believed to destroy large boulders
very efficiently and impact the evolution of grains in the inner region of the
disc. Hence, erosion is another barrier to dust growth from small sizes to
kilometric objects.

In addition to aeolian erosion, fragmentation of dust particles is a process
that can destroy grains in the inner regions of protoplanetary discs. When
dust particles collide at high velocities, they may experience catastrophic
disruptions, leading to the so-called fragmentation barrier (Weidenschilling &
Cuzzi, 1993; Dominik & Tielens, 1997; Blum & Wurm, 2008). Fragmentation
thresholds or material properties to model collisions are still not fully
understood, with many uncertainties remaining. Moreover, dust experiences
radial drift during its growth in the disc due to gas drag. Grains of a few
centimetres to metres drift very efficiently, due to marginal coupling to the
gas, and are accreted onto the star (Whipple, 1972), which defines the radial
drift barrier (Weidenschilling, 1977). In order to prevent radial drift and
the loss of material onto the star, several solutions have been proposed in
order to help the formation of planetesimals. Some rely on the capture of dust
in pressure maxima, others bypass the barriers to dust growth. For instance,
vortices have been explored (Barge & Sommeria, 1995; Meheut et al., 2012;
Loren-Aguilar & Bate, 2015) to trap dust and form clumps directly from
gravitational collapse. Snow lines (Kretke & Lin, 2007; Brauer et al., 2008;
Drażkowska et al., 2014; Vericel & Gonzalez, 2020) and self-induced dust traps
(Gonzalez et al., 2017; Vericel & Gonzalez, 2020; Vericel et al., 2021) form
local pressure maxima, stopping the radial drift and helping grain growth by
the increase of the local dust density. Other properties of dust have also
been investigated like grain porosity (Ormel et al., 2007; Suyama et al.,
2008; Okuzumi et al., 2009, 2012; Kataoka et al., 2013; Garcia & Gonzalez,
2020), allowing grains to grow faster and to larger sizes, while being less
sensitive to fragmentation. Additionally, other processes related to
instabilities have been under investigation these recent years, mostly with
the streaming instability (Youdin & Goodman, 2005; Youdin & Johansen, 2007;
Yang et al., 2017; Schäfer et al., 2017; Auffinger & Laibe, 2018; Li et al.,
2019) which allows dust to directly form boulders from pebbles, bypassing the
different barriers grains might undergo during their evolution. Grishin et al.
(2019) looked at a different solution where planetesimals can be captured in
protoplanetary discs early. This had also been shown to occur in earlier
stages, such as molecular clouds (Pfalzner & Bannister, 2019; Pfalzner et al.,
2021).

In this paper, we study the importance of aeolian erosion in relation to
fragmentation and its importance in disc inner regions. In a previous paper,
we compared the importance of fragmentation and rotational disruption of
porous grains (Michoulier & Gonzalez, 2022), a new barrier introduced by
Tatsuuma & Kataoka (2021). In this paper, however, we will not take into
account porosity as the current model for aeolian erosion of R20 and G20 does
not take into account the evolution of density. We will therefore limit
ourselves to the simpler compact grains formalism.

The paper is built as follows: we first describe our dust growth and
fragmentation model and introduce the model to take aeolian erosion into
account in Sect. 2. We then show the tests performed to make sure the
implementation of the erosion module is correctly done in Sect. 3. Then, we
discuss the results in Sect. 4 using the 1D code
Pamdeas111https://github.com/StephaneMichoulier/Pamdeas.git (Michoulier &
Gonzalez, 2022) and the 3D code Phantom by Price et al. (2018) to show the
unimportance of aeolian erosion with respect to fragmentation. Finally, we
discuss our results about erosion and the limitations of the codes in Sect. 5,
and end with a conclusion in Sect. 6.

##  2 Methods

To determine the importance of different barriers, we describe in the
following how growth, fragmentation and erosion are taken into account in
Pamdeas. When using Phantom (Price et al., 2018), all the details about the
implementation of growth and fragmentation are presented in Vericel et al.
(2021). The erosion model implemented in Pamdeas and Phantom is the same, and
the growth and fragmentation models are identical, but the implementation
differs slightly and are code-related.

###  2.1 Dust grain growth model

In order to model dust growth, we consider a locally uniform mass distribution
of grains where collisions occur exclusively between grains of identical mass.
The equation describing the size variation is given by Stepinski & Valageas
(1997):

$\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)_{\mathrm{grow}}=\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{s}}}v_{\mathrm{rel}}.$
(1)

where $\rho_{\mathrm{d}}$ is the dust local density, $\rho_{\mathrm{s}}$ the
grain intrinsic density and $v_{\mathrm{rel}}$, relative velocity during
collision, is:

$v_{\mathrm{rel}}=\sqrt{2^{3/2}\mathrm{Ro}\,\alpha}\,c_{\mathrm{g}}\frac{\sqrt{\mathrm{St}}}{1+\mathrm{St}}.$
(2)

$\alpha$ is the turbulent viscosity parameter (Shakura & Sunyaev, 1973) and
$c_{\mathrm{g}}$ is the gas sound speed. St is the Stokes number usually
defined as the dimensionless stopping time,
$\Omega_{\mathrm{K}}t_{\mathrm{s}}$, where $\Omega_{\mathrm{K}}$ is the
Keplerian frequency and $t_{\mathrm{s}}$ the drag stopping time. St quantifies
the coupling between gas and dust, which depends on the drag regimes and the
grain size (St $\propto s$ in the Epstein regime). $\mathrm{Ro}$, the Rossby
number, is considered to be a constant equal to 3 (Stepinski & Valageas,
1997). Full details of the model and its derivation are presented in Laibe et
al. (2008); Gonzalez et al. (2015, 2017) or Vericel et al. (2021). Naturally,
the growth rate increases as the relative velocity increases, as this enhances
the probability of collision occurrence. Moreover, the growth rate exhibits a
linear dependence on $\rho_{\mathrm{d}}$, indicating that the settling of dust
grains or their accumulation in dust traps favour dust growth.

###  2.2 Fragmentation

When the relative velocity between grains exceeds a critical threshold known
as $v_{\mathrm{frag}}$, instead of growing, the grains undergo fragmentation,
as observed by Tanaka et al. (1996). The impact’s kinetic energy becomes too
high for the grain’s structure to absorb, causing the bonds between the
constituent monomers of the aggregate to break apart. To quantify the mass
variation during fragmentation, we can adopt a formulation similar to that
proposed by Stepinski & Valageas (1997) for growth. Using a realistic approach
developed by Kobayashi & Tanaka (2010) and Garcia (2018) and further used by
Vericel et al. (2021), the model introduces a concept of progressive or “soft”
fragmentation, taking into account the relative velocity ($v_{\mathrm{rel}}$)
compared to $v_{\mathrm{frag}}$:

$\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)_{\mathrm{frag}}=-\frac{v_{\mathrm{rel}}^{2}}{v_{\mathrm{rel}}^{2}+v_{\mathrm{frag}}^{2}}\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{s}}}v_{\mathrm{rel}}.$
(3)

When the relative velocity is near the fragmentation threshold, the mass loss
becomes less pronounced. Therefore, a fragmenting grain loses approximately
half of its initial mass after a collision time222It is worth noting that our
definition of fragmentation aligns with the point at which half of the mass is
lost. However, it’s important to acknowledge that other definitions exist. For
example, Ringl et al. (2012); Gunkelmann et al. (2016) define fragmentation as
the moment when at least one monomer is ejected from the main aggregate. We
refer to the loss of grain mass when it’s less than half of the initial mass
as erosion. when $v_{\mathrm{rel}}=v_{\mathrm{frag}}$ or more when
$v_{\mathrm{rel}}>v_{\mathrm{frag}}$. When $v_{\mathrm{rel}}\gg
v_{\mathrm{frag}}$, the situation described by Gonzalez et al. (2015) is
recovered, where the entire grain fragments, losing most of its mass within a
collision time.

Figure 1: Left: Time evolution of the size $s$ of different aggregates as they
are eroded. The vertical turnover in the profile gives the characteristic
erosion time. The aggregates are kept at a fixed distance of
$1\leavevmode\nobreak\ \mathrm{au}$ from the star, and the ejected dust has a
size of 100 $\mu$m. Dashed lines represent data from R20, which are in
excellent agreement with our results. Right: Stokes number at the erosion
threshold for aggregates drifting towards the star and four different sizes of
ejected grains. Dotted lines represent the analytical solution presented in
G20.

###  2.3 Aeolian erosion

R20 and G20 considered aeolian erosion by gas as a mechanism to reduce the
size of large aggregates when they are decoupled from the gas. The following
section is inspired by those works. During erosion, grains of size
$s_{\mathrm{ej}}$ are ejected from the original aggregate of size $s$ ranging
from one metre to one kilometre. If we consider a surrounding gas density
$\rho_{\mathrm{g}}$ and a velocity difference between dust and gas $\Delta v$,
the characteristic time for a grain to be ejected is given by:

$t_{\mathrm{ej}}=\frac{\Delta v}{a_{\mathrm{coh}}},$ (4)

where $a_{\mathrm{coh}}$ is the cohesive acceleration required for the grain
to remain attached. The work exerted by the gas on the aggregate and the
energy loss due to erosion are given by R20:

$\displaystyle W$ $\displaystyle=p\Delta vAt_{\mathrm{ej}},$ (5)
$\displaystyle\Delta E$ $\displaystyle=-\Delta m\frac{\Delta v^{2}}{2},$ (6)

where $p=\frac{1}{2}\rho_{\mathrm{g}}\Delta v^{2}$ is the dynamic pressure and
$A$ is the effective shear surface. Since work and energy loss are equal, the
mass loss is then:

$\Delta m=-A\rho_{\mathrm{g}}\frac{\Delta v^{2}}{a_{\mathrm{coh}}}.$ (7)

In the case of an infinitesimal time interval $\mathrm{d}t<t_{\mathrm{ej}}$,
$A\approx s\Delta v\mathrm{d}t$, the mass loss becomes

$\left(\frac{\mathrm{d}m}{\mathrm{d}t}\right)_{\mathrm{eros}}=-\rho_{\mathrm{g}}\frac{\Delta
v^{3}}{a_{\mathrm{coh}}}s.$ (8)

Currently, $a_{\mathrm{coh}}$ is an unknown but Shao & Lu (2000) showed that
the cohesive force could be related to $a_{\mathrm{coh}}$.

$F_{\mathrm{coh}}=m_{\mathrm{ej}}a_{\mathrm{coh}}=\beta_{\mathrm{eros}}s_{\mathrm{ej}},$
(9)

where $m_{\mathrm{ej}}$ is the mass of an ejected grain and
$\beta_{\mathrm{eros}}$ is a parameter to define. The experimental
measurements of $\beta_{\mathrm{eros}}$ by Heim et al. (1999) and Paraskov et
al. (2006) show a strong cohesive force, resulting in a value of
$\beta_{\mathrm{eros}}=0.1\ \mathrm{kg.s^{-2}}$. With this relation between
$a_{\mathrm{coh}}$ and $\beta_{\mathrm{eros}}$, we can finally rewrite Eq. (8)
in terms of size loss as:

$\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)_{\mathrm{eros}}=\frac{-\rho_{\mathrm{g}}}{3}\frac{s^{2}_{\mathrm{ej}}}{s}\frac{\Delta
v^{3}}{\beta_{\mathrm{eros}}}.$ (10)

This equation will be used in the simulations to determine the importance of
erosion. However, for erosion to occur, the erosion threshold must first be
reached. This threshold was determined by Shao & Lu (2000) and is expressed
as:

$\Delta
v_{\mathrm{eros}}=\sqrt{A_{\mathrm{N}}\frac{\gamma_{\mathrm{s}}}{\rho_{\mathrm{g}}s_{\mathrm{ej}}}},$
(11)

where $A_{\mathrm{N}}=0.0123$ is a numerical constant and
$\gamma_{\mathrm{s}}=1.65{\>\times\>}10^{-4}\leavevmode\nobreak\
\mathrm{J\,m^{-2}}$, the surface energy, is the same as in R20. $\Delta
v_{\mathrm{eros}}$ thus depends essentially on $s_{\mathrm{ej}}$.

In this work, $\Delta v=v_{\mathrm{d}}-v_{\mathrm{g}}$, the velocity
difference between dust and gas is given by:

$\displaystyle\Delta v_{\mathrm{r}}$
$\displaystyle=\frac{(1+\varepsilon)\mathrm{St}}{(1+\varepsilon)^{2}+\mathrm{St}^{2}}v_{\mathrm{drift}}-\frac{\mathrm{St}^{2}}{(1+\varepsilon)^{2}+\mathrm{St}^{2}}v_{\mathrm{visc}},$
(12) $\displaystyle\Delta v_{\mathrm{\theta}}$
$\displaystyle=-\frac{\mathrm{St}^{2}}{\left(1+\epsilon\right)^{2}+\mathrm{St}^{2}}\frac{v_{\mathrm{drift}}}{2}-\frac{\left(1+\varepsilon\right)\mathrm{St}}{\left(1+\varepsilon\right)^{2}+\mathrm{St}^{2}}\frac{v_{\mathrm{visc}}}{2},$
(13)

where $\varepsilon$ is the dust-to-gas ratio, $v_{\mathrm{drift}}$ is the
radial drift velocity associated to the gas pressure gradient, and
$v_{\mathrm{visc}}$ is the one due to gas motion caused by viscosity (full
details on the derivation can be found in Michoulier & Gonzalez, 2022).
$\Delta v$ is hence given by $\Delta v=\sqrt{\Delta v_{\mathrm{r}}^{2}+\Delta
v_{\mathrm{\theta}}^{2}}$. and is compared to the erosion threshold $\Delta
v_{\mathrm{eros}}$.

##  3 Tests

To ensure that our implementation of the equations describing erosion works as
intended, we conduct tests to compare our results with those of R20 and G20,
as shown in Fig. 1. To conduct these tests, we use the same disc parameters,
which correspond to a Minimum Mass Solar Nebula model (Perets & Murray-Clay,
2011; Grishin & Perets, 2015). The surface density is
$\Sigma_{\mathrm{g}}=2{\>\times\>}10^{4}\,(r/r_{0})^{-1.5}$
$\mathrm{kg\,m^{-2}}$, and the temperature
$T_{\mathrm{g}}=120\,(r/r_{0})^{-3/7}$ K, with $r_{0}=1\leavevmode\nobreak\
\mathrm{au}$. This gives an aspect ratio $H/r=0.022\,(r/r_{0})^{2/7}$ and a
gas density $\rho_{\mathrm{g}}\approx 3{\>\times\>}10^{-6}\,(r/r_{0})^{-16/7}$
$\mathrm{kg\,m^{-3}}$. The dust-to-gas ratio is small and does not play a role
here. Figure 1 on the left demonstrates that our simulations using Pamdeas
perfectly reproduce the data from Figure 4 of R20. The characteristic erosion
timescale for each aggregate size is accurately reproduced. When $\Delta
v_{\mathrm{eros}}=\Delta v$, as the distance is kept fixed, the erosion
threshold corresponds to only one value of St and hence one size. It should be
noted that the size at which the erosion threshold is reached is slightly
different (127 cm instead of 200 cm in R20). This difference likely arises
from the way $\Delta v$ is computed, which differs between this work and R20
and G20. Although the physical size is different, what determines the relative
velocity is the Stokes number, which is the same, as shown in the right panel.

Figure 2: Left: Evolution of the size $s$ of grains experiencing growth and
erosion only, as they grow and drift inwards from different initial distances
$r$ from the star. The colour represents the Stokes number. The gray dashed
line represents the evolution without erosion, and the green, orange and red
lines represent the evolution with growth and fragmentation using a
fragmentation threshold of 20, 40 and 80 $\mathrm{m\,s^{-1}}$. The size of the
ejected grains is $s_{\mathrm{ej}}=1$ mm, a typical value used by R20 and G20.
Right: Same as the left panel, but with a pressure maximum that traps dust at
$1\leavevmode\nobreak\ \mathrm{au}$.

On the right, Fig. 1 shows the critical Stokes number ($\mathrm{St}$) at the
erosion threshold. In this case, the grains to be eroded have an initial size
of 10 metres, and are initially placed in the outer disc. Grains then drift
toward the star until they cross the erosion threshold. The data are compared
with the analytical solution that give the erosion threshold in terms of
critical Stokes number, a polynomial of degree 5, which takes into account
both laminar and turbulent gas flow (see Eq. (8) in G20, ) around the
aggregate. To achieve this, one writes the sum of $\Delta v$ for laminar and
turbulent gas flow as a function of $\mathrm{St}$, which equals $\Delta
v_{\mathrm{eros}}$ at the erosion threshold, and then numerically solves for
$\mathrm{St}$. As the size of the ejected grains increase, erosion becomes
easier as erosion starts at larger distances $r$ compared to smaller
$s_{\mathrm{ej}}$. The simulations also show excellent agreement with the
analytical solution. The difference occurs at large $\mathrm{St}$ and is due
to the fact that a grain evolving in Pamdeas takes a non-zero time to be
eroded and reach the equilibrium.

##  4 Results

###  4.1 Setup

To be closer to reality than the Minimum Mass Solar Nebula (MMSN) model, we
use a disc model that reproduces observations, presented in Williams & Best
(2014). For Pamdeas, the masses of the star and of the disc are set to
$M_{*}=1\leavevmode\nobreak\ \mathrm{M_{\odot}}$ and
$M_{\mathrm{disc}}=0.01\leavevmode\nobreak\ \mathrm{M_{\odot}}$. The inner and
outer radii are $R_{\mathrm{in}}=0.1\leavevmode\nobreak\ \mathrm{au}$ and
$R_{\mathrm{out}}=300\leavevmode\nobreak\ \mathrm{au}$, while the disc aspect
ratio is $H/R_{0}=0.0895$ with a reference radius
$R_{0}=100\leavevmode\nobreak\ \mathrm{au}$. The density and temperature
profiles exponents are $p=1$ and $q=0.5$. The turbulent viscosity parameter
(Shakura & Sunyaev, 1973) is set to $\alpha=5\times 10^{-3}-5\times 10^{-4}$.
A typical dust-to-gas ratio of 1% is used. We use silicates with
$\rho_{\mathrm{s}}=2.7\leavevmode\nobreak\ \mathrm{kg\,m^{-3}}$ and initial
size $s_{0}=1\leavevmode\nobreak\ \mu$m, with different fragmentation
thresholds $v_{\mathrm{frag}}=20-40-80\leavevmode\nobreak\
\mathrm{m\,s^{-1}}$. For erosion, we use $\beta_{\mathrm{eros}}=0.1\
\mathrm{kg.s^{-2}}$, and the ejected grain size $s_{\mathrm{ej}}=1$ mm, the
largest physical value to maximise the effect of erosion (; ), as smaller
$s_{\mathrm{ej}}$ would reduce
$\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)_{\mathrm{eros}}\propto
s_{\mathrm{ej}}^{2}$. In addition, one of the assumptions in the erosion model
is that the cohesive force $F_{\mathrm{coh}}$ is proportional to the grain
size, which, according to G20 is valid up to 1 mm. Hence, we will only use
this value in this paper. To model a pressure bump centred at
$1\leavevmode\nobreak\ \mathrm{au}$, we simply add to the initial profile a
Gaussian with a width of $0.1\leavevmode\nobreak\ \mathrm{au}$ and a height of
20 times the initial surface at $1\leavevmode\nobreak\ \mathrm{au}$. The local
dust-to-gas ratio in the pressure bump is set to 1, a typical value expected
in dust traps where dust is settled and concentrated. As 3D simulations are
more expensive, we only model the inner disc with Phantom, keeping $M_{*}$ and
$R_{\mathrm{in}}$ the same but taking $R_{\mathrm{out}}=5\leavevmode\nobreak\
\mathrm{au}$ and $R_{0}=1\leavevmode\nobreak\ \mathrm{au}$, which gives
$M_{\mathrm{disc}}=1.64{\>\times\>}10^{-4}\leavevmode\nobreak\
\mathrm{M_{\odot}}$ in order to have the same surface density, with
$\alpha=5\times 10^{-3}$. We use for all 3D simulations $10^{6}$ particles.
Simulations start with gas only, in order to prevent artefacts due to gas
relaxation. After 10 orbits at 5 $\mathrm{au}$ ($\sim 120$ yr), the disc is
relaxed, and dust is added with a uniform dust-to-gas ratio of 1%, and the
grain size is initialized at $s=100\leavevmode\nobreak\ \mu$m, with
$v_{\mathrm{frag}}=80\leavevmode\nobreak\ \mathrm{m\,s^{-1}}$. The other
parameters remain the same.

###  4.2 1D study

In R20 and G20, erosion near the star is presented for certain aggregate
sizes. Aggregates on the order of metres can be easily eroded into centimetre
sized aggregates, particularly with large ejected grains. With
$\alpha=5{\>\times\>}10^{-3}$, the results are shown in Fig. 2, for grains
subject to growth and erosion only. In the left panel, the grains have an
initial size of 0.2 $\mu$m and grow until they begin to drift towards the star
as their Stokes number approaches $0.1$. The grains continue to grow while
drifting until erosion occurs at a distance of $0.19\leavevmode\nobreak\
\mathrm{au}$. Erosion is extremely effective for ejected grains, with
$s_{\mathrm{ej}}=1$ mm (R20). The aggregates, with a maximum size of $4-6$
metres, are rapidly eroded to sizes of a few centimetres. Then, the grains
reach an equilibrium between growth and erosion, and drift slowly from
$0.19\leavevmode\nobreak\ \mathrm{au}$ inwards. For comparison, the green,
orange and red lines represent the evolution of a grain considering growth and
fragmentation, but no erosion. The fragmentation threshold is set to 20, 40
and 80 $\mathrm{m\,s^{-1}}$ respectively. Even a high fragmentation threshold
of 80 $\mathrm{m\,s^{-1}}$ (a value completely unrealistic for most of the
disc, resembling pure growth), the fragmentation threshold is reached before
the erosion threshold. Therefore, fragmentation is always more effective in
destroying dust grains when a realistic value is used.

Figure 3: Left: Same as Fig. 2, but with $\alpha=5{\>\times\>}10^{-4}$ and
only one fragmentation threshold.

We can also consider what would happen if the dust is trapped and can no
longer drift to be accreted onto the star. The answer is shown in the right
panel of Fig. 2. Fragmentation is still effective in limiting the grain size.
In the case without fragmentation, erosion is unable to balance out growth.
There is a small difference around $s=0.1$ m due to erosion, which reduces the
growth rate without completely countering it, affecting the final grain size
somewhat. However, in this case, the final size is of little importance since
the grains are trapped, making it possible for them to form larger objects of
more than one kilometre at later stages.

The reason why erosion is almost absent within a dust trap is simple. Erosion
depends on $\Delta v$, and in a dust trap, the dust-to-gas ratio
($\varepsilon=1$ in our case) is greater than the typical value of 1%. Taking
into account back-reaction, dust tends to make the gas orbit faster and vice
versa, reducing $\Delta v$. Additionally, as the dust density
$\rho_{\mathrm{d}}=\varepsilon\rho_{\mathrm{g}}$ increases, the growth rate is
multiplied by 100 (Eq. (1)). These two factors work against erosion and
prevent it from occurring or at least having a significant impact.

We then perform the same simulations, but with $\alpha=5{\>\times\>}10^{-4}$
shown in Fig. 3. The results are qualitatively the same. In the left panel,
the grains have the same initial size and grow until they begin to drift
towards the star. The grains are eroded at the same distance of
$0.19\leavevmode\nobreak\ \mathrm{au}$. Erosion is also effective, but with
$\alpha=5{\>\times\>}10^{-4}$, grain growth is less efficient because of lower
relative velocities when the viscosity is lower. The aggregates, with a
maximum size of 50 centimetres, are eroded to sizes of a few centimetres,
reach the equilibrium and drift inwards. The green line represents the
evolution when fragmentation is taken into account, with a threshold set to 20
$\mathrm{m\,s^{-1}}$. Fragmentation is still more effective in destroying dust
grains when such a realistic value is used. Higher thresholds with this
$\alpha$ lead to an evolution resembling the pure growth case. Nevertheless,
erosion reduces the size of the grains by one order of magnitude, compared to
roughly 3 when $\alpha=5{\>\times\>}10^{-3}$.

Within the dust trap, shown on the right panel, erosion is almost absent. The
case with erosion does not differ from the case with pure growth, and only two
scenarios are identified. If the threshold is low enough, grains are destroyed
by fragmentation. Otherwise, dust grains can grow freely.

We also performed tests considering porosity (see Michoulier & Gonzalez, 2022,
for its implementation), and we observed no significant change in the
appearance of erosion (not shown). We ran a simulation with
$\alpha=5{\>\times\>}10^{-5}$, but grain growth is so slow that dust barely
reaches sizes of a few centimetres. Erosion still appears, but the dust size
is only divided by a factor of two to three. In this case, the limiting factor
to the grain size is the very low growth rate.

Figure 4: Comparison between simulations in the ($r$, $s$) plane at $t=100$ yr
with growth only top left, growth and fragmentation top right with
$v_{\mathrm{frag,\>Si}}=80\leavevmode\nobreak\ \mathrm{m\,s^{-1}}$, growth and
erosion bottom left and growth, fragmentation and erosion bottom right. The
colour gives the dust-to-gas ratio.

###  4.3 3D study

We want to check now if we obtain the same behaviour with 3D simulations. We
choose a high velocity threshold of $v_{\mathrm{frag}}=80\leavevmode\nobreak\
\mathrm{m\,s^{-1}}$, for which fragmentation is hardly restrictive and our 1D
simulations showed that it still dominates over erosion. Figure 4 shows the
radial grain size distribution coloured with the dust-to-gas ratio
$\varepsilon$, for four simulations at time $t=100$ yr. The top left panel
corresponds to growth only, top right to growth and fragmentation, bottom left
to growth and erosion, and bottom right to growth, fragmentation and erosion.
We see that the simulation with growth is able to form large grains up to the
decimetre between $0.5$ and $2\leavevmode\nobreak\ \mathrm{au}$. Simulations
with fragmentation are identical when erosion is turned on or off (right
panels). This means fragmentation is the dominant mechanism to destroy grains.
When comparing the simulation with growth only and the one with erosion, the
impact of erosion appears only when sizes are larger than $3$ cm, interior to
$1\leavevmode\nobreak\ \mathrm{au}$. This is qualitatively similar to our 1D
simulations, however here grains start to be eroded at larger distances ($\sim
0.9$ $\mathrm{au}$). This is due to larger volume densities
$\rho_{\mathrm{g}}$ in the midplane of the 3D disc compared to the 1D case.
Moreover, fragmentation destroys grains at roughly the same size and at the
same distance both in 1D and 3D. Aggregates of decimetre size are still
present, as in the growth-only simulation. For the simulations with
fragmentation, the peak in the size profile corresponds to the distance where
$v_{\mathrm{rel}}=v_{\mathrm{frag}}$, and the plateau inwards shows an
equilibrium between growth and fragmentation (Vericel et al., 2021).

The simulation with growth only is the one where the inner region is the less
dust-enriched, with $\varepsilon<10^{-1}$. One should note that dust has been
lost to the star due to radial drift in the very inner region. On the
contrary, the ones with fragmentation have $\varepsilon\sim
2{\>\times\>}10^{-1}$ at 0.6 $\mathrm{au}$. Fragmentation helps grains to stay
in the disc because their sizes become smaller the closer they are to the
star, which means they will be more coupled to the gas and less prone to
drift. With erosion only, the region between 0.6 and 1 $\mathrm{au}$ has
$\varepsilon\sim 10^{-1}$. Erosion can thus help the inner region to increase
the local dust-to-gas ratio, but it is not as efficient as fragmentation.
Outside 2 $\mathrm{au}$, the size profiles of all simulations are very
similar, as neither fragmentation nor erosion affect dust growth.

These simulations support the results from our 1D simulations. Moreover, they
show that when accounting for fragmentation, erosion is not present even with
$v_{\mathrm{frag}}=80\leavevmode\nobreak\ \mathrm{m\,s^{-1}}$, which means it
can be ignored during dust evolution and growth of aggregates.

##  5 Discussion

###  5.1 Caveats

The mains limitations arise from our model of dust growth, as we use a 3D code
that eliminates most of the approximations done with the 1D Code Pamdeas.
However, since we use the mono-disperse approximation for dust growth, we do
not track all the smaller grains ejected from larger bodies due to erosion.
This is not very important as we want to track the evolution of the largest
grains, since fragmentation always appears before erosion. In addition, the
mono-disperse approximation only considers collisions of equal-size grains.
Unequal-size grains have relative settling and drifting velocities, acting as
additional sources of collisions. The SPH formalism naturally produces a
spread in $\Delta v$, resulting in a spread in $v_{\mathrm{rel}}$ as well,
similar to the velocity distributions in Windmark et al. (2012b); Garaud et
al. (2013). Nevertheless, our approximation can not account for high-mass-
ratio collisions, which have been found to result in net growth even at
$v_{\mathrm{rel}}$ as high as 70 or 80 m s-1 (Teiser & Wurm, 2009; Kothe et
al., 2010; Windmark et al., 2012a; Wada et al., 2013; Meisner et al., 2013).

Aeolian erosion is a continuous process and one may wonder whether the
discrete nature of particle collision would hinder the comparison between both
processes. This is not a problem in numerical simulations, where time is
incremented by a finite quantity, the timestep. Even when the time between
collisions is long, it is still much shorter than a single timestep (e.g.
Garcia & Gonzalez, 2020). Both processes are thus taken into account
simultaneously in our simulations.

Another issue is the fact that we do not take into account porosity in the
model. This could play a role, as porous grains tend to form larger aggregates
while being less sensitive to fragmentation. A model of erosion taking into
account porosity would be more precise in capturing dust evolution. But we do
not think this will change the results significantly because fragmentation
will remain the limiting factor.

Lastly, in the erosion model, $\beta_{\mathrm{eros}}$ is still not very well
constrained for different kinds of material and more experimental measurements
would be needed, although progress has been made in recent work (Demirci et
al., 2020; Schönau et al., 2023). The model also assumes that all ejected
grains have the same size, while in reality, all sizes should be considered.

###  5.2 Importance of erosion

We see in this paper that fragmentation is always the first process to destroy
grains before erosion for $v_{\mathrm{frag}}=20\leavevmode\nobreak\
\mathrm{m\,s^{-1}}$. Smaller fragmentation thresholds like 10 or 15
$\mathrm{m\,s^{-1}}$ only strengthen fragmentation, thus decreasing the effect
of erosion, while larger fragmentation threshold can be considered less
realistic, both with $\alpha=5{\>\times\>}10^{-3}$ and
$\alpha=5{\>\times\>}10^{-4}$. Moreover, we use the size $s_{\mathrm{ej}}=1$
mm adopted by R20 and G20, which makes erosion more effective. Therefore, we
can safely say that erosion can be completely neglected in dust models when
$\alpha=5{\>\times\>}10^{-3}$. For $\alpha=5{\>\times\>}10^{-4}$, the only
fragmentation considered is $v_{\mathrm{frag}}=20\leavevmode\nobreak\
\mathrm{m\,s^{-1}}$, because higher $v_{\mathrm{frag}}$ are extremely similar
to pure growth. The difference of sizes just before and after erosion in this
case is also smaller, by an order of magnitude, reducing the impact of erosion
on dust evolution. Finally, adding a pressure bump at $1\leavevmode\nobreak\
\mathrm{au}$ did not help erosion to appear, despite the fact that grains grow
up to one kilometre. Erosion can therefore be neglected in all cases when
fragmentation is considered. For a very low-viscosity disc, erosion could
still be ignored, as grains growth would be very slow and large sizes wouldn’t
be reached. However, it is worth mentioning that if larger boulders or
planetesimals are formed or captured, erosion would still be important in
grinding them down in the inner disc region, as discussed by Grishin et al.
(2019); Rozner et al. (2020).

##  6 Summary and conclusion

In this paper, we discuss the importance of erosion in the evolution of dust
in protoplanetary disc. We first present the way we modelled radial drift in a
1D code to capture accurately the effect even if the disc is stationary and
the gas is not evolving. We then present how erosion has been treated based on
R20 and G20, giving the two important equation to implement in the codes. We
also present the model to take into account both growth and fragmentation from
Stepinski & Valageas (1997), Laibe et al. (2008), Garcia (2018), Vericel et
al. (2021), Michoulier & Gonzalez (2022). We perform some tests to be sure
Pamdeas reproduces some of the key results presented in R20 and G20. We then
perform simulations with the same model implemented in both the Pamdeas and
Phantom codes. We show that erosion is negligible when fragmentation is taken
into account with realistic fragmentation thresholds. Both codes give the same
results. When considering a pressure bump, erosion is still not efficient at
destroying grains. We then discuss the main caveats of this study and a
discussion of the insignificance of erosion. To conclude, erosion can be
neglected in models of dust evolution accounting for fragmentation, bouncing,
or any other mechanism that limits the dust sizes to a couple of centimetres
in the very inner region of a protoplanetary disc. Usually, such short
distances of a fraction of $\mathrm{au}$ are not considered when doing
simulations and erosion can be safely neglected from a dust evolution
perspective.

###### Acknowledgements.

We thank Daniel J. Price for useful discussion and advice and the anonymous
referee for their suggestions. The authors acknowledge funding from ANR
(Agence Nationale de la Recherche) of France under contract number
ANR-16-CE31-0013 (Planet-Forming-Disks) and thank the LABEX Lyon Institute of
Origins (ANR-10-LABX-0066) for its financial support within the Plan France
2030 of the French government operated by the ANR. This research was partially
supported by the Programme National de Physique Stellaire and the Programme
National de Planétologie of CNRS (Centre National de la Recherche
Scientifique)/INSU (Institut National des Sciences de l’Univers), France. We
gratefully acknowledge support from the PSMN (Pôle Scientifique de
Modélisation Numérique) of the ENS de Lyon for the computing resources. This
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreements No
210021 and No 823823 (DUSTBUSTERS). Figures were made with the Python library
matplotlib (Hunter, 2007).

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José Martín Castro-Manzanologo_ttlM. Antonia Huertas, João Marcos, María
Manzano, Sophie Pinchinat,
François Schwarzentruber54th International Conference on Tools for Teaching
Logic119TTL2015

#  dasasap, an App for Syllogisms

J. Martín Castro-Manzano  Faculty of Philosophy and Humanities, UPAEP
<EMAIL_ADDRESS>Verónica Reyes-Meza  Faculty of Psychology, UPAEP
<EMAIL_ADDRESS>Jorge Medina-Delgadillo  Faculty of Philosophy and
Humanities, UPAEP
<EMAIL_ADDRESS>
###### Abstract.

The main goal of this contribution is to introduce a cross-platform
application to learn-teach syllogistic. We call this application dasasap for
develop all syllogisms as soon as possible. To introduce this application we
show the logical foundations for the game with a system we call
$\mathcal{L}_{\square}$, and its interface developed with LiveCode.

###### Key words and phrases:

Application, Software, Syllogistic, LiveCode

###### 1991 Mathematics Subject Classification:

F.4.1 Mathematical Logic, K.8 Personal Computing

##  1\. Introduction

Syllogistic has a remarkable impact in our education and culture: most
undergraduate logic courses, books and manuals around the world cover or
include a fragment of syllogistic not just to teach logic, but to provide
formæ mentis. Therefore, whether studying syllogistic is relevant or not is
beyond doubt; the problem, if any, has to do with the different treatments
used to learn-teach syllogistic.

The main goal of this contribution is to introduce a cross-platform
application to learn-teach syllogistic. We call this application dasasap for
develop all syllogisms as soon as possible. To introduce this application we
show the logical foundations for the game with a system we call
$\mathcal{L}_{\square}$ (Section 2), and its interface developed with LiveCode
(Section 3). We close this work with comments about future work (Section 4).

##  2\. Logical foundations

###  2.1. Syllogisms

A categorical proposition is a proposition with a defined quantity (universal
or particular), a subject $\mathsf{S}$, a quality (affirmative or negative),
and a predicate $\mathsf{P}$. The adequate combinations of these elements
produce four categorical propositions, each one with a name:

  * •

Proposition $\mathsf{A}$ states $All\ \mathsf{S}\ is\ \mathsf{P}$.

  * •

Proposition $\mathsf{E}$ states $No\ \mathsf{S}\ is\ \mathsf{P}$.

  * •

Proposition $\mathsf{I}$ states $Some\ \mathsf{S}\ are\ \mathsf{P}$.

  * •

Proposition $\mathsf{O}$ states $Some\ \mathsf{S}\ are\ not\ \mathsf{P}$.

A categorical syllogism is a sequence of three categorical propositions
ordered in such a way that the two first propositions are called premises
(major and minor) and the last one is called conclusion. Within the premises
there is a term $\mathsf{M}$, known as middle term, that works as a link that
gathers the remaining terms, namely, $\mathsf{S}$ and $\mathsf{P}$:

######  Example 1.

  1. (1)

Major premise: All $\mathsf{M}$ is $\mathsf{P}$.

  2. (2)

Minor premise: All $\mathsf{S}$ is $\mathsf{M}$.

  3. (3)

Conclusion: All $\mathsf{S}$ is $\mathsf{P}$.

The previous example can be represented as a string of characters using the
terms and proposition names stated previously in an infix notation:
$\mathsf{MAP}\mathsf{SAM}\therefore\mathsf{SAP}$, where $\therefore$ indicates
that the premises are to the left and the conclusion to the right. According
to the position of the middle term $\mathsf{M}$ we can build four figures of
the syllogism that encode all the valid and only the valid syllogisms. These
have been named using mnemonics [1] (Table 1).

Table 1. Valid categorical syllogisms Figure 1 | Figure 2 | Figure 3 | Figure 4
---|---|---|---
Barbara | Cesare | Disamis | Camenes
$\mathsf{MAP}\mathsf{SAM}\therefore\mathsf{SAP}$ | $\mathsf{PEM}\mathsf{SAM}\therefore\mathsf{SEP}$ | $\mathsf{MIP}\mathsf{MAS}\therefore\mathsf{SIP}$ | $\mathsf{PAM}\mathsf{MES}\therefore\mathsf{SEP}$
Celarent | Camestres | Datisi | Dimaris
$\mathsf{MEP}\mathsf{SAM}\therefore\mathsf{SEP}$ | $\mathsf{PAM}\mathsf{SEM}\therefore\mathsf{SEP}$ | $\mathsf{MAP}\mathsf{MIS}\therefore\mathsf{SIP}$ | $\mathsf{PIM}\mathsf{MAS}\therefore\mathsf{SIP}$
Darii | Festino | Bocardo | Fresison
$\mathsf{MAP}\mathsf{SIM}\therefore\mathsf{SIP}$ | $\mathsf{PEM}\mathsf{SIM}\therefore\mathsf{SOP}$ | $\mathsf{MOP}\mathsf{MAS}\therefore\mathsf{SOP}$ | $\mathsf{PEM}\mathsf{MIS}\therefore\mathsf{SOP}$
Ferio | Baroco | Ferison |
$\mathsf{MEP}\mathsf{SIM}\therefore\mathsf{SOP}$ | $\mathsf{PAM}\mathsf{SOM}\therefore\mathsf{SOP}$ | $\mathsf{MEP}\mathsf{MIS}\therefore\mathsf{SOP}$ |

###  2.2. Jigsaw puzzles

A jigsaw puzzle is a tiling array composed by a finite set of tessellating
pieces that require assembly by way of the interlocking of knobs and sockets
(Figure 1). When finished, a jigsaw puzzle mechanically produces a complete
picture by the adequate interlocking of all the pieces; when the pieces cannot
be interlocked, the jigsaw puzzle cannot be completed, and thus the picture
cannot be mechanically produced.

Figure 1. Pieces of a typical jigsaw puzzle

These types puzzles date back as far as Archimedes [2], so the historical
paths of logic and jigsaw puzzles have a long tradition, although the typical
pictorial jigsaw puzzles we are familiar with have their roots in Europe when
John Spilsbury’s first dissected maps in the 1760s primarily for educational
purposes [3]. We suggest that syllogisms can be developed as jigsaw puzzles
that require assembly by way of interlocking.

###  2.3. The system $\mathcal{L}_{\square}$

$\mathcal{L}_{\square}$ is a system that uses adequate combinations of
polygons as jigsaw puzzles in order to decide the (in)validity of categorical
syllogisms. Its vocabulary is defined by two elementary diagrams (i.e.,
pieces), socket and knobs (Figure 2).

Figure 2. Vocabulary for $\mathcal{L}_{\square}$

Syntax is given by a single rule: if a diagram belongs to the vocabulary of
$\mathcal{L}_{\square}$, the diagrams depicted in Figure 3, and only those,
are well formed diagrams.

Figure 3. Well formed diagrams for $\mathcal{L}_{\square}$

Semantics is given by Figure 4.

Figure 4. Semantics for $\mathcal{L}_{\square}$

With this system we can represent the categorical propositions as in Figure 5.

Figure 5. Categorical propositions in $\mathcal{L}_{\square}$

We can arrange this system in a Square of Opposition (Figure 6). It should be
clear that the only rules preserved in this Square are the rules for
contradiction between $\mathsf{A}$ ($\mathsf{E}$) and $\mathsf{O}$
($\mathsf{I}$); the rules for contraries, subalterns, and subcontraries do not
work. Thus, this system’s Square behaves as a Modern Square of Opposition
rather than a Traditional Square. Also, conversion, contraposition, and
obversion are valid transformations in this Square.

Figure 6. Square of Opposition in $\mathcal{L}_{\square}$

###  2.4. Decision Algorithm in $\mathcal{L}_{\square}$

Using a single term, in this case the middle term, we can represent the
Identity Principle ($\mathsf{IP}$) in this system:

  * •

Proposition $\mathsf{A}$ states $All\ \mathsf{M}\ is\ \mathsf{M}$.

  * •

Proposition $\mathsf{E}$ states $No\ \mathsf{M}\ is\ \mathsf{M}$.

  * •

Proposition $\mathsf{I}$ states $Some\ \mathsf{M}\ are\ \mathsf{M}$.

  * •

Proposition $\mathsf{O}$ states $Some\ \mathsf{M}\ are\ not\ \mathsf{M}$.

Figure 7 shows the four categorical propositions using the middle term
$\mathsf{M}$. We can observe that from these four cases only $\mathsf{A}$
encodes a logical truth, that is, a true statement for any structure, provided
it is a weak version of the $\mathsf{IP}$ as in $\forall
x(\mathsf{M}x\Rightarrow\exists y\mathsf{M}y)$.

Figure 7. Identity Principle in $\mathcal{L}_{\square}$

Using the $\mathsf{IP}$ we suggest a decision algorithm for syllogistic in
$\mathcal{L}_{\square}$ with complexity of $O(N)$ (Algorithm 1).

[htbp]  Input: syllogism $\sigma$ Output: decision $\delta$ for _$\sigma$_ do
if _middle terms of $\sigma$ == $\mathsf{IP}$_ then $\delta\leftarrow valid$;
else $\delta\leftarrow invalid$; end end Decision algorithm in
$\mathcal{L}_{\square}$

Hence, this algorithm not only is time efficient but sound and complete:

######  Proposition 1.

(Soundness) If $\mathcal{A}(\sigma_{i,j})=valid$ then $\sigma_{i,j}$ is valid.

Proof. Let us denote the application of Algorithm 2.4 to a given syllogism
$\sigma_{i,j}$ from figure $i\in\\{1,2,3,4\\}$ and row $j\in\\{1,2,3,4\\}$ in
Table 1 by $\mathcal{A}(\sigma_{i,j})$ (thus, for instance, the application of
Algorithm 2.4 to a Dimaris syllogism is $\mathcal{A}(\sigma_{4,2})$). We prove
soundness by cases. Since there are four figures, we need to cover each valid
syllogism from each figure. Here we only show the first case that covers the
syllogisms of figure 1 (Barbara, Celarent, Darii, and Ferio) in Section 2.5.
The remaining syllogisms can be proven similarly. Thus, we have that for every
$\sigma_{i,j}$, when $\mathcal{A}(\sigma_{i,j})=valid$, $\sigma_{i,j}$ is
valid. $\blacksquare$

######  Proposition 2.

(Completeness) If $\sigma_{i,j}$ is valid then
$\mathcal{A}(\sigma_{i,j})=valid$.

Proof. We prove this by contradiction. Suppose that for all $i,j$, the
syllogism $\sigma_{i,j}$ is valid but for some valid syllogism $\sigma_{k,j}$,
$\mathcal{A}(\sigma_{k,j})=invalid$. Now, we know $\sigma_{1,j}$ is valid and
if we apply $\mathcal{A}(\sigma_{1,j})$ we obtain
$\mathcal{A}(\sigma_{1,j})=valid$, as we can see in Section 2.5. So, since all
valid syllogisms $\sigma_{n>1,j}$ can be reduced to the valid syllogisms of
figure 1 by the rules of equivalence (conversion, contraposition, and
obversion), it follows that $\mathcal{A}(\sigma_{n>1,j})=valid$, and thus, for
all valid syllogisms $k$, $\mathcal{A}(\sigma_{k,j})=valid$, which contradicts
our assumption. $\blacksquare$

###  2.5. Example

As an example of how $\mathcal{L}_{\square}$ works we prove the four valid
syllogisms of figure 1 (Figure 8), but it should be clear that the other
syllogisms can be proven similarly.

Figure 8. Barbara, Celarent, Darii, Ferio. In order to represent and build a
categorical syllogism in $\mathcal{L}_{\square}$ we just stack up two well
formed diagrams that represent the premises. When the middle terms interlock
each other forming the $\mathsf{IP}$ in a jigsaw puzzle style (which is a step
denoted by the arrows), the inference is valid, thus allowing the terms
$\mathsf{S}$ and $\mathsf{P}$ interlock in the third diagram, i.e., the
conclusion, which is below the turnstile.

##  3\. Interface

Our application, dasasap, is motivated by the impact of syllogistic in our
education and culture, is inspired upon $\mathcal{L}_{\square}$, and is
implemented with LiveCode, which is an event-oriented programming language
that allows the development of cross-platform applications.

At this moment dasasap comes with two modes: arcade and learning. In the
arcade mode the user tries to develop valid syllogisms as quick as possible
thinking in a jigsaw puzzle style. The objective of this mode is to fasten the
recognition of valid syllogisms. The learning mode is a tutorial mode that
helps users understand how to develop syllogisms in a jigsaw puzzle style
assuming the ideas of $\mathcal{L}_{\square}$ and thus showing the basic
concepts of logic (propositions, truth values, form-content distinction, and
so on).

In Figure 9 we can see the game’s home screen running in Ubuntu 12.04. We also
depict fragments of the learning and arcade modes (Figures 10 and 11).

Figure 9. dasapap’s home snapshot

The learning mode includes four options. In the option What’s logic about? the
application explains the basic concepts of logic in order to provide an
operative definition of logic: proposition, argument, form-content
distinction, and truth-valid distinction are some of the concepts included.
The option So you think you are logical? leads to a mini-game that produces a
ranking: the user is given random syllogisms and the user must decide whether
they are valid or invalid. The options What’s a syllogism and What’s dasasap?
are self-explanatory.

Figure 10. Learning mode snapshot

The arcade mode keeps track of the ranking of the user and allows the
construction of syllogisms by measuring time efficiency and accuracy in the
development of syllogisms.

Figure 11. Arcade mode snapshot

##  4\. Conclusion

We have introduced an application we call dasasap for develop all syllogisms
as soon as possible. We showed its logical foundations with a system we call
$\mathcal{L}_{\square}$, which is a sound and complete system that uses
adequate combinations of polygons as jigsaw puzzles in order to decide the
(in)validity of categorical syllogisms. We have implemented the mechanics of
this system with LiveCode in order to develop a cross-platform application to
aid the teaching-learning process of basic logic.

Since this application is a small fragment of a larger project we are
currently developing, at the moment we are refining the implementation and the
overall design. We are also adding a reward system and different match-three
puzzle styles.

#### Acknowledgements

The authors would like to thank the reviewers for their precise corrections
and useful comments. Financial support given by UPAEP Grant 30108-1008.

## References

  * [1] Peter of Spain, Tractatus called afterwards Summule logicales, first critical edition from the manuscripts, with an Introduction by L.M. de Rijk. Assen, van Gorcum & Co., 1972
  * [2] Slocum, J., Botermans, J., Puzzles Old and New. Seattle University of Washington Press, 1986
  * [3] Williams, A.D., Shortz, W., The jigsaw puzzle: piecing together a history, Berkley Books, 2004


and spread across multiple sectors. The middle panel depicts the second stage
$t>t_{\\#}$ with $p\approx p_{\\#}$. The charge variance at this stage is
peaked at zero and $0.25$, indicating the quantum state is either projected to
a unique state or a superposition of two consecutive charge sectors $(N,N+1)$,
respectively. The third panel is for late time dynamics $t\gg t_{\\#}$ deep in
the charge sharp phase. In this regime, only the peak near zero remains,
indicating the long-time evolved quantum state has a unique sharp charge as
expected.

Figure 16: Critical dynamics of charge-sharpening transition in qubit chains.
Comparison of the dynamics of the ancilla entanglement entropy $S_{1,Q}$ and
the quantity $1-N_{0}$, where $N_{0}$ is the fraction of trajectories with
$\delta\mathcal{Q}^{2}<0.01$ and the ancilla entanglement entropy $S_{1,Q}$.
After rescaling the overall amplitude, both observables collapse to the same
exponential function in the long-time limit.

We now focus on the critical dynamics of the charge-sharpening phase
transition. We first present a strong evidence to show that the critical
dynamics is only about two consecutive sectors $(N,N+1)$. In Fig 16, we
compare the dynamics of $1-N_{0}$ to that of the ancilla-system entanglement
entropy $S_{1,\mathcal{Q}}$ (see Sec. III.2 for the definition) in the
vicinity of the transition point. The former involves multiple charge sectors,
while the latter only involves two consecutive sectors $(N,N+1)$ (with
$N=L/2$). After rescaling the overall amplitude, the long-time dynamics of
$S_{1,\mathcal{Q}}$ almost perfectly matches with the second stage dynamics of
$1-N_{0}$, both going like $\sim e^{-t/\xi_{t}}$. Furthermore, due to the
absence of other sectors, the ancilla probe saturates to the critical behavior
much earlier than $1-N_{0}$. We thus conclude that the long-time critical
dynamics only involves two consecutive sectors. In the long-time limit
$t>t_{\\#}$, both observables decay with the same exponential function,
suggesting that the critical dynamics is exponential as in Eq.(49) and
Eq.(50).

Figure 17: Universal decay rate of charge-sharpening transition in qubit
chains. Data and collapse of the decay rate in the vicinity of the charge-
sharpening phase transition. The transition point $p=0.088$ and the critical
exponent $\nu=2.15$ established in the main text is used to collapse the
curves.

We use the ancilla dynamics to establish the universal scaling functions
$A_{O}(x)$ and $B(x)$ in Eq.(49) and Eq.(50). The scaling function for the
amplitude $A_{O}(x)$ depends on the choice of observable. It has been
extracted for $N_{0}$ in Fig. 3(a) and for the ancilla probe $E_{1,Q}$ in Fig.
3(b). In this section, we extract the scaling of $B(x)$ for the universal
decay rate. In Fig.17, we calculate the long-time dynamics of
$S_{1,\mathcal{Q}}$ for various $p$ and system sizes $L$, then fit the tail to
extract the decay rate $\xi_{t}$. We find $\xi_{t}(p,L)$ cross at the
transition point $p=p_{\\#}$, indicating the existence of a scaling function
$\xi_{t}(x)/L=B(x)$. To extract this function, we collapse the curves for
different $L$ using the transition point $p_{\\#}=0.088$ and the critical
exponent $\nu_{\\#}=2.15$ established in Fig.3(b).

##  Appendix D Finite size scaling

In the case of qubit chains ($d=1$), the numerical simulations are performed
for system sizes $L\leq 24$. We rely on finite-size scaling protocols to
extract the critical properties in the thermodynamic limit. We briefly explain
the protocols in this section.

The quantities we studied in the main text, including the tripartite mutual
information, the probability of a trajectory with certain charge variance and
the ancilla probes shown in Figs. 2 and 3, all have zero scaling dimension. In
the vicinity of the transition point, they share the same finite size scaling
ansatz,

$R(p,L)=f\left((p-p_{0})L^{1/\nu},vL^{-\omega}\right)+...$ (51)

where $L$ is the system size, the measurement probability $p$ is the relevant
scaling field with a critical exponent $\nu>0$. The transition point $p_{0}$
is either the entanglement phase transition $p_{c}$ or the charge-sharpening
phase transition $p_{\\#}$.

To make the analysis systematic, we keep the leading irrelevant scaling
variable $v$ in the above scaling ansatz. In thermodynamic limit
$L\rightarrow\infty$, it is suppressed by a non-negative exponent $\omega$. In
finite size scaling, however, this field may play an important role.
Numerically, we find that the variable $v$ is significant for the ancilla
probes while negligible for other quantities. We therefore have to use a more
involved finite size scaling protocol as explained below to analyze the
ancilla probes Deng and Blöte (2003).

Since our numerics indicates that the scaling function $f(x,y)$ in Eq.(51) is
analytic for both $x=(p-p_{0})L^{1/\nu}$ and $y=vL^{-\omega}$, one can
approximate $f$ with its Taylor expansion near the critical point,

$R(p,L)=a_{R}+b_{R}(p-p_{0})L^{1/\nu}+c_{R}(p-p_{0})^{2}L^{2/\nu}+d_{R}/L^{\omega}+...$
(52)

where we assume $p$ is close to the critical point $p_{0}$ and the system size
is sufficiently large so that both $x=(p-p_{0})L^{1/\nu}$ and $y=vL^{-\omega}$
are small. We also redefine the Taylor coefficient $d_{R}$ to absorb the
unknown amplitude $v$ of the irrelevant variable. In the ideal case $d_{R}=0$,
$R(p_{0},L)$ collapses to the universal constant $a_{R}$ at the critical
point. It indicates that the curves $R(p,L)$ for different system sizes
perfectly cross at $p=p_{0}$. However, in realistic models, $d_{R}$ is non-
zero. The irrelevant field then shifts the crossing points as the system size
increases.

The ansatz Eq.(52) allows us to extract the phase transition point $p_{0}$ and
the critical exponent $\nu$ with the presence of a non-negligible irrelevant
scaling variable. In practice, we collect dozens of data points for different
$p$ and system size $L$ in the vicinity of the critical points. We then
perform a non-linear fitting with the ansatz Eq. (52) by taking the
coefficients $a,b,c,d$ and critical properties $p_{0}$ and $\nu$ as the
fitting parameters. We then try to drop some of the parameters to make sure
the fitting is robust and the error bars are reliable.

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