  * January 2021

# Diffusion and escape from polygonal channels: extreme values and geometric
effects

Jordan Orchard1, Lamberto Rondoni2,3, Carlos Mejía-Monasterio4, Federico
Frascoli1 1 Department of Mathematics, School of Science, Swinburne University
of Technology, H38, PO Box 218, Hawthorn, Victoria 3122, Australia  2
Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli
Abruzzi 24, I-10129 Torino, Italy  3 Istituto Nazionale di Fisica Nucleare,
Sezione di Torino, Turin, Italy,  4 School of Agricultural, Food and
Biosystems Engineering, Technical University of Madrid, Av. Complutense s/n,
28040 Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>,,
,

###### Abstract

Polygonal billiards are an example of pseudo-chaotic dynamics, a combination
of integrable evolution and sudden jumps due to conical singular points that
arise from the corners of the polygons.  Such pseudo-chaotic behaviour, often
characterised by an algebraic separation of nearby trajectories, is believed
to be linked to the wild dependence that particle transport has on the fine
details of the billiard table. Here we address this relation through a
detailed numerical study of the statistics of displacement in a family of
polygonal channel billiards with parallel walls. We show that transport is
characterised by strong anomalous diffusion, with a mean square displacement
that scales in time faster than linear, and with a probability density of the
displacement exhibiting exponential tails and ballistic fronts. In channels of
finite length the distribution of first-passage times is characterised by fat
tails, with a mean first-passage time that diverges when the aperture angle is
rational. These findings have non trivial consequences for a variety of
experiments.

††: JSTAT

Keywords: Polygonal billiards, Anomalous diffusion, Strong anomalous
diffusion, First-passage times, Survival probability.

##  1 Introduction

The transport properties of assemblies of highly confined particles in
nanochannels greatly differ from those of macroscopic fluid systems. In
confined geometries, the interactions between the particles and the channel
walls occur with frequency and magnitude that are comparable (when not
greater) than those associated with particle-particle interactions, hindering
the realisation of Local Thermodynamic Equilibrium (LTE) on which the standard
thermodynamic laws like Fick’s law of diffusion are based [1, 2]. When
confinement is particularly pronounced, the conditions for the validity of
kinetic theory fail as well. For instance, macroscopic containers may keep a
gas in different equilibrium states even if they are connected by channels
that allow the flow of particles but prevent the interactions among them due
to geometry, like in Knudsen gases [3, 4]. Geometry effects due to confinement
may also prevent the decay of microscopic correlations that is required for
locality to hold, allowing hydrodynamic interactions that commonly lead to an
enhancement in the fluid flow [5, 6, 7, 8, 9].

When the dimensions of the container are comparable to the molecular
characteristic scales, transport drastically slows down, with single-file
diffusion being a prototypical example [10, 11, 12]. There are other cases
where instead densely packed systems yield a strong enhancement of the rate of
diffusion [13, 14], and where anomalies arise as the result of collective
mediated interactions due to confinement, external driving and macromolecular
crowding [15, 16]. Colloidal systems out of equilibrium and small biological
systems [17, 18] share similar features.

A prototypical example of transport in confined geometries is that of
diffusion through narrow channels. Understanding whether transport through
nano channels is diffusive or anomalous has gained increasing interest as it
would allow the control of a wide variety of phenomena and applications, such
as ion transport across the cell membrane [19], drug delivery [20], nanopore
sensing [21], ion transport control for energy harvesting [22], water flow in
carbon nanotubes [23], diffusion of waves in corrugated channels [24], single
photons in optic fibers [25] among many others.

Denoting by $x(t)$ the position of a tagged particle at time $t$, transport of
matter is customarily defined in terms of the fluctuations of the displacement
$\Delta x=x(t)-x(0)$ of the particles constituting the fluid. Macroscopic
transport within stationary media is commonly diffusive, and the displacement
is described by the diffusion equation, whose solution for a localised initial
density is a time dependent Gaussian distribution

$P(\Delta x,t)=1/\sqrt{2\pi dDt}\exp\left(-\Delta x^{2}/2dDt\right),$ (1)

where $d$ is the spatial dimension, and $D$ is the diffusion coefficient. The
size of the fluctuations of the particles’ displacement is then given by the
variance of the displacement, i.e. the Mean Square Displacement (MSD). In the
case of Eq. (1), the MSD is given by:

$\langle(x(t)-x(0))^{2}\rangle=2dDt\ .$ (2)

If the MSD is not described by Eq. (2), then transport is called anomalous.
One of the aims of the present study is to characterise the anomalous
transport associated with a specific family of billiards and discuss the role
of geometry in a number of important properties that describe the billiards’
dynamics.

In this paper we revisit the problem of particle transport in polygonal
billiard channels. Billiards constitute an attractive basis to study a broad
range of fundamental problems in statistical mechanics, in particular
transport in confined geometries [26, 27]. Mathematically, a billiard is a
$d$-dimensional dynamical system, representing one point particle that moves
freely inside a compact domain, the so-called table
$\mathcal{B}\subset\mathbb{R}^{d}$, and that is specularly reflected at the
boundary $\partial\mathcal{B}$ of $\mathcal{B}$. The boundary is required to
be piecewise smooth, except in a countable number of points. Multiple
extensions of this simple scenario have been considered, including interacting
particles [26, 28], time-dependent boundaries [29], thermostated dynamics [30,
31], stochastic boundaries [32], and boundary with holes [33].

Normal diffusive transport as described by (2) is usually encountered in
dispersing and semi-dispersing billiards [34, 35], whose dynamics is
characterised by positive Lyapunov exponents. In constrast, polygonal
billiards have zero Lyapunov exponents. The separation of nearby trajectories,
which is always slower than exponential, is due to the corners of the polygon
associated to the existence of singular conical points in the billiard flow.
The question on whether diffusive transport can be obtained in polygonal
channels was  thoroughly analysed in several works [7, 8, 9, 36, 37].
Transport was found to range from diffusive to ballistic, with abrupt and
unpredictable variations of its details under variations of the geometric
parameters.

We focus on those fine details for a particular family of polygonal billiard
channels with parallel boundaries defined in Section 2. The statistics of the
particle displacement is analysed in Section 3. Later in Section 4, we analyse
the transmission of particles through channels of finite length and discuss
the statistics of survival probabilities and first-passage times to the
channel’s exits. Finally in Section 5 we report our conclusions.  Our main
results are on the interplay between ballistic fronts and the wild dependence
that transport properties have on the geometric parameters. Additionally, we
show that the distribution of first passage times for channels of finite width
is broad, with a mean first passage time that diverges for simple rational
billiards.

##  2 Polygonal billiard channels

We consider periodic polygonal billiard channels, made of identical copies of
a fundamental cell with parallel walls, vertically shifted of a distance $d$
with respect to each other, as shown in Fig. 1. The fact that the walls are
parallel is an important feature of our systems and will have interesting
consequences, as will be seen shortly.

For all geometries, the horizontal width of the cell is set to $2\delta x=1$.
Therefore, two parameters suffice to specify the shape of the channel: the
vertical shift $d>0$ and the aperture angle $\alpha\in(0,\pi)$. When the
channel width $d>\delta x\cot(\alpha/2)$ the horizon is infinite, meaning that
a particle trajectory can move for arbitrarily long distances without hitting
the channel boundaries. Inside the channel, point particles move with constant
speed along straight lines until they elastically collide with the boundary,
where their trajectories are specularly reflected.

As noted earlier, the dynamics of polygonal billiards have zero Lyapunov
exponents. Whether this should imply faster or slower transport is not
obvious. In the case of dispersive billiards, which have positive Lyapunov
exponents, a well-understood example is the Lorentz gas, which was intensively
investigated  a few decades ago. In that system, transport can be understood
in terms of unstable periodic orbits [38, 39], which subdivide in confined and
ballistic orbits, the latter being periodic only modulo the periodic lattice
spacing. Confined orbits do not contribute to transport, whereas ballistic
ones do: their combined contribution results in normal diffusion if the
horizon is finite. The strict hyperbolicity of the system causes generic
trajectories to always come close to some periodic one, but only for a short
time in the vicinity of a given periodic orbit, thus efficiently mixing
velocities.

Figure 1: Polygonal channel made of (three) identical copies. The geometry of
the fundamental cell is determined by the aperture angle $\alpha$ and the cell
width $d$, with a cell length fixed to $2\delta x=1$. The boundary height is
given by $\delta y=\delta x\cot\alpha/2$. Setting the central cell at the
origin, a channel made of $L$ cells extends to $x=\pm\mathcal{L}$, with
$\mathcal{L}=\lfloor\frac{L}{2}\rfloor+\frac{1}{2}$, and $\lfloor\cdot\rfloor$
being the floor function.

Contrary to the  Lorentz gas, intuition on polygonal channels’ dynamics is not
straightforward, one reason being that parallel walls cause families of
bouncing ball modes to be present. These are periodic trajectories that
collide with the same pair of parallel wall segments, giving  a vanishing
contribution to transport. Differently from the chaotic case, escape from a
narrow neighbourhood of a bouncing ball orbit is very slow and jeopardises the
mixing of velocities.  Only near the polygon’s corners trajectories are
separated, and velocities substantially change. This implies that trajectories
initially in the neighbourhood of a bouncing ball orbit, move slowly away from
that orbit, hence, at least initially, they give a small contribution to the
transport properties. This behaviour is more prominent in rational polygons,
where the number of possible velocity orientations is finite, e.g. just four
for $\alpha=\pi/2$. A similar phenomenon, however, occurs for the ballistic
orbits, i.e. those that are periodic only up to periodic lattice vector
translation. These trajectories are very fast, and do not quickly decorrelate
from orbits that start in their vicinity, which overall results in a large
contribution to transport. Which of the two effects dominates transport and
how it is related to the geometry of the system is a hard open question, which
we partly unravel in this paper.

What one observes is that the velocity vector explores a larger set of
discrete orientations in irrational polygons than in rational ones. With
parallel walls, the set of orientations of the particle velocity does not
contract or expand upon successive collisions with the billiard boundary, as
they instead do with non parallel walls. This leads to a propensity of sets of
trajectories to persist along highly correlated unidirectional flights. While
it is not clear whether the set of orientations is dense, the exploration of
possible orientations is instead extremely slow [40, 7].

This polynomial separation of trajectories inspired a simple map, the Slicer
Map [41], to understand the properties of transport. The slicer map exhibits
strong anomalous diffusion [42] and, as recently shown, the map and polygonal
billiard channels belong to a universality class of dynamics asymptotically
dominated by a ballistic front [37]. It is important to note that trajectories
travelling arbitrarily long distances before reversing do act effectively as
ballistic fronts. Interestingly, such ballistic fronts do not require an
infinite horizon but exist with finite horizon as in the case of periodic
dispersive billiards. But, unlike that case, they are only weakly unstable
because collisions with the boundary are not defocusing: the result is an
overall poorly mixing dynamics that is much harder to characterise and
globally more unpredictable than chaotic dynamics [7].

One of our main goals in this work is to study the statistics of particle
displacement and its relation with the two free parameters $d$ and $\alpha$
that determine the billiard’s geometry. There are two main characteristics of
the geometry that have a strong impact on transport:  whether the angle
$\alpha$ is a rational or irrational multiple of $\pi$ and whether the horizon
is finite or infinite. Hereafter, when referring to $\alpha$ as being rational
or irrational, we will mean that $\alpha/\pi$ is rational or irrational.
Unless otherwise specified, for each rational angle
$\alpha_{\mathrm{R}}=\frac{p}{q}\pi$, with $p,q\in\mathbb{Z}$ we will consider
its irrational counterpart defined as
$\alpha_{\mathrm{I}}=\left(\varphi-\frac{3}{5}\right)\alpha_{\mathrm{R}}$
where $\varphi=\left(\sqrt{5}+1\right)/2$, so that $\alpha_{\mathrm{R}}$ and
$\alpha_{\mathrm{I}}$ differ by less than $2\%$.

Furthermore and unless otherwise specified, we consider only two different
channel widths: $d=2\delta y$ for the infinite horizon and $d=\delta y/2$ for
the finite horizon case, where $\delta y=\delta x\cot\alpha/2$ is the
geometric width of the polygonal boundary (see Fig. 1). When $\delta y=1$ the
$y$-coordinate of the lower corners located at the integer values $x=n$
coincide with that of the upper corners at $x=n+1/2$ and the horizon is
critical.

##  3 Particle displacement

In this Section we study the statistics of the displacement $\Delta
x(t)=x(t)-x(0)$ for noninteracting particles moving in a parallel polygonal
channel of infinite length. At the initial time, $t=0$, the particles
positions $(x(0),y(0))$ are uniformly distributed inside the central cell, and
the velocity vectors all have unit magnitude, with their orientation forming
an angle with the horizontal direction that is uniformly distributed over
$[0,2\pi)$.

###  3.1 Probability distribution

We focus first on the probability distribution function of the particle
displacement $P(\Delta x,t)dt$, which is the probability that one particle is
found at a distance $\Delta x$ from its initial position and at a time between
$t$ and $t+dt$. We have numerically reconstructed $P(\Delta x,t)$, by
computing the displacement $\Delta x(t)$ at different times $t$ for an
ensemble of particles. The shape of the probability density $P(\Delta x,t)$,
and thus the properties of the particle transport in the channel, depend on
the two free parameters $d$ and $\alpha$.

Transport is called scale invariant when the probability density scales in
time as:

$P(\Delta x,t)=\frac{1}{t^{\gamma}}P\left(\frac{\Delta x}{t^{\gamma}}\right)$
(3)

with $\gamma$ a positive fixed parameter. Here $t^{-\gamma}$ defines the
proper scaling length of the process. For processes with constant speed,
$\gamma$ must take values in $[0,1]$. Then, a generalised diffusion
coefficient can be defined as:

$\mathcal{D}:=\lim_{t\to\infty}\frac{\mathrm{Var}\left[\Delta
x(t)\right]}{2dt^{2\gamma}}\ ,\quad\mathrm{Var}\left[\Delta
x(t)\right]=\langle\Delta x^{2}(t)\rangle-\langle\Delta x(t)\rangle^{2}.$ (4)

Normal diffusion corresponds to $\gamma=1/2$, sub-diffusion to $0<\gamma<1/2$
and super-diffusion to $1/2<\gamma<1$. For $\gamma=1$ the transport is called
ballistic. As a consequence of scale invariance, the moments of the
displacement obey the following relation:

$\displaystyle\left\langle|\Delta x(t)|^{p}\right\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}|\Delta x|^{p}P(\Delta
x,t)\mathrm{d}(\Delta x)$ (5) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}|\Delta x|^{p}P\left(\frac{\Delta
x}{t^{\gamma}}\right)\mathrm{d}\left(\frac{\Delta x}{t^{\gamma}}\right)$
$\displaystyle=$ $\displaystyle
t^{p\gamma}\int_{0}^{\infty}|u|^{p}P(u)\mathrm{d}u\ ,$

where $u=\Delta x/t^{\gamma}$. Assuming that the distribution $P(u)$ is not
broad, so that all its moments exist, the integral in the last line of Eq. (5)
is finite and we obtain

$\left\langle|\Delta x(t)|^{p}\right\rangle\sim t^{p\gamma}\ .$ (6)

In particular, the mean-square displacement scales in time as
$\left\langle\left(\Delta x(t)\right)^{2}\right\rangle\sim t^{2\gamma}$.

In order to analyse the scaling of the probability distribution function of
the particles’ displacement, we have numerically followed an ensemble of
$10^{6}$ trajectories, for a time $t=10^{5}$, for four different polygonal
channels: a rational polygon with $\alpha_{R}={\pi}/{2}$ with finite
$(d=\delta y/2)$ and infinite $(d=2\delta y)$ horizons, as well as an
irrational polygon with
$\alpha_{I}=\frac{\pi}{2}\left(\varphi-\frac{3}{5}\right)$, for the same wall
separation $d$. In Fig. 2 we show the probability distribution of the particle
displacement computed at six different times:
$t=\\{1000,10000,25000,50000,75000,100000\\}$. Within the fairly large space
and time scales that we have explored, we found that $P(\Delta x,t)$ is indeed
not broad, and has exponential or multi exponential tails, that largely
dominate the distribution. Moreover, $P(\Delta x,t)$ is also scale-invariant
and, interestingly, the scaling parameter $\gamma$ substantially depends only
on the aperture angle $\alpha$, whereas its dependence on $d$ appears to be
negligible.  We find that the scaled distributions do asymptotically collapse
into the same master function with $\gamma=\gamma_{R}\approx{5}/{6}$ and
$\gamma=\gamma_{I}\approx 0.91$ for the rational and irrational polygons
defined above. In the following section, we will consider the dependence on
the aperture angle in more detail.

At finite times and large displacements, we observe that the distribution
contains structures that are characteristic of ballistic fronts, with peaks of
higher probability that move away  from the center as time elapses.
Interestingly, these ballistic fronts are present for all four cases,
regardless of the horizon being finite or infinite. The position of the
ballistic fronts is indicated by arrows in the upper left panel of Fig. 2.

The existence of ballistic fronts means that the scaling of Eq. (3) has to be
complemented by a second contribution $\mathcal{F}(\Delta x,t)$ such that

$\mathcal{F}(\Delta x,t)=\frac{1}{t}\mathcal{F}\left(\frac{\Delta
x}{t}\right)$ (7)

satisfies a ballistic scaling (see _e.g._ Ref. [43]). Clearly,
$\mathcal{F}(\Delta x,t)$ is a sub leading contribution describing the
behaviour at large displacements, _i.e._ $t^{\gamma}\ll\Delta x<t$, with
appropriate scaling velocities. The collapse described by the leading scale-
invariant contribution of Eq. (3) is then observed for displacements $\Delta
x/t^{\gamma}$ smaller than the scaled position of the ballistic front, as
expected.

Furthermore, we also observe that the bulk of the distributions are different:
while for the rational polygons the centers of the distributions are
spikelike, for the irrational polygons they are rounded. We notice that the
bulk of the distribution, which in scaled units is defined by $\Delta
x/t^{\gamma}<1$, has the same dependence as its tails. These quantitative and
qualitative differences between rational and irrational polygons have been
found for all geometries considered here.

Figure 2: Probability density of the particle displacement $P(\Delta x,t)$
multiplied by $t^{\gamma}$, plotted as a function of scaled variable $\Delta
x/t^{\gamma}$ obtained from $10^{6}$ particle trajectories. Each histogram
corresponds to $t=1000$ (black), $10000$ (red), $25000$ (green), $50000$
(blue), $75000$ (orange), and $100000$ (grey). The probability density is
shown for four different billiards as indicated by the labels:  rational
polygon $\alpha_{R}={\pi}/{2}$ and the corresponding irrational polygon
$\alpha_{I}$, finite horizon with $d={\delta y}/{2}$ and infinite horizon for
$d=2\delta y$. The scaling used was $\gamma_{R}=5/6$ for the rational polygon
and $\gamma_{I}=0.91$ for the irrational polygon. The arrows in the upper left
panel indicate the position of the ballistic front at different times (see
Fig. 3).

Looking at the evolution of the distributions one intuitively realises that in
polygons with infinite horizon the distribution spreads more rapidly than
those with finite horizon, because the progression in time is essentially
determined by the speed of the ballistic front.

Figure 3: Ballistic scaling of the probability density of the particle
displacement $P(\Delta x,t)$ shown in Fig. 2. The dashed vertical lines are a
guide to the eye, indicating the value of the speed of the ballistic front
$v_{bal}=\Delta x/t$. The arrow in the upper left panel indicates the non-
collisional ballistic front with unit speed.

In Fig. 3 we plot the  probability density of the particle displacement using
a ballistic scaling, obtained with an exponent $\gamma=1$. The scaling
variable $\Delta x/t$ corresponds in this case to the speed of the front that
we denote as $v_{\mathrm{bal}}$: this is useful to distinguish the features of
the distribution that scale ballistically.

Let us discuss first the rational polygon (left column of Fig. 3). In scaled
units $\Delta x/t$ the position of the different peaks indicated by the arrows
in Fig. 2 do collapse (as indicated in Fig. 3 by the vertical dashed lines),
thus confirming that they correspond to a ballistic front. For the rational
polygon with infinite horizon, the speed of the ballistic front is
approximately $v_{\mathrm{bal}}\approx 2/3$. With finite horizon, the
existence of a ballistic front is also confirmed, with a different speed
$v_{\mathrm{bal}}\approx 1/2$. Moreover, the fastest widening trajectories
with finite horizon correspond precisely to the ballistic front, while in the
case of infinite horizon there exist trajectories that spread faster: those
that undergo no collisions and have speed $\Delta x/t=1$ (indicated by the
arrow in the upper left panel of Fig. 3). Without acceleration, no trajectory
can be faster than this free-flight front, which arise as soon as the
channel’s horizon is infinite and independently of the angle $\alpha$. In
addition, the fraction of such fastest trajectories decreases with $t$, since
it approaches the set of exactly horizontal trajectories.

Turning our attention to the irrational polygon, we do indeed see the free-
flight ballistic front when the horizon is infinite, in addition to the slower
ballistic front. The latter varies with the geometry of the billiard, but it
exists whether the horizon is finite or infinite. We speculate that this
ballistic front is due to the trajectories that initially are in a close
neighbourhood of the shortest periodic orbit of the billiard. As such, other
fronts are expected to exist and indeed we observe certain features of the
probability distribution that collapse in position under the ballistic
scaling. This may also explain the different velocity of the front for
different geometries of the billiard. Given the differences between rational
and irrational polygons, it comes as a surprise that the front speed seems to
depend strongly on the channel’s width, but apparently only mildly on whether
the polygon is rational or not.

To study this further, we have numerically evaluated the speed of the
ballistic front for the rational $\alpha=\pi/2$ polygonal channel and its
irrational counterpart, for different widths. The results are shown in Fig. 4,
where the dark blue squares refer to the rational polygon, and the light
yellow circles to the irrational one. Opening the horizon leads to a sharp
transition in the behaviour of the speed of the ballistic front: for finite
horizon $d<\delta y$, the speed of the ballistic front does saturate to
approximately $1/2$. When the horizon opens, the speed $v_{\mathrm{bal}}$
grows monotonically toward its upper bound $v_{\mathrm{bal}}=1$. From a fit to
power-law, we obtain that the speed approaches unity as

$v_{\mathrm{bal}}=1-0.52\left(\frac{d}{\delta y}\right)^{-2/3}\ ,\ \
\mathrm{for}\quad\frac{d}{\delta y}>1\ ,$ (8)

as shown by the solid curve in Fig. 4. As a final remark we note that the
ballistic front of the polygonal channel is mildly affected by whether the
angle is rational or irrational. However, recalling that
$\left(\varphi-3/5\right)\approx 1$, and that the angles compared in Fig. 4
are very similar in value, we cannot discard the possibility that
$v_{\mathrm{bal}}$ could depend on the value of the aperture angle in a
stronger fashion. Moreover, conjecturing that the speed of the ballistic front
depends on the length of the simplest periodic orbit, $v_{\mathrm{bal}}$
should depend evenly on all the parameters of the geometry of the billiard.

Figure 4: Velocity of the ballistic front as a function of the width of the
channel $d$ in units of $\delta y$ for the irrational angle
$\alpha=\frac{\pi}{2}\left(\phi-\frac{3}{5}\right)$ (light yellow circles) and
the rational angle $\alpha=\frac{\pi}{2}$ (dark blue squares). The dashed line
indicates the value $v_{\mathrm{bal}}=1/2$. The solid curve corresponds to the
function $v_{\mathrm{bal}}=1-0.52(d/\delta y)^{-2/3}$.

###  3.2 Mean-square displacement

In the previous section we have analysed the properties of the probability
distribution of the particle displacement and established that transport is
anomalous, with $P(\Delta x,t)$ having exponential tails and a super-diffusive
scaling, with an exponent that depends on whether the angle is rational or
not. In this section we further explore the dependence of transport on the
aperture angle $\alpha$, focussing on the mean-square displacement. For a
scale-invariant probability distribution, the mean-square displacement scales
asymptotically as a power-law of time $\langle\left(\Delta
x(t)\right)^{2}\rangle\sim t^{\eta}$, where the exponent $\eta$ is related to
the scaling exponent of the probability distribution as $\eta=2\gamma$ (see
Eq. (11)).

Figure 5: MSD scalings $\eta$ for rational (solid circles) and irrational
billiards (open squares). Simulations involved $10^{6}$ trajectories of length
$t=10^{6}$ for irrational $\alpha$’s and $t=10^{5}$ for the rational
$\alpha$’s. A finite horizon channel with $d=\delta y/2$ was used for all
cases, but a similar behaviour occurs for infinite horizons. The horizontal
dashed lines indicate the expected values for the MSD for $\alpha=\pi/2$ from
the corresponding scalings $\gamma_{\mathrm{R}}$ and $\gamma_{\mathrm{I}}$ of
the probability distribution function. Statistical errors are of the same
magnitude as the symbols’ size.

We have numerically computed the mean-square particle displacement for
polygonal channels with a finite horizon, $d=\delta y/2$, and different
rational and irrational values of the angle $\alpha$. The scaling exponent
$\eta$ is obtained from a fit to power-law of $\langle x^{2}(t)\rangle$ in the
large time limit. The angles were chosen as rational values between $\pi/12$
and $11\pi/12$.

We show in Fig. 5 the power exponent $\eta$ of the mean-square particle
displacement for different angles $\alpha$, in blue crosses rational polygons
and in red circles their irrational counterparts. For $\alpha=\pi/2$ and
$\alpha=\frac{\pi}{2}\left(\varphi-\frac{3}{5}\right)$ we obtain
$\eta=1.662\pm 0.006$ and $\eta=1.825\pm 0.006$ respectively, in agreement
with the values of $\gamma_{R}$ and $\gamma_{I}$ obtained in the previous
section.

Moreover, for all the parameters considered, the transport in parallel walls
polygonal channels is found to be super-diffusive. Restricted to rational
polygons, $\eta$ seems to monotonically increase for $\alpha\geq\pi/3$. At
smaller angles the power scaling $\eta$ concentrates at values around $1.78$.
We recall that for $\alpha=\pi$ the corners of the polygonal channel disappear
and the channel becomes rectilinear, yielding $\eta=2$. Therefore, we expect
that $\lim_{\alpha\rightarrow\pi}\eta=2$ irrespective if the angle is rational
or not, as confirmed by Fig. 5. In the irrational polygons, transport appears
to be more stable than in the rational polygons, with a power scaling
concentrated inside a small interval around $\eta\approx 1.84$.

As a final remark we notice that the numerical convergence of the MSD
statistics of the irrational polygons for small aperture angles
$\alpha\lesssim\pi/4$, is drastically slower than for larger angles. The
properties of the convergence to the asymptotic transport and how they are
affected by the geometry deserves further investigation.

###  3.3 Higher order moments

To complete the picture we have numerically computed higher order moments of
the particle displacement $\langle\left|\Delta x(t)\right|^{p}\rangle$.

In Section 3.1 we found that the probability distribution of the particle
displacement is scale invariant (see Eq. (3)), with a scaling $x/t^{\gamma}$
that depends on the geometry of the billiard. The scale-invariance of
$P(\Delta x,t)$ exhibiting exponential tails does hold for displacements that
are shorter than the position of the ballistic front, namely in scaled units
for

$\frac{\Delta x}{t^{\gamma}}<v_{\mathrm{bal}}\ t^{1-\gamma}\ .$ (9)

Different scales determining the evolution of $P(\Delta x,t)$ commonly lead to
strong anomalous diffusion characterised by a non-homogeneous scaling of the
moments of the displacement with time. Strong anomalous diffusion, first
described in Ref. [42], is characterised by the relation:

$\langle\left|\Delta x(t)\right|^{p}\rangle\sim t^{\nu(p)}\ ,$ (10)

where the scaling exponent $\nu(p)$ is not linear in the moment’s order $p$.

Strong anomalous diffusion was observed in Ref. [9] for general polygonal
channels with finite and infinite horizon. There, the authors found that the
moments of displacement can be subdivided in two branches: a scaling
$\langle\left|\Delta x(t)\right|^{p}\rangle\sim t^{\nu_{\mathrm{low}}}$ for
low order moments $p<p^{\star}$, and $\langle\left|\Delta
x(t)\right|^{p}\rangle\sim t^{\nu_{\mathrm{high}}}$ for $p>p^{\star}$. This
piecewise behaviour is characteristic of systems with two different dynamic
scales [42] and was recently shown [37] that polygonal channels with infinite
horizon belong to a broader class of universality of transport that is
asymptotically dominated by ballistic fronts.

Figure 6: Spectrum of the moments of displacement of Eq. (11) for the same
channels as in Fig. 2, rational polygon with $\alpha={\pi}/{2}$, irrational
polygon with $\alpha=\frac{\pi}{2}\left(\phi-\frac{3}{5}\right)$, and finite
horizon with $d=\frac{\delta y}{2}$ (dark blue circles), and infinite horizon
for $d=2\delta y$ (light yellow squares). In the left panel the lines are
$5p/6$ (dashed), and $p-1/2$, $p-5/6$ (in solid). In the right panel the lines
are $0.91p$ (dashed), and $p-0.36$, $p-0.54$ (in solid). Solid lines are shown
with the same colour as the corresponding symbols.

Here we find that, independently of the geometry of the polygon, transport is
strongly anomalous, with a piecewise linear spectrum characterised by
superdiffusion for moments of order $p\leq p^{\star}$ and ballistic branch for
higher moments. This is shown in Fig. 6, where we plot the scaling exponent

${\color[rgb]{0,0,0}\nu(p)=\lim_{t\rightarrow\infty}\frac{\log\langle\left|\Delta
x(t)\right|^{p}\rangle}{\log t}}\ $ (11)

for the same polygonal channels investigated in section 3.1. Consistently with
the previous results, low order moments of the displacement with $p<p^{\star}$
and $p^{\star}>2$, scale as

$\nu(p)=\left\\{\begin{array}[]{ll}\frac{5}{6}\ p&\textrm{for the rational
polygon}\\\ \\\ 0.91\ p&\textrm{for the irrational polygon}\end{array}\right.\
$ (12)

for arbitrary horizons. Higher order moments scale ballistically, with a
threshold order $p^{\star}$ that depends on the geometry, as shown in Table 1.
We conjecture that this behaviour is qualitatively observed for arbitrary
values of $\alpha$ and of $d$. Moreover, Fig. 5 shows that for $\alpha<\pi$
the MSD scales slower than ballistic, which means that the threshold moment
order is $p^{\star}>2$ independently of the geometry.

polygon | horizon | $\alpha$ | $d$ | $p^{\star}$ | $\nu(p)$
---|---|---|---|---|---
rational | finite | $\frac{\pi}{2}$ | $\delta y/2$ | $3$ | $p-\frac{1}{2}$
rational | infinite | $\frac{\pi}{2}$ | $2\delta y$ | $5$ | $p-\frac{5}{6}$
irrational | finite | $\left(\varphi-\frac{3}{5}\right)\frac{\pi}{2}$ | $\delta y/2$ | $4$ | $p-0.36$
irrational | infinite | $\left(\varphi-\frac{3}{5}\right)\frac{\pi}{2}$ | $2\delta y$ | $6$ | $p-0.54$
Table 1: Asymptotic ballistic spectrum of the moments of the displacement
$\nu(p)$ for $p>p^{\star}$, for the four combinations investigated.

##  4 Transmission through polygonal channels of finite length

In the previous section we have shown that the transport across parallel walls
polygonal channels is super-diffusive and dominated asymptotically by a
(possibly more than one) ballistic front, at different speeds. In this section
we investigate how transport is determined in channels of finite length for
which the asymptotic regime is not available.

We have explored channels of finite length with ends located at positions
$x=\pm\mathcal{L}=\left(\lfloor\frac{L}{2}\rfloor+\delta x\right)$, and act as
absorbing boundaries. We recall that $L\geq 1$ is the number of polygonal
cells composing the channel, $\lfloor\cdot\rfloor$ refers to the greatest
integer that is less than or equal to the argument, and $\delta x=1/2$. When a
particle reaches one of the two ends, it escapes the channel and is removed
from the dynamics.

As particles have constant speed, finite size effects will appear effectively
at times larger than $\tau^{\star}$, the shortest time at which particles
escape from the channel. If the horizon is infinite then
$\tau^{\star}=\lfloor\frac{L}{2}\rfloor$ exactly. At finite horizon, the
shortest absorption time is determined by the fastest ballistic front for the
given geometry, and is approximatively
$\tau^{\star}=\mathcal{L}/v_{\mathrm{bal}}$.

For finite channels, the moments of the particle displacement can be defined
at time $t$ as a conditional average over the particles that are still inside
the channel. For a total number $M$ of particles we define

$\langle|\Delta
x(t)|^{p}\rangle=\frac{1}{\mathcal{N}_{t}}\sum_{j=1}^{M}\chi_{j}(t)\left|\Delta
x_{j}(t)\right|^{p}\ ,$ (13)

where the indicator function $\chi_{j}(t)=1$ if by time $t$ the $j$-th
particle has not yet been absorbed, and $\chi_{j}(t)=0$ otherwise. Then

$\mathcal{N}_{t}=\sum_{j=1}^{M}\chi_{j}(t)\ $ (14)

is the total number of particles that by time $t$ have not escaped from the
channel.

We have numerically computed Eq. (13) for a channel composed of $L=2001$
rational cells with $\alpha=\pi/2$ and finite horizon $d=\delta y/2$. The
first four moments are shown in Fig. 7, indicating a growth similar to that of
the infinite channel (shown in dotted lines) upon saturation at
$t=\tau^{\star}$.

The saturation arises from the definition of the conditional sum in Eq. (13),
and gives quantitative bounds for the moments, in particular for the mean-
square displacement. In a channel of semi length $\mathcal{L}$, each particle
contributes at most $|\mathcal{L}-x(0)|^{2}$ to the MSD: since the remaining
particles in the channel give a smaller contribution, the MSD saturation value
is of order $O(\mathcal{L}^{2})$.

Figure 7: First four moments of the particle displacement $p=1,\ldots,4$ (from
dark blue to light yellow), for a channel composed of $L=2001$ cells,
$\alpha=\pi/2$ and $d=\delta y/2$ as per Eq. (13) (solid curves). The dashed
curves correspond to the moments for the channel with infinite length. The
dotted vertical line indicates the value of the shortest first-passage time
$t=\tau^{\star}$.

###  4.1 Statistics of first-passage times

The time $\tau^{\star}$ corresponds to the shortest time at which a particle
escapes the channel. In this section we explore the properties of this escape
process in terms of the statistics of the first-passage time.

The concept of first-passage underlies a number of diverse stochastic
processes, when the relevant information is represented by the first time at
which the value of a random variable reaches a preset value [44]. The escape
time of a given trajectory $x(t)$ can be seen as a first-passage time defined
by:

$\tau=\inf\\{t:|x(t)|=\mathcal{L}\\}\ ,$ (15)

where the $\inf$ is defined over the set of randomly initialised trajectories.

Consider a large set of trajectories parametrised by their initial position
$\\{(x_{j}(0),y_{j}(0))\\}$ and initial velocity vector
$\\{\vec{v}_{j}\mathrel{}:\mathrel{}|\vec{v}_{j}|=1\\}$, and denote by
$\tau_{j}$ the time $t$ at which the $j$-th particle exits from one of the two
channel’s ends. Then borrowing the concept from stochastic dynamics, the
normalised distribution $\Psi(\tau)$ of the escape times $\\{\tau_{j}\\}$ can
be called the First-Passage Time Distribution (FPTD). For a randomly chosen
initial condition, $\Psi(\tau)\mathrm{d}\tau$ is the odds that such a
trajectory will escape at a time between $\tau$ and $\tau+\mathrm{d}\tau$. The
mean escape time is then the Mean First-Passage Time (MFPT) defined as

$\langle\tau\rangle=\int_{0}^{\infty}\tau\Psi(\tau)\mathrm{d}\tau\ .$ (16)

An interesting question is whether or not the Mean First-Passage Time (MFPT)
characterises the typical escape time. As we learned in the previous section,
the spreading of the trajectories occurs in different fashions, of which only
part are ballistic. Therefore, it is not a priori clear if the MFPT is enough
to describe the escaping process.

We have numerically computed the distribution of the first-passage time in
channels of finite length and the same geometry as those studied in Section
3.1. As for the infinite channel, a large set of trajectories are randomly set
in the central polygon cell, and for each of them we measure the time $\tau$
at which each trajectory reaches the channel ends. The histogram of this set
of times is then used to numerically approximate the FPTD. The results are
shown in Fig. 8 as solid dark blue curves.

At short times, the distribution $\Psi(\tau)$ has a cutoff at
$\tau=\tau^{\star}$, namely $\tau^{\star}=\lfloor\frac{L}{2}\rfloor$
corresponding to the shortest free-flight trajectory in channels with infinite
horizon, while for channels with finite horizon
$\tau^{\star}=\mathcal{L}/v_{\mathrm{bal}}$, where $v_{\mathrm{bal}}$ is the
speed of the fastest ballistic front. At later times, we observe an
oscillating behaviour due to the escape of other slower ballistic fronts. More
importantly, we note that $\Psi(\tau)$ decays asymptotically as a power-law of
$\tau$, meaning that the FPTD is broad and thus only a finite number of its
lowest moments exist.

It is physically reasonable to describe our numerical results for $\Psi(\tau)$
with the distribution

$\Psi(\tau)=\frac{\tau_{0}^{\mu}}{\Gamma(\mu)}\frac{\mathrm{e}^{-\tau_{0}/\tau}}{\tau^{1+\mu}}$
(17)

where $\Gamma(\cdot)$ is the gamma function, $\tau_{0}$ sets the cutoff of the
distribution at short absorption times, and $\mu\geq 0$ is the so-called
persistence exponent [45]. At short times, this distribution has a exponential
cutoff that is a consequence of finite speed processes and, as our present
results indicate, it is asymptotically dominated by a power-law tail. Note
that the distribution in Eq. (17) is exact for a one-dimensional Brownian
motion in the semi-infinite line, with $\mu=1/2$ and $\tau_{0}=x_{0}^{2}/2D$,
where $x_{0}$ is the starting position and $D$ the diffusion coefficient [44].
The light yellow dashed curves in Fig. 8 correspond to the fit to Eq. (17),
and the fitting parameters are reported in Table 2.

Figure 8: Distribution of the first-passage time $\Psi(\tau)$ (solid dark blue
curves) to exit a finite channel of $L=41$, for the same geometries as in Fig.
2: rational polygon with $\alpha=\frac{\pi}{2}$, irrational polygon with
$\alpha=\frac{\pi}{2}\left(\phi-\frac{3}{5}\right)$, finite horizon with
$d=\frac{\delta y}{2}$ and infinite horizon for $d=2\delta y$. Light yellow
dashed curves are fits to Eq. (17) with fitting parameters as in Table 2.
Insets: Numerically obtained distribution of the similarity index $P(\omega)$
for the corresponding geometries (solid black curves). The dashed light yellow
curves correspond to Eq. (19).

The sharper than exponential cutoff at short times, for the transmission
across a polygonal channel, could be smoothed out by considering a higher
power of the exponent $\tau_{0}/\tau$. Moreover, the short times oscillations
are the result of the ballistic fronts and ultimately, to the underlying
skeleton of the deterministic periodic orbits. Although Eq. (17) is not
expected to be an exact description for our systems, the comparison is
physically appealing, given its broad applicability. For instance, taking
$\mu=1-1/\alpha$, Eq. (17) provides a good approximation for the distribution
of first-arrivals for Lévy flights, with Lévy index $\alpha$, $1<\alpha<2$
[46]. Also, for $\mu=1-H$, Eq. (17) is a reasonable approximation for the FPTD
of fractional Brownian motion with Hurst index $H$, where $0<H<1$ [47] (for a
more detailed discussion about Eq. (17) see _e.g._ Ref. [48]).

Furthermore, the distribution in Eq. (17) is normalised, but for $0\leq\mu<1$
its first moment, the MFPT, does not exist. For $1\leq\mu<2$, the MFPT exists
but the variance of $\tau$ diverges, an so on. Observing the decay of the
tails of $\Psi(\tau)$ in Table (2), we realise that for the rational polygonal
channels, the MFPT, i.e. the mean time a particle takes to escape from the
channel, diverges. This has important consequences for the transmission
through a finite rational polygonal channel: even for a finite number of
randomly initialised particles for which the MFPT to escape typically exists,
a large variability in the absorption times is to be expected, yielding a
meaningless MFPT [49]. This calls for a characterisation of the
transmissibility that is not based on the mean escape time. In contrast, for
irrational polygonal channels, the tail of $\Psi(\tau)$ decays faster than
$\tau^{-2}$, namely with $1<\mu<2$ so that the MFPT exists but the variance of
the escape time diverges.

To study the variability in the escape times, we consider the similarity index
$\omega$ studied in the context of first-passage times in Ref. [48]. Consider
a set of $N$ trajectories, and let $\\{\tau_{j}\\}$ denote the set of escape
times corresponding to these trajectories. Then the similarity index is
defined by the ratio between any of the escape times, say $\tau_{j}$ with
$j\in\\{1,\ldots,N\\}$, and mean escape time averaged over the set of $N$
trajectories:

$\omega_{N}=\frac{1}{N}\frac{\tau_{j}}{\langle\tau\rangle_{N}}\ ,$ (18)

where $\langle\tau\rangle_{N}=\frac{1}{N}\sum_{j=1}^{N}\tau_{j}$. This
quantity probes the escape time of any given trajectory relative to the
ensemble average over $N$ trajectories. The domain of the similarity index is
$\omega_{N}\in[0,1]$ and is a random variable if the escape times
$\\{\tau_{j}\\}$ are random variables. A low variability means that
$\tau_{j}\approx\langle\tau\rangle_{N}$ for all the $N$ trajectories, yielding
$\omega_{N}\approx\frac{1}{N}$.

Here we explore the simplest expression for $\omega$, namely for $N=2$.
Consider two trajectories with randomly chosen initial conditions. The first
trajectory exits the channel at a time $\tau_{1}$ while the second trajectory
does it at a time $\tau_{2}$. Then, we have
$\omega_{2}=\tau_{1}/(\tau_{1}+\tau_{2})$ 111Alternatively one can define
$\omega_{2}=\tau_{2}/(\tau_{1}+\tau_{2})$, which ultimately indicates that the
probability distribution $P(\omega_{2})$ is symmetric over its domain.. When
all trajectories are absorbed at approximately the same time,
$\omega_{2}\approx 1/2$. When variations are large, i.e. $\tau_{1}\gg\tau_{2}$
or $\tau_{2}\gg\tau_{1}$ , $\omega_{2}$ attains a value close to $0$ or to
$1$. For clarity in what follows we refer to $\omega_{2}$ simply as $\omega$.

polygon | horizon | $\alpha$ | $d$ | $\tau_{0}$ | $\mu$
---|---|---|---|---|---
rational | finite | $\frac{\pi}{2}$ | $\delta y/2$ | $60$ | $0.536$
rational | infinite | $\frac{\pi}{2}$ | $2\delta y$ | $80$ | $0.498$
irrational | finite | $\left(\varphi-\frac{3}{5}\right)\frac{\pi}{2}$ | $\delta y/2$ | $400$ | $1.38$
irrational | infinite | $\left(\varphi-\frac{3}{5}\right)\frac{\pi}{2}$ | $2\delta y$ | $960$ | $1.2176$
Table 2: Parameters $\tau_{0}$ and $\mu$ obtained from the fit to Eq. (17),
shown in Fig. 8 as dashed light yellow curves. Note the striking difference
between rational and irrational channels.

The distribution of $\omega$ for the FPTD of Eq. (17) is given by [48]

$P(\omega)=\frac{\Gamma(2\mu)}{\Gamma^{2}(\mu)}\omega^{\mu-1}(1-\omega)^{\mu-1}\
.$ (19)

In the insets in Fig. 8 we show the corresponding numerically computed
distributions $P(\omega)$ as solid black curves. For the rational polygonal
channel, $P(\omega)$ has a characteristic “U” shape, with most probable values
of $\omega$ at $0$ and $1$. Strikingly, the mean $\langle\omega\rangle=1/2$ is
the least probable value of the distribution. Such large variations of the
absorption times are a consequence of the broadness of the FPTD.

For the irrational polygonal channels, the distribution of $\omega_{2}$ is
maximal at the mean $\langle\omega\rangle=1/2$ for the infinite horizon, and
is shaped like a “M” for the finite horizon. While the variability of the
absorption times is markedly smaller than for the rational polygons,
variations remain wild, as indicated by the almost flat profile of $P(\omega)$
near the center of the domain.

While the description in terms of $P(\omega)$ is only qualitative, it clearly
reveals a remarkable difference between the rational polygon with
$\alpha=\pi/2$ for which the MFPT diverges, and its irrational counterpart for
which the MFPT exist but its variability is large to the extend that it does
not characterise the escape time of a typical trajectory. In the next section
we will generalise this observation to any rational and the irrational
polygons.

As a final observation, we note that with the exception of the irrational
polygon with finite horizon, Eq. (19), shown as the dashed light yellow curves
in the insets of Fig. 8, is a reasonably good approximation of the numerically
obtained $P(\omega)$. However, the “M” shape obtained for the irrational
polygon with finite horizon cannot be described by Eq. (19), but is
characteristic of a FPTD with power-law tails with an additional exponential
cutoff at large $\tau$ [48], and its origin deserves further investigation.

###  4.2 Survival probability

In the previous Section we have seen that it is not possible to use the mean
first-passage time to characterise the transmission through finite polygonal
channels. In this Section we explore a global characterisation of the
transmission process in terms of the survival probability. The survival
probability $S(t)$ is the fraction of particles that have not yet escaped at
time $t$, and is related to the FPTD through

$\Psi(\tau)=-\frac{\mathrm{d}S(\tau)}{\mathrm{d}\tau}\ .$ (20)

In our settings, the trajectories initially prepared in the central cell of
the channel can indeed be physically seen as a short pulse of particles
released in the center of the channel. By measuring the time at which the
particles escape from the channel, we compute the survival probability $S(t)$
and characterise the transmission of the initial pulse, by defining a channel
escape time $\tau_{\epsilon}$ as the hitting time at which the survival
probability decays to a value smaller than a given $\epsilon>0$, namely

$\tau_{\epsilon}=\inf\left\\{t|S(t)<\epsilon\right\\}\ .$ (21)

We hence define the channel escape time as the time at which a fraction
$(1-\epsilon)$ of the total particles has escaped from the channel.  Being an
observable integrated over a finite time it is not affected by the divergence
of the moments of $\Psi(\tau)$ and, interestingly, is one of the observables
frequently measured in short pulse-response experiments [50].

Indeed, the inverse problem of inferring details of the geometry from the
survival probability is a standard problem of scattering processes. In recent
years it has been investigated in the so-called stochastic billiards, that are
random processes derived from deterministic billiards [51, 52].

Figure 9: Top panel:  Persistence exponent $\mu$ of the FPTD of Eq. (17) as a
function of the aperture angle $\alpha$, for a finite $d=\delta y/2$ (dark
blue curve) and infinite $d=2\delta y$ (light yellow curve) horizons. The
horizontal dashed line indicates the value of $\mu$ above which the MFPT does
not exist. Bottom panel: Corresponding escape time $\tau_{\epsilon}$ of Eq.
(21) for $\epsilon=0.1$, as a function of the angle $\alpha$. The results in
this figure were derived from the survival probability obtained for each
$\alpha$, in a channel made of $L=101$ cells.

From Eqs. (17) and (20), the survival probability decays asymptotically as
$S(\tau)\sim\tau^{\mu-2}$. This is in contrast to polygonal channels with non
parallel walls, for which $S(\tau)$ was instead found to decay as a stretched
exponential [36].

To explore the transmission in polygonal channels, we computed the statistics
of the first-passage time as a function of the aperture angle $\alpha$, for
finite and infinite horizons. For all values of $\alpha$, the survival
probability (and thus the FPTD) decay asymptotically as a power-law of $\tau$.
From a fit to power-law, we obtained the exponent $\mu$ for the decay of the
FPTD in Eq. (17), as a function of $\alpha$, with the results shown in the top
panel of Fig. 9.

Strikingly, for all simple rational polygons with aperture angle
$\alpha=\pi/n$, and $n$ an integer, the decaying exponent of the FPTD is
$\mu<1$, meaning that for all simple rational polygons the mean first-passage
time diverges. The same is observed for $\alpha=2\pi/3$ and $3\pi/4$. In
contrast, other generic values, including the irrational polygons, have
$1<\mu<2$ meaning that the MFPT exists, but the variance of the escape time
diverges. This result holds irrespectively of whether the channel’s horizon is
finite or infinite. We argue that the divergence of the MFPT is characteristic
of rational polygons, and the reason why we do not observe this for all
rational aperture angles $\alpha_{R}$, lies in the infinite numerical accuracy
needed to distinguish rational and irrational angles.

In the lower panel of Fig. 9, we show the channel escape time
$\tau_{\epsilon}$ for $\epsilon=0.1$, namely the time at which the $90$% of
the initial particles have left the channel. The escape time is about an order
of magnitude larger for simple rational polygons than for irrational ones,
indicating that in rational polygonal channels transport drastically slows
down, presumably because of the dense skeleton of periodic orbits
characterising the channels’ dynamics [53].

##  5 Conclusions

We have addressed the nature of mass diffusion in a family of open polygonal
channels with parallel walls, analysing the statistics of particle
displacement. While parallel channels have been subject to implicit or partial
examination in previous works, the key questions of those studies concern the
implications of normal transport in the absence of chaotic dynamics. This
paper has instead a broader scope and has  enabled insight into the fine
tuning of transport properties with the details of the channel’s geometry.

The probability distribution function of the particle displacement $P(\Delta
x,t)$ is a combination of two contributions: a leading scale-invariant
contribution with exponential tails describing the short fluctuations of the
displacement, and a sub leading ballistic contribution, describing the
evolution of the ballistic fronts. The scale of the leading contribution is
$\Delta x/t^{\gamma}$, with a characteristic scale $\gamma<1$ that depends on
the geometry of the channel.

The ballistic front exists independently of the geometry of the channel. In
particular, it is present irrespective of whether the horizon is finite or
infinite. The geometry determines the speed of the front. For an aperture
angle $\alpha=\pi/2$ the speed seems to saturate when the horizon is finite,
namely for $d<\delta y$, and converges to its upper bound $v_{\mathrm{bal}}=1$
as $d\rightarrow\infty$. We argue that this behaviour is qualitatively shown
by arbitrary values of the aperture angle. Moreover, we speculate that the
ballistic front is due to the trajectories that initially are in a close
neighbourhood of the shortest periodic orbit of the polygonal billiard. That
would explain not only the dependence of the speed of the ballistic front on
the geometry of the channel but would also shed light on the strong
variability observed on the transport properties. This question will be
addressed in a forthcoming research.

In addition to the geometry dependent ballistic fronts, a free-flight
ballistic front of unit speed appears, as soon as the horizon becomes
infinite. However, the fraction of such fastest trajectories does necessarily
decrease in time, since it approaches the set of exactly horizontal
trajectories. With infinite horizon it would be possible in principle to
describe the transport in polygonal channels in terms of the single big jump
principle [43, 54]. However, our results suggest that the sub leading
ballistic contribution vanishes rapidly enough to affect the properties of
transport.

For arbitrary geometry mass transport is super-diffusive, meaning that the MSD
scales faster than linear in time. Polygons with rational aperture angle
$\alpha$ show a MSD with a power scaling $\gamma$ that increases smoothly with
$\alpha$, in contrast to irrational angles for which the power scaling is
roughly constant. At large aperture angles both families converge to a
ballistic behaviour as $\alpha\rightarrow\pi$. It still remains to be
understood whether the coefficient for the MSD scaling of irrational parallel
billiards, which appears to be almost constant throughout the range of the
aperture angles we simulate, is an indication of a form of structural
stability for these kinds of systems.

Exploring the spectrum of higher moments of the displacement we obtained that
transport is characterised by strong anomalous diffusion, with moments higher
than $p^{\star}$ scaling ballistically and lower moments exhibiting the same
scaling as the MSD. For the geometries investigated, the threshold moment is
$p^{\star}>2$ in all cases, meaning that the contribution of the ballistic
fronts to the MSD vanishes sufficiently fast. In any case, the spectrum of
moments of the displacement suggest that the transport in these polygonal
channels belong to the universality class recently described in Ref.[37].

Borrowing concepts from stochastic dynamical systems, we have analysed the
transmission of particles through finite channels under the light of first-
passage time statistics. The distribution of FPT to leave the channel is
dominated at short times by the ballistic fronts, and asymptotically decays as
power-law, with a scaling that is determined by the channel’s geometry. Fat
tailed FPT distributions have important consequences on the variability of
escape times. Strikingly, our numerical results suggest that when the aperture
angle $\alpha$ is a simple rational, all moments of the distribution of FPT
diverge, in particular the  mean first passage time. For other angles the mean
escape time exists, but its value does not represent the typical transmission
due to the divergence of the escape time variance. The striking difference
between statistics of the FPT for rational and irrational channels is
interesting, pointing to highly dissimilar dynamics associated to
heterogeneous structures of periodic and ballistic trajectories for the two
cases. Notably, the consequences for experiments involving pulses and
assemblies of particles are non trivial.

Findings discussed in the present work do not only pertain to highly idealised
systems. Our investigation of transport properties, and in particular their
dependence on the geometry of the channel, may help us elucidate the
properties of realistic systems, for example those of transport and solution
kinetics, or a variety of settings that involve diffusion and propagation of
pulses, cells and tracers.

CMM does also acknowledges financial support from the Spanish Government grant
PGC2018-099944-B-I00 (MCIU/AEI/FEDER, UE). LR acknowledges that the present
research has been partially supported by MIUR grant Dipartimenti di Eccellenza
2018-2022 (E11G18000350001). JO and FF finally acknowledge financial support
from the Australian Research Grant DP180101512 and thank Swinburne University
of Technology for the generous allocation of CPU time on the OzSTAR HPC
cluster.

##  6 References

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