#  Following your nose: Autochemotaxis and other mechanisms for spinodal
decomposition in flocks

Maxx Miller<EMAIL_ADDRESS>Department of Physics and Institute for
Fundamental Science, University of Oregon, Eugene, OR $97403$ John Toner
<EMAIL_ADDRESS>Department of Physics and Institute for Fundamental Science,
University of Oregon, Eugene, OR $97403$

###### Abstract

We develop the hydrodynamic theory of dry, polar ordered, active matter
(“flocking”) with autochemotaxis; i.e., self-propelled entities moving in the
same direction, each emitting a substance which attracts the others (e.g.,
ants). We find that sufficiently strong autochemotaxis leads to an instability
to phase separation into one high and one low density band. This is very
analogous to both equilibrium phase separation, and “motility induced phase
separation” (“MIPS”) and can occur in flocks due to any microscopic mechanism
(e.g., sufficiently strong attractive interactions) that makes the entities
cohere.

Two of the principle mechanisms of ordered biological motion are “taxis”Keller
and Segel (1971a, 1970, b); Horstmann (2003); Stark (2018) and
“flocking”Vicsek  _et al._ (1995); Toner and Tu (1995); Tu  _et al._ (1998);
Toner and Tu (1998); Toner  _et al._ (2005); Toner (2012) . “Taxis” is motion
directed preferentially along the gradient of some quantity. In
“autochemotaxis”, this quantity is the concentration of a substance secreted
by the moving creatures themselves. This can lead to aggregation of those
creaturesKeller and Segel (1971a, 1970, b); Horstmann (2003); Stark (2018).

“Flocking” is the collective coherent motion of a large number of organisms
(e.g., flocks of birds) that occurs when creatures follow their neighbors.
Strikingly, even purely short ranged interactions generate long-ranged order
in an arbitrarily large flock in two dimensionsToner and Tu (1995); Tu  _et
al._ (1998); Toner and Tu (1998); Toner  _et al._ (2005); Toner (2012), which
is forbidden in equilibrium systems by the Mermin-Wagner-Hohenberg
TheoremMermin and Wagner (1966).

Hydrodynamic theories of autochemotaxisKeller and Segel (1971a, 1970, b);
Horstmann (2003); Stark (2018) and flockingToner and Tu (1995); Tu  _et al._
(1998); Toner and Tu (1998); Toner  _et al._ (2005); Toner (2012) separately
already exist. In this paper, we develop and study a hydrodynamic theory for
systems in which both effects are present. Specifically, we consider active,
self-propelled particles (hereafter called “boids”) moving over a frictional
substrate, so that momentum is not conserved and time reversal symmetry is
broken. The underlying dynamical rules are assumed to be rotation and
translation invariant. This combination, as we show in this paper, can lead to
trails (e.g., ant trails).

More specifically, we investigate autochemotaxis in an “active homogeneous
polar ordered state”, which spontaneously breaks the underlying rotation
invariance by having all of the boids move, on average, in the same direction
(i.e., “flock”). The homogeneity of the state means that translation
invariance is not broken.

A linear stability analysis of the hydrodynamic theory shows that, like
disordered active systems undergoing only autochemotaxis Stark (2018), the
flock becomes unstable when the autochemotaxis is sufficiently strong. The
instability, near threshold, is extremely anisotropic. Specifically, plane
wave modulations of the density with wavevector $\mathbf{q}$ are only unstable
in a narrow region of region of $\mathbf{q}$ space, extended in the direction
perpendicular to the direction of flock motion, as illustrated in figure 1a.
As a result, the instability is towards forming “bands” in real space running
parallel to the direction of mean flock motion (denoted by $\hat{x}$), as
illustrated in figures 1b and 2.

The instability we find here is completely different from the “banding
instability” Bertrand and Lee (2022); Bertin  _et al._ (2009); Mishra  _et
al._ (2010); Toner (2012), which occurs in a completely different region of
parameter space is driven by completely different physicsBertrand and Lee
(2022); Bertin  _et al._ (2009); Mishra  _et al._ (2010); Toner (2012), and is
characterized by the formation of bands perpendicular to the direction of mean
flock motion.

The results we obtain herein also apply to any dry polar ordered active matter
system in which any sufficiently strong microscopic mechanism drives the
inverse compressibility negative, since any such system will be described, at
the longest length and time scales, by a Toner-Tu model with a negative
inverse compressibility, which is the basis for all of our predictions here.

Figure 1:  a) The region of instability in $\mathbf{q}$ space, for
autochemotaxic systems very close to the instability threshold on the unstable
side. The instability is towards forming “bands” in real space running
parallel to the direction of mean flock motion, as illustrated in b) and
figure 2. b) The “band” structure of the instability at intermediate times.
The density is only modulated along one of the directions (which we call $y$,
and which is indicated in the figure) perpendicular to the direction $\hat{x}$
of mean flock motion. The mean speeds within each band also differ. Figure 2:
A plot of the initial (dashed blue curve) and final (solid orange curve)
density for a typical numerical solution of the hydrodynamic equations for 1d
configurations, as described in the text, at intermediate times. Here $\ell$
is the linear spatial extent of our system in the $y$-direction (which is
periodic).

Motivated by the geometry of the instability, we analytically and numerically
solved our equations of motion for “one-dimensional” configurations, by which
we mean configurations in which all of the fields depend on only one of the
$d-1$ Cartesian coordinates perpendicular to $\hat{x}$ ($d$ being the
dimension of space).

Our numerical solutions show that these bands quickly evolve into a set of
well-separated parallel “plateaus”. In each plateau, the density is almost
constant, as illustrated in figure 2. The evolution then continues via band
merging, ultimately leading to full phase separation into a single high
density band, and a single low density band.

Figure 3: Phase diagram of flock phase-separation. Solid blue and dashed
orange lines are the binodal and spinodal lines respectively. The orange
filled region, labeled “unstable”, corresponds to the coexistence of the
’liquid’ (high density) and ’gas’ (low density) phases.

Our analytic steady-state solution implies the phase diagram shown in figure
(3), in which the vertical axis is a model parameter defined later which
decreases with increasing autochemotaxis. This analytic solution is
asymptotically valid sufficiently close to the critical point in this figure.
In the region above the curve labelled “Binodal line”, the uniform state is
the only stable steady state of the flock. Below the curve labelled “Spinodal
line”, only the phase separated state is stable. Between these two curves,
either state can be stable.

There is a fundamental dissimilarity between our result and equilibrium
liquid-vapor phase separation. In equilibrium, a homogeneous state in this
part of the phase diagram would be meta-stable; that is, it would occupy a
local, but not a global minimum in the energy landscape. If fluctuations were
sufficiently large, it could reach the global minimum, and phase separate.
Because flocks are non-equilibrium systems, we have no analog to this energy
argument, and thus can only say that the homogeneous and phase separated
states are locally stable between the binodal and spinodal curves. There is no
criterion known to us that unambiguously identifies one state as truly stable,
and the other as meta-stable. We therefore simply call this region the
bistable region.

Despite this distinction, those familiar with equilibrium phase separation
will see the similarities with our system, although ours also differs from the
equilibrium case both because of its strong anisotropy, and because of non-
equilibrium effects.

One of these effects is to change the familiar “common tangent construction”
for the two coexisting densities to an uncommon tangent construction. This is
very similar to the behavior found in motility-induced phase separation (MIPS)
in disordered active systems Cates and Tailleur (2015). A common tangent
construction does become asymptotically exact as the system approaches the
critical point.

Our treatment of this phase separation is entirely “mean-field”: that is, it
ignores fluctuations in the local density and velocity. Fluctuations in the
density are well-known to radically change the scaling behavior of the density
and dynamics near the critical point in equilibrium systemsChaikin and
Lubensky (1995); Ma (1995). Furthermore, even stable (i.e., non-phase-
separating) flocks experience divergent fluctuation corrections to their
dynamicsToner and Tu (1995); Tu  _et al._ (1998); Toner and Tu (1998); Toner
_et al._ (2005); Toner (2012). We therefore expect the true scaling behavior
close to the critical point we find here to differ from our mean field
results.

The hydrodynamic theory combines the Keller-Segel equationsKeller and Segel
(1971a, 1970, b); Horstmann (2003); Stark (2018) for autochemotaxis with the
Toner-Tu equationsToner and Tu (1995); Tu  _et al._ (1998); Toner and Tu
(1998); Toner  _et al._ (2005); Toner (2012) into a unified set of
hydrodynamic equations for autochemotaxic flocking. This a continuum model
which takes as its variables three fields: the number density of boids
$\rho(\mathbf{r},t)$, the boid velocity field $\mathbf{v}(\mathbf{r},t)$, and
the concentration $\eta(\mathbf{r},t)$ of the chemical “signal” (or “chemo-
attractant”). The hydrodynamic equations governing the time evolution of these
fields are constrained by the symmetries (rotation and translation invariance)
and conservation law (boid number) of the underlying dynamics to take the form
vde :

$\displaystyle\partial_{t}\mathbf{v}+\lambda_{1}(\mathbf{v}\cdot\nabla)\mathbf{v}+\lambda_{2}(\nabla\cdot\mathbf{v})\mathbf{v}+\lambda_{3}\nabla(|\mathbf{v}|^{2})$
$\displaystyle=$ $\displaystyle U(|\mathbf{v}|,\rho,\eta)\mathbf{v}-\nabla
P_{1}(|\mathbf{v}|,\rho,\eta)-\mathbf{v}(\mathbf{v}\cdot\nabla
P_{2}(|\mathbf{v}|,\rho,\eta))$ (1) $\displaystyle+$ $\displaystyle
D_{B}\nabla(\nabla\cdot\mathbf{v})+D_{T}\nabla^{2}\mathbf{v}+D_{2}(\mathbf{v}\cdot\nabla)^{2}\mathbf{v},$
$\displaystyle\partial_{t}\rho$ $\displaystyle=$
$\displaystyle\nabla\cdot\bigg{(}k_{1}(|\mathbf{v}|,\rho,\eta)\nabla\rho-
k_{2}(|\mathbf{v}|,\rho,\eta)\rho\nabla\eta+k_{1a}(|\mathbf{v}|,\rho,\eta)\mathbf{v}(\mathbf{v}\cdot\nabla\rho)-k_{2a}(|\mathbf{v}|,\rho,\eta)\rho\mathbf{v}(\mathbf{v}\cdot\nabla\eta)\bigg{)}-\nabla\cdot(\mathbf{v}\rho),$
(2) $\displaystyle\partial_{t}\eta$ $\displaystyle=$ $\displaystyle
D_{\eta}\nabla^{2}\eta+k_{6}(|\mathbf{v}|,\rho,\eta)-\lambda_{\eta
1}(\mathbf{v}\cdot\nabla)\eta-\lambda_{\eta
2}(\nabla\cdot\mathbf{v})\eta{-\zeta_{\eta}\nabla\cdot(\mathbf{v}\rho)}+D_{\eta\rho}\nabla^{2}\rho+D_{\eta
v}\nabla^{2}|\mathbf{v}|$ (3) $\displaystyle+$ $\displaystyle D_{\eta
a}\nabla\cdot[\mathbf{v}(\mathbf{v}\cdot\nabla)\eta]+D_{\eta\rho
a}\nabla\cdot[\mathbf{v}(\mathbf{v}\cdot\nabla)\rho]+D_{\eta
va}\nabla\cdot[\mathbf{v}(\mathbf{v}\cdot\nabla)|\mathbf{v}|]\,.$

Here, $k_{1}$ is the motility of the boids, $k_{2}$ the sensitivity to the
chemical signal , and $k_{6}$ describes the production and decay of the
chemical signal Hillen and Painter (2009). We have introduced anisotropic
analogs $k_{1a}$ and $k_{2a}$ of $k_{1}$ and $k_{2}$, since we expect the
response along the direction of the flock velocity $\mathbf{v}$ to be
different, in general, from the response perpendicular to that direction. The
numbering of the $k$’s follows the convention of Keller and SegelKeller and
Segel (1971a, 1970, b); Horstmann (2003); Stark (2018); Hillen and Painter
(2009).

Our flock is on a frictional substrate, which destroys Galilean invariance and
momentum conservation, and allows the friction-like
$U(|\mathbf{v}|,\rho,\eta)$ term. Because our system is active and has a
moving steady state, $U(|\mathbf{v}|,\rho,\eta)$ should be positive for small
$|\mathbf{v}|$, and negative for large $|\mathbf{v}|$, thereby ensuring that
the flock settles down to a steady state with a non-zero average velocity
$\langle\mathbf{v}\rangle$.

For a discussion of the physical significance of the other terms in (1) and
(2), seeToner and Tu (1995); Tu  _et al._ (1998); Toner and Tu (1998); Toner
_et al._ (2005); Toner (2012).

The quantities $k_{1,2,1a,2a,6}$, $P_{1,2}(|\mathbf{v}|,\rho,\eta)$,
$U(|\mathbf{v}|,\rho,\eta)$, $D_{B,T,2}$, and $\lambda_{1,2,3}$ are in general
functions $|\mathbf{v}|$, $\eta$, and $\rho$. They can not depend of the
direction of $\mathbf{v}$ due to rotation invariance. We expand all of these
in powers of the departures of $|\mathbf{v}|$, $\eta$, and $\rho$ from their
values in an assumed uniform steady state $\rho(\mathbf{r},t)=\rho_{0}$,
$\eta(\mathbf{r},t)=\eta_{0}$, and $\mathbf{v}=v_{0}\hat{x}$. Here the
direction $\hat{x}$ is spontaneously selected by the system, thereby
spontaneously breaking the rotational symmetry of the underlying dynamics.

We can simplify these equations by “eliminating” the fields $\delta v_{x}$ and
$\delta\eta$, which are “fast” in the sense that their time derivatives do not
vanish in the limit of extremely slowly spatially varying fields. See Miller
and Toner  for details. This “elimination” gives a closed pair of equations
for $\rho$ and the projection $\mathbf{v}_{\perp}$ of the velocity field
perpendicular to the direction $\hat{x}$ of mean flock motion. These equations
take exactly the form of the Toner-Tu equationsToner (2012) or flocks without
chemotaxis, which are identical in form to equations (1) and (2) with the
chemical signal concentration $\eta$ set to zero, with one crucial difference:
the effective inverse compressibility (or, equivalently, the bulk modulus)
$B\equiv{\partial P_{1}\over\partial\rho}\bigg{|}_{\rho_{0}}$ in these
equations is reduced by the autochemotaxis, and can be driven negative for
sufficiently strong chemotaxis. The development of a negative compressibility
is also the mechanism which drives equilibrium liquid-gas phase separation;
however, the nature of the ensuing instability here is different, as we’ll
see.

To investigate this instability, we linearized the equations of motion, and
determined the complex eigenfrequencies of Fourier modes with wavevector
$\mathbf{q}$. We found that these modes become unstable when $B$ become
negative, for $\mathbf{q}$’s inside the $\mathbf{q}$ space lemniscate depicted
in figure 1a, whose shape is given by

$\displaystyle
q_{x}^{2}=\frac{|B|q_{\perp}^{2}-D_{\rho\perp}D_{{}_{L\perp}}q_{\perp}^{4}}{V^{2}},$
(4)

where $V$ and $D_{\rho\perp,L}$ are constant model parameters with the
dimensions of speed and diffusion constants, respectively. (All of these
parameters can be expressed in terms of those of the original equations of
motion (1-3); the detailed expressions for them are given in the ALPMiller and
Toner ). Here $q_{x}$ and $q_{\perp}$ are the magnitudes of the projection of
the wavevector $\mathbf{q}$ along, and perpendicular to, the direction
$\hat{x}$ of mean flock motion, respectively.

Note that as $B\to 0^{-}$ from below, this lemniscate becomes vanishing small,
and infinitely narrow. Specifically, its length along the $\perp$ direction
vanishes like $\sqrt{|B|}$, while its width along $q_{x}$ vanishes like $|B|$,
as the reader can verify for herself from (4). Another measure of the
narrowness of the lemniscate is the “opening angle” $\theta_{c}$ of the self-
crossing at $\mathbf{q}={\bf 0}$, which vanishes like $\sqrt{|B|}$.

The most unstable mode lies at $q_{x}=0$; i.e., wavevector exactly
perpendicular to the direction of mean flock motion $\hat{x}$.

The origin of this instability, and the reason it is confined to modes nearly
orthogonal to the direction of mean flock motion, is the competition between
convection and pressure forces.

By “convection”, we mean that, in the absence of pressure forces and
diffusion, any instantaneous local configuration of $\rho(\mathbf{r},t)$ or
$\mathbf{v}(\mathbf{r},t)$ is simply be transported along the direction of
flock motion at speeds $v_{\rho}$ and $v_{v}$ respectively. (These speeds can
be derived from equations (1\- (3), as is done explicitly in the ALPMiller and
Toner .) The two speeds, $v_{\rho}$ and $v_{v}$, are not, in general, equal to
one another, nor to the mean speed $|\langle\mathbf{v}\rangle|$ of the flock,
due to the aforementioned absence of Galilean invariance. Indeed, their
inequality plays a crucial role in limiting the instability.

To understand this, consider the evolution of a small fluctuation in which
both the density and velocity depart from their mean values in a band
perpendicular to the direction of mean flock motion, as illustrated in figure
4. This corresponds to a fluctuation with wavevectors making an angle
$\theta_{\mathbf{q}}=0$ with the mean velocity $\langle\mathbf{v}\rangle$. The
fluctuation in the density creates a pressure gradient, which, because the
inverse compressibility $B$ is negative, causes the velocity field to increase
in the direction that tends to increase the fluctuation in the density. That
is, if the density is higher in our band, the pressure forces tend to increase
the velocity into the band, making the density grow even more. The opposite
will happen if the density is lower than average in our band: pressure forces
will tend to make the velocity flow out, carrying boids out of the band,
thereby reducing the density even further.

This positive feedback mechanism is responsible for the familiar instability
of negative compressibility equilibrium systems. In our system, however, this
mechanism must compete against the aforementioned convection. For the band
direction just discussed, convection stabilizes the system by separating the
band of perturbed velocity from the band of perturbed density, because they
propagate at different speeds. This is illustrated in the figure 4a. Once the
velocity fluctuation is separated from the density fluctuation, both simply
decay away by diffusion. Therefore, this orientation of the bands is stable.

Now consider the other extreme case, in which the band runs parallel to the
direction of mean flock motion, as illustrated in figure 4b. Now the relative
displacement of the velocity and density fluctuations makes no difference: the
two bands of each still overlap, since the bands are infinitely extended in
the direction of that relative motion. Hence, the velocity fluctuation
continues to sit right on top of the density fluctuation, allowing the
pressure induced instability to occur.

We can even derive the opening instability angle $\theta_{c}$ by this
argument, as we show in the ALPMiller and Toner .

Figure 4: Heuristic illustration of why the bands always form perpendicular to
the direction of mean flock motion. See the text after equation (4) for
explanation.

We’d now like to determine the final steady state of the system when the
instability occurs. Since the instability is sharply focused in the directions
perpendicular to the bulk motion, we will seek solutions that depend only on
one Cartesian component of position, with that component perpendicular to
$\hat{x}$, the direction of mean flock motion. We’ll call this direction $y$.
Specifically, we’ll search for solutions of the form

$\mathbf{v}_{\perp}=v_{y}(y,t)\hat{y}\ \ \ ,\ \ \
\rho=\rho_{ref}+\Delta\rho(y,t)\,,$ (5)

where we expand around a “reference density” $\rho_{ref}\neq$ the mean density
$\rho_{0}$ , to be able to treat the behavior of flocks with different
$\rho_{0}$’s in a single unified framework.

Since our system is linearly unstable, we must go beyond linear order in our
expansion to find a stable state. It proves to be sufficient to expand our
equations of motion to quadratic order in the fluctuations $v_{y}(y,t)$, and
cubic order in $\Delta\rho(y,t)$. The most important non-linearities arise
from the expansion of $P_{1}(\rho)$ to third order in $\Delta\rho$:

$P_{1}=P_{0}+M\Delta\rho+w\Delta\rho^{2}+u\Delta\rho^{3}$ (6)

We’ll begin by determining the stability boundary in the $\rho$-$M$ plane. As
discussed above, and shown in the ALPMiller and Toner , this boundary is where
the inverse compressibility vanishes; that is, when

$B\equiv\left({dP_{1}\over d\rho}\right)\bigg{|}_{\rho_{0}}=0\,.$ (7)

We will define the value of $M$ that satisfies this condition to be
$M_{spinodal}$, since it is the precise analog of the spinodal line in
equilibrium phase separationChaikin and Lubensky (1995): that is, the line on
which the uniform density state becomes unstable.

Inserting our expansion (6) into the condition (7) gives the spinodal line:

$M_{\rm spinodal}=M_{c}-3u(\rho_{0}-\rho_{c})^{2}\,,$ (8)

where we’ve defined the “critical” value $M_{c}$ of $M$ and the “critical”
density as

$M_{c}\equiv{w^{2}\over 3u}\ \ \ ,\ \ \ \rho_{c}\equiv\rho_{ref}-{w\over
3u}\,.$ (9)

The critical values $M_{c}$ and $\rho_{c}$ of $M$ and $\rho$ prove to be the
analogs of the critical temperature and critical density of an equilibrium
phase separating system.

To determine the final steady state of the flock, we looked for steady state
solutions to our equations of motion that have two plateaus of constant
density, separated by a localized interface. As explained in detail in the
ALPMiller and Toner , we found that such solutions with mean density
$\rho_{0}$ exist only if $M$ is below $M_{binodal}$, where

$M_{\rm binodal}=M_{c}-u(\rho_{0}-\rho_{c})^{2}\,.$ (10)

Note that this is above the “spinodal” line (10). Hence, between the two lies
a region in which both the uniform state, and the phase separated state, are
stable.

We also found that sufficiently close to the critical point (9), the two
plateau densities can be obtained from a “common tangent construction” on the
effective “Free energy” $F$ defined by $P_{1}\equiv{dF\over d\rho}$, which
implies

$F(\rho)=F_{0}+P_{0}\Delta\rho+{M\over 2}\Delta\rho^{2}+{w\over
3}\Delta\rho^{3}+{u\over 4}\Delta\rho^{4}\,.$ (11)

That is, the two plateau densities $\rho_{\pm}$ are the densities at which a
plot of $F(\rho)$ has a common tangent. This is illustrated in figure (5a).

This analogy to equilibrium statistical mechanics only applies close to the
critical point (9). As also explained in detail in the ALPMiller and Toner ,
we found that, as we move away from the critical point, the densities are
determined by an “uncommon tangent construction”: the slope of the free energy
(11) remains the same at both coexisting densities, but the tangents to the
free energy curve at those two densities do not coincide, as illustrated in
figure (5b). In the ALPMiller and Toner , we calculate the offset of these two
tangents, thereby determining the two coexisting densities.

Figure 5:  a) Illustration of the common tangent construction. b) Illustration
of the uncommon tangent construction.

Finally, we note that since the mean local speed of the flock should depend on
the local density, which is different in the two plateaus, the speeds in each
plateau, in general, will be different as well, as shown in the second panel
of figure (1).

All of these predictions are fully borne out by numerical solution of the
time-dependent equations of motion in a periodic space, as illustrated in
figure 2, and in the animation Movie (1) in the supplemental materials.

All of our simulations started with initial conditions in which the departure
from a uniform state was small. Furthermore, we chose the mean density
$\rho_{0}$ to be the critical density $\rho_{c}$, so our simulations were the
analog of a “critical quench” right through the critical point (9).

At small times this system quickly forms into multiple well-defined plateaus,
with density and velocity profiles that agree well with our analytic solution.
Adjacent plateaus merge quickly, while distant plateaus do not merge on the
time scales of our simulations.

This slowing down is a consequence of the exponential approach to the uniform
plateau state of both the density and the velocity. It is only the overlap of
these exponential tails that leads to any interaction at all between
neighboring interfaces. As a result, the attraction between neighboring
interfaces falls off exponentially with their separation. Hence, the time for
neighboring domains of high density to merge also grows exponentially with
their separation.

We believe that this slowing down is an artifact of both the one-
dimensionality and the noiselessness of our simulations. We expect that in a
noisy system, the bands will begin to undulate, causing them to bump into
their neighbors, at which point the bands can start to merge. We expect this
mechanism to reduce the phase separation time to grow only algebraically with
system size.

In conclusion, we have shown that a dry active autochemotaxic flock, or, more
generally, any flock with sufficiently strong attractive interactions (i.e.,
strong enough to drive the inverse compressibility negative)between the
flockers, can become unstable to the formation of density bands running
parallel to the mean direction of motion of the flock. This behavior may be
connected to the formation of, e.g., unidirectional ant trails. The final
state of this anisotropic instability is a single high density plateau and a
single low density plateau. The coexisting plateau densities of this non-
equilibrium phase separation, near its “critical point”, are determined by a
common tangent construction on an effective free energy, while further from
the critical point, an uncommon tangent construction must be used, as in, e.g,
“motility-induced phase separation” (“MIPS”)Cates and Tailleur (2015). Our
analytical theory is in agreement with long-time evolution numerical
solutions, even far from the critical point.

Our work has focused on noiseless dry active autochemotaxic flocks. Two
possible extensions are first, including fluctuation effects on phase
separation, and second, developing a theory of flock interface dynamics to
understand the late stage phase separation dynamics.

##  I Acknowledgements

We thank Tristan Ursell and Justin Kittel for stimulating discussions of their
experiments on the dynamics of ant trails; those experiments and discussions
motivated our work. We also thank the MPIPKS for their hospitality and
financial support while a portion of this work was underway. Finally, we thank
Ants for their interesting behavior.

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