# Active particles in reactive disordered media: how does adsorption affects
diffusion?

R. Salgado-García<EMAIL_ADDRESS>Centro de Investigación en Ciencias-IICBA,
Universidad Autónoma del Estado de Morelos. Avenida Universidad 1001, Colonia
Chamilpa, 62209, Cuernavaca Morelos, Mexico.

###### Abstract

In this work we study analytically and numerically the transport properties of
non-interacting active particles moving on a $d$-dimensional disordered media.
The disorder in the space is modeled by means of a set of non-overlapping
spherical obstacles. We assume that obstacles are _reactive_ in the sense that
they react in the presence of the particles in an attractive manner: when the
particle collides with an obstacle, it is attached during a random time
(_adsorption time_), i.e., it gets adsorbed by an obstacle; thereafter the
particle is detached from the obstacle and continues its motion in a random
direction. We give an analytical formula for the effective diffusion
coefficient when the mean adsorption time is finite. When the mean adsorption
time is infinite, we show that the system undergoes a transition from a normal
to anomalous diffusion regime. We also show that another transition takes
place in the mean number of adsorbed particles: in the anomalous diffusion
phase all the particles become adsorbed in the average. We show that the
fraction of adsorbed particles, seen as an order parameter of the system,
undergoes a second-order-like phase transition, because the fraction of
adsorbed particles is not differentiable but changes continuously as a
function of a parameter of the model.

###### pacs:

05.40.-a,05.60.-k,05.10.Gg

##  I Introduction

Active particles have the ability to take energy from the environment and
drive themselves out of equilibrium. This characteristic endows a system of
active particles with novel properties such as the emergence of collective
behavior Ramaswamy (2010); Marchetti  _et al._ (2013); Bechinger  _et al._
(2016). Active particles models have been used to model several systems of
real life such as microswimmers Elgeti  _et al._ (2015), cell tissues Giavazzi
_et al._ (2018), microtubuli Sanchez  _et al._ (2012), flocks of birds Bialek
_et al._ (2012) insects swarms Sinhuber and Ouellette (2017); Attanasi  _et
al._ (2014), or shoaling fish Herbert-Read  _et al._ (2011); Jhawar  _et al._
(2020), just to mention a few examples. The study of these systems has
revealed several remarkable properties in both, interacting and non-
interacting active particle models. For instance, on one hand the interacting
active particle systems have shown to exhibit the emergence of large scale
collective motion Vicsek and Zafeiris (2012); Ramaswamy (2010); Sevilla
(2016); Chepizhko  _et al._ (2013) as well as large-scale instabilities
Marchetti  _et al._ (2013). On the other hand, non-interacting active particle
systems have shown to exhibit complex non-equilibrium transients in the mean
square fluctuations Peruani and Morelli (2007) and anomalous velocity
distributions Golestanian (2009) among other other interesting features Biswas
_et al._ (2020); Bechinger  _et al._ (2016); Su  _et al._ (2017).

Particularly some attention has been addressed to the problem of non-
interacting active particles in disordered and complex media Bechinger  _et
al._ (2016); Chepizhko and Peruani (2013); Bertrand  _et al._ (2018);
Jakuszeit  _et al._ (2019); Zeitz  _et al._ (2017). For instance, in Ref.
Chepizhko and Peruani (2013) it was studied the transport properties of active
particles in a heterogeneous two-dimensional space in which the heterogeneity
is modeled by randomly distributed obstacles. In that work, the authors
reported the occurrence of spontaneous trapping of active particle at large
obstacle density. In Ref. Bertrand  _et al._ (2018) it was considered a system
of run-and-tumble particles, which is a prototypical model of self-propelled
particles, in crowded environments. They found that this simplified version of
active particle systems can be dealt with in an analytical way in
$d$-dimensions, thus providing explicit formulas for the diffusivity of run-
and-tumble particles.

In this work we are mainly interested in the study of the diffusion of non-
interacting active particles in a disordered $d$-dimensional reactive medium.
The latter means that the disordered medium interacts (in this case,
attractively) with the particles. This interaction is such that the particles
are adhered to the obstacles surface, a phenomenon which is commonly referred
to as _adsorption_. In our model the disorder of the medium is modeled by
means of a set of randomly located obstacles in a $d$-dimensional space, and
the obstacles have the property of exerting a force over the particles via an
“attractive potential”. The result of this attractive potential is that the
particles get attached during a time that we consider as a random variable.
Once the particle is detached from the obstacle, it continues its motion in a
random direction. We are interested in describing the diffusion properties of
this model and the effect of the adsorption over the diffusivity of the active
particles. Particularly we will show that there is a transition from normal to
anomalous diffusion when the mean adsorption time goes to infinity. This
transition is accompanied by a transition in which all the particles, in the
average, are adsorbed, which is a kind of trapping effect for the active
particles. The trapping transition we report here is induced by different
mechanism of the one reported in Ref. Chepizhko and Peruani (2013); Morin  _et
al._ (2017) where the transition depends on the obstacle density; here the
trapping transition is a consequence of the adsorption properties of the
medium rather than the obstacle density.

This paper is organized as follows. In Sec. II we present the model of active
particles, the disorder and the adsorption properties of the medium. In Sec.
III we compute the effective diffusion coefficient when the mean adsorption
time is finite, which is the condition to have normal diffusion for the active
particles. We test our formulas for the diffusion coefficient in two a three
dimensions by means of numerical simulations. In Sec. IV we assume that the
mean adsorption time diverges, thus the active particles no longer diffuses in
a normal way, leading to a transition from normal to anomalous subdiffusion.
In this case we are interested in computing the anomalous diffusion exponent,
for which we obtain an analytical expression. We also analyze the fraction of
adsorbed particles and we compared our predictions with numerical simulations.
Finally in Sec. V we give the main results and conclusions of this work.

##  II Model

The model we will consider consists of an ensemble of non-interacting active
particles moving on a $d$-dimensional space. The space is assumed to be non-
homogeneous and the inhomogeneities are modeled by means of a set of non-
overlapping hard hyper-spherical obstacles of radius $a$. We will denote by
$\phi$ the _packing fraction_ parameter, which is the fraction of volume
occupied by the obstacles. To be precise, if we consider a finite hyper-cube
$\mathcal{D}\subset\mathbb{R}^{d}$ volume in $\mathbb{R}^{d}$ of length $L$,
and if denote by $M$ the number of obstacles randomly placed in $\mathcal{D}$,
then the packing fraction $\phi$ is defined as

$\phi:=\frac{MV(\mathcal{S})}{V(\mathcal{D})}$ (1)

where $V(\mathcal{S})$ is the volume occupied by a single obstacle in
$\mathcal{D}$ and $V(\mathcal{D})=L^{d}$ is the volume ob the hyper-cube
$\mathcal{D}$. Since we are considering hyper-spheres as obstacles it is clear
that $V(\mathcal{S})=S_{d}(a)$ where $S_{d}(a)$ stands for the volume of a
$d$-dimensional hyper-sphere of radius $a$,

$S_{d}(a):=\frac{\pi^{d/2}a^{d}}{\Gamma\left(1+\frac{d}{2}\right)},$ (2)

being $\Gamma$ the well-known Euler Gamma function.

The particles are active in the sense that they move with a constant speed $v$
along a direction that might change due to the collision with the obstacles.
When an active particle collides with an obstacle it gets adsorbed for a
while. The time span during which the particles remains adsorbed will be
referred to as the _adsorption time_. Our model assumes that the adsorption
times are drawn at random with a given distribution characterized by a
probability density function (PDF) denoted by $\rho_{\tau}(t)$. In the
following sections we will made some assumptions about the nature of this
probability distribution. Once the particle is desorbed, it continues its
motion with the same constant speed, but in a random direction. The new
direction is chosen at random with uniform distribution over all the possible
directions in the $d$ dimensional space. Thereafter, the particle moves in a
straight line along this new direction, with the constant speed $v$, until a
new collision of the particle with an obstacle takes place. Fig. 1 shows a
schematic representation of the active particle motion in a non-homogeneous
two-dimensional space.

Figure 1:  Active particles in a 2D disordered medium. The disorder is
introduced in the model by means of a set of hard obstacles, fixed in space
having the role of scatters for the particle motion. The particle move freely
outside the obstacles with constant speed $v$. The trajectory of the particle
is described as a sequence of successive displacements. The (random) vectors
$\Delta\mathbf{r}_{1}$, $\Delta\mathbf{r}_{2}$, $\Delta\mathbf{r}_{2}$, …are
displacements of the particle between successive collisions. When the particle
collides with an obstacle, it remains on the obstacle surface. After a random
time the particle leaves the surface and continues its motion outside the
obstacles in a straight line. The net displacements performed during the
adsorption stage are denoted by $\Delta\tilde{\mathbf{r}}_{1}$,
$\Delta\tilde{\mathbf{r}}_{2}$, $\Delta\tilde{\mathbf{r}}_{2}$, …. Clearly the
sum of all these displacements describe the trajectory of the particle.

In order to describe the transport properties of the system we will consider
the motion of a single particle. This can be done because we assume that the
active particles do not interact each other. Actually, a given particle just
interacts with the ensemble of hard hyper-spheres (obstacles) or, in other
words, the particles only interact with the disordered medium.

Let $\mathbf{r}(t):=(x_{1}(t),x_{2}(t),\dots,x_{d}(t))$ denote the position of
the particle at time $t$ and let $\mathbf{v}(t)$ be its velocity. At time
$t=0$ the particle starts its motion at
$\mathbf{r}_{0}:=\mathbf{r}(0)=(x_{1}(0),x_{2}(0),\dots,x_{d}(0))$ with
velocity $\mathbf{v}(0)=v\mathbf{e}_{1}$ where $\mathbf{e}_{1}$ is a unitary
vector in $\mathbb{R}^{d}$ drawn at random with a uniform distribution.

Once the particle starts its motion, it moves in a straight line with constant
speed $v$ until a collision with an obstacle takes place. Then the particle
gets _adsorbed_ by the obstacle surface and surrounds the obstacle during a
random time (the adsorption time) $\tau$ which is drawn at random with density
$\rho_{\tau}(t)$. Once the particle is detached from the obstacle, it moves
again with constant speed $v$ in a new random direction $\mathbf{e}_{1}$,
which is a unitary vector in $\mathbb{R}^{d}$ drawn at random with uniform
distribution. This is the main process by which the particle spreads in space.
Thereafter, the particle moves in a straight line with constant speed until a
new collision occurs. It can be observed that when the particle is not
“attached” to the obstacles, the magnitude $v:=|\mathbf{v}(t)|$ of the vector
velocity is constant, and its direction is randomly changing every time a
collision occurs. In Fig. 1 it can be appreciated graphically the main
features of the model.

At this point it is important to state that we consider all the physical
quantities as dimensionless. This can be done as follows. First we choose
(conveniently) a length $L_{c}$ and a time $T_{c}$. Then, if $\tilde{v}$ is
the magnitude of the velocity of a particle and $v$ is the corresponding
dimensionless velocity, then

$v:=\frac{T_{c}}{L_{c}}\tilde{v}.$ (3)

All the lengths and times appearing in the model can also be put in its
dimensionless version by dividing them by $L_{c}$ and $T_{c}$ respectively.
For instance if $\tilde{L}$ is the length of the box $\mathcal{D}$, then, the
dimensionless length of the box is defined as $L:=\tilde{L}/L_{c}$ and if
$\tilde{t}$ is the time, the dimensionless time $t$ is defined as
$t:=\tilde{t}/T_{c}$. Notice that the choice of $L_{c}$ and $T_{c}$ is
arbitrary, thus we can say that the value of $v$, in certain sense, fix the
units to be used in the system.

The main goals in this work is to study the diffusion properties as well as
the adsorption properties of this model. In particular when the conditions for
the occurrence of normal diffusion are guaranteed we will be interested in the
effective diffusion coefficient. If we denote by $\mathbf{R}(t)$ the total
displacement of the particle from its initial condition, i.e.,
$\mathbf{R}(t):=\mathbf{r}(t)-\mathbf{r}(0)$, then the effective diffusion
coefficient is defined as,

$D_{\mathrm{eff}}:=\lim_{t\to\infty}\frac{\langle|\mathbf{R}(t)|^{2}\rangle-|\langle\mathbf{R}(t)\rangle|^{2}}{2\,d\,t},$
(4)

where the expected value $\langle\cdot\rangle$ is computed by averaging over
an ensamble of independent trajectories $\mathbf{r}(t)$.

##  III Normal diffusion

Now we shall present some calculations for obtaining an analytical expression
for the effective diffusion coefficient. To this end we will made use of the
central limit theorem in order to obtain the asymptotic statistical properties
of the net displacement $\left|\mathbf{r}(t)-\mathbf{r}(0)\right|$ for large
$t$. This procedure is similar to the one used for obtaining the transport
properties (such as the diffusion coefficient and the effective particle
current) in one-dimensional disordered systems Salgado-García and Maldonado
(2013); Salgado-García (2014, 2016); Hidalgo-Soria and Salgado-García (2017).

In this section we will assume that the distribution of adsorption time has
finite mean and variance, which makes valid the central limit theorem. Then,
for the subsequent calculations it is not necessary to state the explicit form
of the distribution $\rho_{\tau}$. Rather, it is enough to assume that

$\displaystyle\bar{\tau}$ $\displaystyle:=$
$\displaystyle\langle\tau\rangle=\int t\rho_{\tau}(t)dt,$ (5)
$\displaystyle\sigma_{\tau}^{2}$ $\displaystyle:=$
$\displaystyle\langle\tau^{2}\rangle-\langle\tau\rangle^{2}=\int
t^{2}\rho_{\tau}(t)dt-\left(\int t\rho_{\tau}(t)dt\right)^{2}.$ (6)

are both finite.

###  III.1 Diffusion coefficient

The goal in this section is to describe the motion of an active particle on
the disordered medium. As stated above, we consider a particle starting its
motion at the point $\mathbf{r}(0)=\mathbf{r}_{0}$ with an initial random
direction $\mathbf{e}_{1}$. Then the particle moves in a straight line which
can be written as

$\mathbf{r}(t)=\mathbf{r}_{0}+vt\,\mathbf{e}_{1},$ (7)

where $\mathbf{e}_{1}$ is a unitary vector in $\mathbb{R}^{d}$ and $v$ is the
constant speed of the active particle. Eq. (7) is the trajectory of the
particle before the particle collides for the first time with an obstacle. Let
$t_{1}$ the time at which the first collision occurs and
$\ell_{1}:=|\mathbf{r}(t_{1})-\mathbf{r}(0)|$ the distance traveled by the
particle from the initial position until the first collision. We will denote
by $\Delta t_{1}:=t_{1}-t_{0}$ (with $t_{0}:=0$) the time elapsed from the
beginning of the motion until the first collision takes place. Clearly
$\ell_{1}$ and $\Delta t_{1}$ are related by

$\ell_{1}=v\Delta t_{1}.$ (8)

Moreover, it is also clear that the total displacement
$\Delta\mathbf{r}_{1}:=\mathbf{r}(t_{1})-\mathbf{r}(0)$ can be written as
$\Delta\mathbf{r}_{1}=v\mathbf{e}_{1}\Delta t_{1}$.

Next, once the collision has occurred, the particle gets attached to the
obstacle during a time $\tau_{1}$ which, as we said before, is a random
variable with a PDF $\rho_{\tau}$. After that, the particle is detached from
the obstacle and moves with a constant speed $v$ in a new independent random
direction defined by $\mathbf{e}_{2}$. We assume that the particle leaves the
obstacle surface at some point which is different from the point at which it
was adsorbed. This mimics in some way a diffusion process occurring at the
surface of the obstacle, thus endowing the particle with a kind of
“scattering” mechanism by it interaction with the obstacle. Therefore, during
the adsorption stage, the particle also performs a displacement, as we
illustrate in Fig. 1. We will denote such a displacement by
$\Delta\tilde{\mathbf{r}}_{1}$.

The above described process repeats again: after the particle is detached from
the first obstacle, it moves in a straight line until it reaches the next
obstacle, and so on. This process defines sequences of displacements
$\\{\Delta\mathbf{r}_{j}\,:\,1\leq j\leq n\\}$ and
$\\{\Delta\tilde{\mathbf{r}}_{j}\,:\,1\leq j\leq n\\}$ which describe the
trajectory of the particle up to certain time $t$. The total displacement of
the particle $\mathbf{R}(t)=\mathbf{r}(t)-\mathbf{r}(0)$ can be written as

$\mathbf{R}(t)=\sum_{j=0}^{n}\left(\Delta\mathbf{r}_{j}+\Delta\tilde{\mathbf{r}}_{j}\right).$
(9)

In Appendix A we show that the displacement $\Delta\tilde{\mathbf{r}}_{j}$ is
a random vector of zero mean,
$\langle\Delta\tilde{\mathbf{r}}_{j}\rangle=\mathbf{0}$ and variance
$\langle\Delta|\tilde{\mathbf{r}}_{j}|^{2}\rangle-\big{|}\langle\Delta\tilde{\mathbf{r}}_{j}\rangle\big{|}^{2}=2a^{2}$.
On the other hand, the random vector $\Delta\mathbf{r}_{j}$ can be written as,

$\Delta\mathbf{r}_{j}:=\ell_{j}\mathbf{e}_{j},\qquad\mbox{for }1\leq j\leq n,$
(10)

where $\ell_{j}$ is the distance traveled by the particle from the $(j-1)$th
to the $j$th collision. Such a distance turns out to be a random variable
whose distribution is known as the _free path distribution_ Bourgain  _et al._
(1998). It is known that the free path distribution $f(x)$ of a particles
moving linearly in a $d$-dimensional space is approximately distributed
according to an exponential distribution Adib (2008); Lu and Torquato (1993),
whose PDF can be expressed as

$f(x)=\left\\{\begin{array}[]{r@{\quad\mbox{ if }
\quad}l}\frac{1}{\lambda}e^{-x/\lambda}&x>0\\\ 0&x<0,\end{array}\right.$ (11)

where $\lambda$ is known as the _mean free path_ and can be written in terms
of the parameters of the obstacle distribution as follows Adib (2008); Lu and
Torquato (1993)

$\lambda=a\frac{s_{d}}{s_{d-1}}\frac{1-\phi}{\phi},$ (12)

where $s_{d}$ is defined as the volume of a hyper-sphere of radius one, i.e.,
$s_{d}:=S_{d}(1)$.

It is clear that the free paths $\ell_{j}$ between collisions are independent
because we assume that $i$) the obstacle distribution are uniformly and
_independently_ distributed and $ii$) the random directions randomly drawn
after the adsorption time are independent each other. These two assumptions
imply that no correlations are present among successive free paths. In other
words, we can say that the random variables $\\{\ell_{j}\,:\,1\leq j\leq n\\}$
are independent and identically distributed with the PDF $f(x)$. Therefore,
the displacements $\\{\Delta\mathbf{r}_{j}\,:\,1\leq j\leq n\\}$ are
independent and identically distributed random vectors. The latter is a
consequence of the assumption that the random directions, along which the
particle continues its motion after a collision, are also independent.

On the other hand, the total time $t$ necessary for the particle to perform
the net displacement $\mathbf{R}(t)$ can be written as,

$t=\sum_{j=1}^{n}\Delta t_{j}+\sum_{j=1}^{n}\tau_{j},$ (13)

which, as we see, relates the variable $n$ (the number of collisions) and the
time $t$. By definition we have that $\\{\tau_{j}\,:\,1\leq j\leq n\\}$ are
independent and identically distributed random variables. As we saw above, the
random variable $\Delta t_{j}$ is related to the random variable $\ell_{j}$ by
a linear transformation,

$\ell_{j}=v\Delta t_{j},$ (14)

which makes $\Delta t_{j}$ to have, up to a constant factor, the same
statistical properties of $\ell_{j}$ which we described above. To be precise,
$\Delta t_{j}$ has exponential distribution with a parameter
$\theta:=v/\lambda$, i.e., the PDF of $\Delta t_{j}$ is given by

$g(t)=\left\\{\begin{array}[]{r@{\quad\mbox{ if }
\quad}l}\frac{1}{\theta}e^{-t/\theta}&t>0\\\ 0&t<0.\end{array}\right.$ (15)

The next step for obtaining the asymptotic behavior of the trajectory
$\mathbf{r}(t)$ is to apply the limit theorems to the sum of random variables
appearing in Eqs. (9) and (13). This technique has been proved to be useful
for determining the exact diffusion coefficients in one-dimensional disordered
media Salgado-García and Maldonado (2013); Salgado-García (2014, 2016);
Hidalgo-Soria and Salgado-García (2017).

Now we proceed to compute the effective diffusion coefficient. The first step
consists in applying the central limit theorem to the sum of random vectors
appearing in Eq. (9) and the sum of random variables in Eq. (13).

The central limit theorem Gnedenko and Kolmogorov (1968); Meerschaert and
Scheffler (2001) states that the sum of random vectors $\mathbf{S}_{n}$,
defined as,

$\mathbf{S}_{n}:=\sum_{j=1}^{n}\left(\Delta\mathbf{r}_{j}+\Delta\tilde{\mathbf{r}}_{j}\right),$
(16)

converges to a normal distribution in the limit of $n\to\infty$ if it is
appropriately normalized. To be precise,

$\lim_{n\to\infty}\frac{\mathbf{S}_{n}-\langle\mathbf{S}_{n}\rangle}{\sqrt{n}}=\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\left(\Delta\mathbf{r}_{j}-\Delta\tilde{\mathbf{r}}_{j}\right)
}{\sqrt{n}}=\mathbf{Z},$ (17)

where $\mathbf{Z}$ is a random vector with multivariate normal distribution of
zero mean, $\langle\mathbf{Z}\rangle=\mathbf{0}$ whose variance matrix is
denoted by $\mathbf{\Sigma}$ and is given in Appendix A. This result allows to
approximate the finite sum of displacements as,

$\mathbf{R}(t)=\sum_{j=0}^{n}\left(\Delta\mathbf{r}_{j}-\Delta\tilde{\mathbf{r}}_{j}\right)\approx\sqrt{n}\mathbf{Z},\qquad\mbox{for}\quad
n\gg 1.$ (18)

Notice that $n$ is the number of collisions occurred up to time $t$, thus, if
$n\to\infty$ then $t\to\infty$.

Next, we will use the same technique in order to approximate the sum of random
times,

$\sum_{j=1}^{n}\left(\Delta t_{j}+\tau_{j}\right).$ (19)

In order to apply the central limit theorem, we denote by $\mu$ the mean of
the random variable $t_{j}+\tau_{j}$ and by $\sigma$ its variance, i.e.,

$\displaystyle\mu$ $\displaystyle:=$ $\displaystyle\langle\Delta
t_{j}+\tau_{j}\rangle=\langle t_{j}\rangle+\langle\tau_{j}\rangle,$ (20)
$\displaystyle\sigma^{2}$ $\displaystyle:=$ $\displaystyle\langle(\Delta
t_{j}+\tau_{j})^{2}\rangle-\mu^{2}.$ (21)

Some direct calculations show that we can write $\mu$ and $\sigma^{2}$ as,

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{\lambda}{v}+\bar{\tau},$ (22) $\displaystyle\sigma^{2}$
$\displaystyle=$ $\displaystyle\frac{\lambda^{2}}{v}+\sigma_{\tau}^{2}.$ (23)

Therefore, the sum of random times en Eq. (19) comply with the central limit
theorem as follows

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\left(\Delta
t_{j}+\tau_{j}\right)-n\mu}{\sqrt{n}\sigma}=Z_{0},$ (24)

where $Z_{0}$ is a random variable with standard normal distribution, i.e., a
random variable with normal distribution with zero mean, $\langle
Z_{0}\rangle=0$ and variance $\langle Z_{0}^{2}\rangle-\langle
Z_{0}\rangle^{2}=1$ .

Then, the finite sum of times can be approximated by

$\sum_{j=1}^{n}\left(\Delta t_{j}+\tau_{j}\right)\approx n\mu+\sqrt{n}\sigma
Z_{0},\qquad\mbox{for}\quad n\gg 1.$ (25)

It is important to remark that Eqs. (18) and (25) allow us to obtain the
statistical properties of the net displacement $\mathbf{R}(t)$ for large $t$.
First notice that, as expressed in Eq. (9), $\mathbf{R}(t)$ is not explicitly
a function of time $t$. Actually the information of the total time $t$
necessary for the particle to perform the net displacement $\mathbf{R}(t)$ is
contained in Eq. (13). Then, the strategy followed here is to use the
approximation (25) for obtaining an expression for the number of collisions
$n$ as a function of the total time $t$. Hence, this result is used within
(18) in order to have the total displacement $\mathbf{R}(t)$ explicitly in
terms of the time $t$. Notice also that, the larger time $t$, the larger
number of collisions $n$. Therefore, expressions (18) and (25), which are
valid for large $n$, are also valid for large time $t$.

Within this approximation we have that the total time $t$ as a function of the
number of collisions $t$ is given by

$n\mu+\sqrt{n}\sigma Z_{0}\approx t,\quad\mbox{for large t}.$ (26)

It is clear that, if the time $t$ is fixed, then the number of collisions
performed by the particle is a random variable. The above equations relates
these variables in the limit of large $t$ by means of the central limit
theorem. According to Ref. Salgado-García and Maldonado (2013), Eq. (26) can
be solved for $n$ giving

$n\approx\frac{t}{\mu}-\frac{\sigma\,t^{1/2}}{\mu^{3/2}}Z_{0}.$ (27)

The next step consists in substituting $n$ given in the above expression, into
the asymptotic expression for the net displacement (18), which results in,

$\mathbf{R}(t)\approx
n^{1/2}\mathbf{Z}=\left(\frac{t}{\mu}-\frac{\sigma\,t^{1/2}}{\mu^{3/2}}Z_{0}\right)^{1/2}\mathbf{Z}.$
(28)

We can see that, the leading order as $t\to\infty$ in the above expression is
$t^{1/2}$, thus obtaining the following asymptotic expression for the net
displacement,

$\mathbf{R}(t)\approx\left(\frac{t}{\mu}\right)^{1/2}\mathbf{Z},$ (29)

for $t\to\infty$.

From Eq. (29) we see that $\langle\mathbf{R}(t)\rangle=0$, which allows us to
write the diffusion coefficient as

$\displaystyle D_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{\langle|\mathbf{R}(t)|^{2}\rangle}{2\,d\,t}$
(30) $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{\langle|t^{1/2}\mu^{-1/2}\mathbf{Z}|^{2}\rangle}{2\,d\,t}=\frac{\langle|\mathbf{Z}|^{2}\rangle}{2\,d\,\mu}.$

In Appendix A we prove that,

$\langle|\mathbf{Z}|^{2}\rangle=2\lambda^{2}+2a^{2}.$ (31)

which leads to the following expression for the diffusion coefficient,

$\displaystyle D_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}+a^{2}}{d\mu}.$ (32)

Recalling Eq. (22), the definition of $\mu$, the above expression for
$D_{\mathrm{eff}}$ can be rewritten as,

$\displaystyle D_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{d}\frac{\lambda^{2}+a^{2}}{\bar{\tau}+\lambda/v},$ (33)

where $\lambda$ is the mean free path given in Eq. (12).

###  III.2 Numerical experiments

Figure 2:  Simulation of active particles diffusing in a two-dimensional
disordered medium. The particles move in a straight line at constant speed
outside the obstacles. When a particle collides with an obstacle it gets
adsorbed during a random time interval. Once the particles gets desorbed, they
continue their motion in random directions. For this simulation the packing
fraction is $\phi=0.3927$.

In order to test our formula for the diffusion coefficient we perform
numerical simulations of the system of active particles in two and three
dimensions. For the two-dimensional case we consider a system of $N=10^{3}$
particles and $M=20\times 10^{4}$ obstacles which are placed randomly with
uniform distribution in a two-dimensional square box $\mathcal{D}$ of wide
$L=10^{4}$. We set the particle velocity to $v=1$ and the distribution of the
adsorption times is chosen as an exponential distribution with mean
$\bar{\tau}=1$. We use periodic boundary conditions for the particle dynamics
in $\mathcal{D}$. Notice that the volume of $\mathcal{D}$ is
$V(\mathcal{D})=10^{8}$. We choose several values for the radius of the
obstacles $a$, ranging from $a=2.5$ to $a=50$. These values result in several
values for the packing fraction $\phi$.

In Fig. 2 we show some examples of trajectories of active particles with
different initial conditions. We use a packing fraction $\phi\approx 0.3927$
corresponding to $a=25$. We see that the time that the particles remains
“adhered” to the obstacle surface during a random time, before the particles
continues its motion in an arbitrary direction. We interpret this
characteristic of our model as if the particle were performing a surface
diffusion while it is adsorbed. The latter can be though as similar, for
instance, to the mechanism of facilitated diffusion in DNA Lomholt  _et al._
(2009); Mirny  _et al._ (2009), in which the involved particles might execute
diffusion in different dimensions.

Figure 3: Effective diffusion coefficient for active particles in two
dimensions. The effective diffusion coefficient estimated from numerical
experiments are obtained for several values of packing fraction $\phi$, which
are shown in filled circles. The approximated formula given in Eq. (33) is
represented by the solid line. We observe that the numerical experiments and
the theoretically computed diffusion coefficient are consistent within the
accuracy of our numerical simulations. The value $\phi_{\mathrm{max}}$
corresponds to the maximal packing fraction which is nearly $\phi\approx
0.907$ Torquato (2002).

Next we perform the simulation of the $N=10^{3}$ particles obtaining a the
same number of trajectories. The total simulation time is set to $T=5\times
10^{5}$. Once we obtain all the trajectories we estimate the mean square
displacement averaging over all the simulated trajectories which allows to
estimate the effective diffusion coefficient. In Fig. 3 we show the effective
diffusion coefficient for several values of the packing fraction, which
corresponds to the radii chosen for the simulations: $a=2.5\,m$ for
$m=1,2,\dots,10$, corresponding to the filled circles shown in Fig. 3. In the
same figure we show our analytical formula (solid black line) given in Eq.
(33).

For the three dimensional case we consider a system of $N=10^{3}$ particles
and $M=20\times 10^{4}$ obstacles which are placed randomly with uniform
distribution in a three dimensional cube $\mathcal{D}$ of wide $L=10^{4}$. We
set the particles velocity to $v=1$ and the adsorption times are distributed
according to an exponential distribution with mean $\bar{\tau}=1$ . We use
periodic boundary conditions for the particle dynamics in $\mathcal{D}$. We
choose several values for the radius of the obstacles $a$, ranging from $a=40$
to $a=180$. These values fix the packing fraction. The total simulation time
is set to $T=5\times 10^{6}$. Once we obtain all the trajectories, we estimate
the mean square displacement averaging over all the trajectories which allows
to estimate the effective diffusion coefficient. In Fig. 4 we show the
effective diffusion coefficient for several values of the packing fraction
corresponding to the radii chosen for the simulations: $a=40+5\,m$ for
$m=0,1,2,\dots,14$, corresponding to the filled circles shown in Fig. 4. In
the same figure, solid line represents the analytical formula given in Eq.
(33).

Figure 4:  Effective diffusion coefficient for active particles in three
dimensions. The effective diffusion coefficient estimated from numerical
experiments are obtained for several values of packing fraction $\phi$, which
are shown in filled circles. The approximated formula given in Eq. (33) is
represented by the solid line. We observe that the numerical experiments and
the theoretically computed diffusion coefficient are consistent within the
accuracy of our numerical simulations. The value $\phi_{\mathrm{max}}$
corresponds to the maximal packing fraction which is nearly $\phi\approx
0.740$ Torquato (2002).

We can observe that in both cases the analytical formula predicts
satisfactorily the behavior of the diffusion coefficient. As we see, our
formula seems to be less satisfactory as we increase the packing fraction.
This phenomenon has been also observed in the study of the effective diffusion
in disordered Lorentz gas. This loss of accuracy might have its origin in the
fact that the free path distribution given in Eq. (11) is actually an
approximation (valid for $\phi\ll 1$) to the real free path distribution Adib
(2008); Lu and Torquato (1993). This, together with the statistical errors in
numerical simulations might cause this effect and requiere a further detailed
study. Nevertheless, these numerical simulations shows that the analytical
approach presented here is satisfactory enough to analyze other cases that
deserve our attention.

##  IV Anomalous diffusion

In this section we explore the case in which the mean adsorption time is no
longer finite. As we mentioned above, in this case the system no longer
exhibits normal diffusion and instead, the square displacement grows, in the
average, lower than linear in time,

$\langle|\mathbf{R}|^{2}\rangle-\big{|}\langle\mathbf{R}\rangle\big{|}^{2}\propto
t^{\beta},$ (34)

with $0<\beta<1$. In the following we will refer to $\beta$ as the _anomalous
diffusion exponent_.

It is known that a sum of random variables having no finite variance nor
finite mean, does not fulfill the central limit theorem. Nevertheless, such a
sum of independent random variables still converges (appropriately normalized)
to a stable law Gnedenko and Kolmogorov (1968); Meerschaert and Scheffler
(2001). When the central limit theorem is broken in this class of systems, it
has been show the occurrence of a transition from normal to anomalous
diffusion Salgado-García and Maldonado (2013); Salgado-García (2014, 2016);
Hidalgo-Soria and Salgado-García (2017) and the limit theorems to stable laws
allow the computation of the anomalous diffusion exponent $\beta$ in Eq. (34).

###  IV.1 Anomalous exponent

When we deal with the case in which the mean adsorption time is no longer
finite. In order to analyze such a case, we introduce the model for the
distribution of adsorption times, which consists in choosing the PDF
$\rho_{\tau}$ as a power-law,

$\rho_{\tau}(t)=\left\\{\begin{array}[]{r@{\quad\mbox{ if } \quad}l}\alpha
t^{-\alpha-1}&t\geq 1\\\ 0&t<1.\end{array}\right.$ (35)

Notice that the mean adsorption time diverges if $\alpha\leq 1$, otherwise the
mean adsorption time is given by

$\bar{\tau}:=\int_{-\infty}^{\infty}t\rho(t)dt=\int_{1}^{\infty}\alpha
t^{-\alpha}dt=\frac{\alpha}{\alpha-1}.$ (36)

The variance of $\tau$ can be computed in a similar way, resulting in

$\sigma_{\tau}^{2}=\frac{\alpha}{\alpha-2}-\left(\frac{\alpha}{\alpha-1}\right)^{2},$
(37)

which is clearly finite only if $\alpha>2$.

In Appendix B we explore the cases $i$) $0<\alpha<1$, $ii$) $1<\alpha<2$,
$iii$) $\alpha=1$ and $iv$) $\alpha=2$ separately. The case $\alpha>2$ is not
considered here since the central limit theorem is valid for these values of
$\alpha$, thus leading to normal diffusion as it was stated in Sec. III. The
reason to treat all these cases separately is due to the fact that the
normalization of the sum of random variables is different in every case, and
converges to different limit distributions. However, the diffusion properties
only depends on the behavior of the mean adsorption time leading to only two
different regimes. Here we only summarize the results for the behavior of
square displacement which, according to Appendix B, is given by

$\langle|\mathbf{R}|^{2}\rangle-\big{|}\langle\mathbf{R}\rangle\big{|}^{2}\propto\left\\{\begin{array}[]{r@{\quad\mbox{
if } \quad}l}t^{\alpha}&0<\alpha\leq 1\\\ \frac{t}{\ln(t)}&\alpha=1\\\
t&\alpha>1.\end{array}\right.$ (38)

This result shows that the system undergoes a transition from normal
($\alpha>1$) to anomalous ($\alpha\leq 1$) diffusion. The anomalous diffusion
phase is characterized by the behavior of the mean square displacement which
grows in time lower that linear. For $0<\alpha<1$ this growth is a power of
$t$ with exponent $\beta=\alpha$. For $\alpha=1$ the grow in time of the mean
square displacement is still lower than linear but not a power law. Indeed, in
this case such a grow is proportional to $t$ but screened by a logarithm,
behavior which is known as _marginal subdiffusion_. Interestingly, the case
$1<\alpha\leq 2$ leads to a normal diffusive behavior. This fact is
interesting because for these values of $\alpha$ the random adsorption times
have no finite variance, i.e., even having a heavy-tailed distribution for the
random adsorption times with infinite variance, the system yet exhibit normal
diffusive behavior. There is a plethora of examples in the literature which
infinite variance (of the random variables involved in the underlying dynamics
of the system) is enough for the system to exhibit anomalous behavior Bouchaud
and Georges (1990); Metzler and Klafter (2000); Salgado-García (2016);
Hidalgo-Soria and Salgado-García (2017).

###  IV.2 Mean number of adsorbed particles

In this section we will compute the asymptotic behavior of the mean number of
adsorbed particles in the case of adsorption times distributed according to a
power law with the distribution given in Eq. (34). Consider an ensemble of $N$
non-interacting active particles moving in the disordered medium. Let
$\mathbf{r}_{k}(t)$ be the trajectory of the $k$th particle for $1\leq k \leq
N$. Let $S_{k}(t)$ denote the “state” of the particle, i.e., $S_{k}(t)=1$ if
the particle is adsorbed and $S_{k}(t)=0$ if not. If we denote by
$N_{\mathrm{ads}}(t)$ the number of adsorbed particles at time $t$ then it is
clear that,

$N_{\mathrm{ads}}(t)=\sum_{k=1}^{N}S_{k}(t).$ (39)

Then, the fraction $\gamma$ of adsorbed particles is given by

$\gamma_{N}:=\frac{N_{\mathrm{ads}}(t)
}{N}=\frac{1}{N}\sum_{k=1}^{N}S_{k}(t).$ (40)

By the law of large numbers Gnedenko and Kolmogorov (1968) we have that, for
$N\to\infty$, the last sum converge to the mean of $S_{k}(t)$, which is a
consequence of the fact that the $\\{S_{k}(t)\\}$ is a collection of
independent random variables for a fixed $t$. Explicitly we have that the
fraction $\gamma$ of adsorbed particles for $N\to\infty$,

$\gamma:=\lim_{N\to\infty}\gamma_{N}=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}S_{k}(t),$
(41)

can be written as

$\gamma=\langle S_{k}(t)\rangle.$ (42)

The expected value in the above expression is taken with respect to the
ensemble in the thermodynamic limit. If we further assume that the process
$\\{S_{k}(t)\,:\,t>0\\}$ is a stationary (uniquely) ergodic process we would
have that $\gamma$ does not depend on time and the ensemble average can be
written as a time-average,

$\gamma=\langle
S_{k}(t)\rangle=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}S_{k}(t^{\prime})dt^{\prime}.$
(43)

The last expression allows us to write the fraction of adsorbed particles in
terms of the sums of random adsorption times,

$\gamma=\lim_{t\to\infty}\frac{1}{t}\sum_{j=1}^{n}\tau_{j}(t).$ (44)

In Appendix C we show that for the the model of random adsorption times ruled
by a power law distribution, the fraction of the adsorbed particles in the
thermodynamic-like limit is given by,

$\gamma=\left\\{\begin{array}[]{r@{\quad\mbox{ if } \quad}l}1&0<\alpha\leq
1\\\
\frac{\bar{\tau}}{\bar{\tau}+\frac{\lambda}{v}}&\alpha>1.\end{array}\right.$
(45)

Notice that the above expression is a continuous function of $\alpha$. This
can be seen as follows: notice that $\bar{\tau}=\bar{\tau}(\alpha)$ (cf. Eq.
(36)) and that $\tau\to\infty$ as $\alpha\to 1$. Thus,

$\lim_{\alpha\to 1}\frac{\bar{\tau}}{\bar{\tau}+\frac{\lambda}{v}}=1,$ (46)

which shows that $\gamma(\alpha)$ is continuous at $\alpha=1$. This fact is
important because, despite the fraction of adsorbed particles is continuous,
it is not differentiable. This means that the system undergoes as kind of
second-order phase transition manifested in the order parameter $\gamma$.

###  IV.3 Numerical experiments

In this section we test our formulas for the anomalous exponent $\beta$ and
the fraction of adsorbed particles. For this purpose we simulated the system
of active particles in two dimensions. In this case we considered a system of
$N=10^{3}$ particles and $M=20\times 10^{4}$ obstacles which were placed
randomly with uniform distribution in a two-dimensional square box
$\mathcal{D}$ of wide $L=10^{4}$. We used periodic boundary conditions for the
dynamics in $\mathcal{D}$ and set the particle velocity to $v=1$. Notice that
the volume of $\mathcal{D}$ is $V(\mathcal{D})=10^{8}$, which, together with
the choice of radius of the obstacles $a=10$, fix the packing fraction $\phi$,
whose value is approximately $\phi\approx 0.01571$. For the random adsorption
times we used the distribution given in Eq. (35) and perform the simulations
for several values of $\alpha$.

Next our goal is to characterize, from the numerically simulated trajectories,
the nature of the diffusivity. To this end we estimated the exponent $\beta$
as follows. First, from the simulated trajectories we obtain the time-
dependent mean square displacement
$\langle|\mathbf{R}|^{2}\rangle-\big{|}\langle\mathbf{R}\rangle\big{|}^{2}$
which, according to our hypothesis, it behaves in time as a power law $\propto
t^{\beta}$. Once we we obtain the mean square displacement we made use the
least squares method to estimate $\beta$.

Figure 5: Exponent of the mean square displacement. We show the behavior of
the exponent $\beta$ of the mean square displacement as a function of
$\alpha$, the parameter of the distribution of the adsorption times. The solid
black line represents the theoretically obtained exponent $\beta(\alpha)$
given in Eq. (38). The red filled circles represent the estimated exponent
$\beta$ obtained from the numerical simulations. We can see that the system
undergoes a kind of second order transition, which in this case is a
transition from normal ($\alpha>1$) to anomalous diffusion ($\alpha\leq 1$).

In Fig. 5 we can see the estimated $\beta$ for several values of $\alpha$ (red
filed circles). In the same figure we also display the theoretically computed
exponent $\beta$ as a function of $\alpha$. We can appreciate that the
numerically estimated exponent is consistent with its analytical counterpart
at least for values of $\alpha$ below $\alpha=0.5$ and above $\alpha=1$. The
deviations we observe for intermediate values of $\alpha$ can actually be
explained as an effect of the slow convergence, which is in turn a consequence
of the power law behavior. Below we will perform a simple analysis to estimate
how slow is the convergence in this system.

Now we turn to the analysis of the mean number of adsorbed particles. From the
simulated trajectories we estimated the mean fraction of adsorbed particles
$\gamma$ for large times. In Fig. 5 we show the result of the estimated
$\gamma$ from the numerical simulations (red filled circles) as well as the
analytically computed (solid black line) fraction of adsorbed particles
$\gamma(\alpha)$ for several values of $\alpha$. As in the case of the
exponent $\beta(\alpha)$ we can see that the numerical simulations are
consistent with the theoretical predictions, specially for values of $\alpha$
below $\alpha=0.5$ and above $\alpha=1$. The underestimation of the mean
fraction of adsorbed particles can be explained as a finite-time effect, i.e.,
due to a slow convergence of the averaged quantities. First consider a fixed
value of $\alpha$ and recall that, according to Eq. (44), $\gamma$ can be
obtained as a limit as follows,

$\gamma=\lim_{t\to\infty}\gamma_{t}=\lim_{t\to\infty}\frac{1}{t}\sum_{j=1}^{n}\tau_{j}(t).$
(47)

Figure 6:  Fraction of adsorbed particles. We show the fraction of adsorbed
particles $\gamma$ as a function of $\alpha$ obtained from numerical
simulations (red filled circles) and computed from the analytical formula (45)
(black solid line). We see that the numerically estimated $\gamma$ is
consistent with the theoretical prediction. Notice that the numerical
estimations are less satisfactory for $\alpha$ in the interval $(0.5,1)$, an
effect that occurs due to the slow convergence of $\gamma$ in the limit
$t\to\infty$.

According to Appendix C, if we assume that $0<\alpha<1$ we have that
$\gamma_{t}$ can be written alternatively as,

$\gamma_{t}\approx\frac{Wn^{1/\alpha}}{\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}+Wn^{1/\alpha}},$
(48)

where $n=n(t)$ is the number of collisions at time $t$. It is not hard to see
that, as $t\to\infty$ (and therefore as $n\to\infty$) we have that
$\gamma_{t}$ goes as,

$\displaystyle\gamma_{t}\approx 1-\frac{\lambda}{vW}n^{1-1/\alpha}.$ (49)

In Appendix C we also showed that $n(t)\approx t^{\alpha}/W^{\alpha}$.
Substituting this identity into the above expression we can obtain
$\gamma_{t}$ explicitly as a function of $t$. Then we have

$\displaystyle\gamma_{t}\approx
1-\frac{\lambda}{vW}\left(\frac{t^{\alpha}}{W^{\alpha}}\right)^{1-1/\alpha}\approx
1-\frac{\lambda}{vW^{\alpha}}t^{\alpha-1},$ (50)

for $0<\alpha<1$ and large $t$.

Expression (50) shows that as $t$ goes to infinity, the fraction of adsorbed
particles attains the value $\gamma=1$. However, as we can also appreciate in
Eq. (50) the rate of convergence is extremely slow for $\alpha$ near (and
below) $\alpha=1$ since it goes as $t^{\alpha-1}$. Certainly, this behavior
explains why the estimations obtained from numerical simulations are less
accurate for $0.5<\alpha<1$ than the ones made outside this interval.

##  V Conclusions

In work we have studied a system of non-interacting active particles in a
$d$-dimensional disordered reactive medium. The disorder has been modeled as a
set of non-overlapping obstacles which limits the movement of the particles
throughout a hard core interaction. That the medium be reactive means that the
particles interact with the medium in an attractive manner: once a particle
collides with an obstacle, it gets adsorbed to it during a time which we
assume to be randomly drawn with certain distribution. Next we were interested
in studying the diffusion as well as the adsorption properties of this system.
In particular we showed that the diffusion coefficient is finite if the
distribution of the adsorption times has finite mean; actually we provided an
analytical formula for the diffusion coefficient which satisfactorily fits the
diffusion coefficient obtained by means of numerical simulations in two a
three dimensions. Thereafter we analyzed the behavior of the square
displacement when the mean adsorption is no longer finite. We showed that in
this case the system undergoes a transition from normal to anomalous
subdiffusion and we were able to obtain an analytical expression for the
anomalous diffusion exponent. The occurrence of this transition from normal to
anomalous diffusion is also accompanied by a transition in the fraction of
adsorbed particles. When the mean adsorption time is finite the fraction of
adsorbed particles is strictly less than one. However, when the distribution
of adsorption times passes to a regime in which the mean adsorption time
diverges, the fraction of adsorbed particles equals one. The latter means that
in such a case all the particles become adsorbed by the obstacles in the
average, thus collapsing all in the surface of the obstacles. This phenomenon
in interesting because this transition can also seen at the level of
distributions. In the phase in which $\gamma<1$, the motion of the particles
is not restricted, they move in a $d$-dimensional space outside the obstacles.
However, when the system reaches the phase $\gamma=1$, then all the particles
collapse their motion into a region of lower dimension. This means that the
distribution of the particles passes from being absolutely continuous to
singular in $\mathbb{R}^{d}$ with respect to Lebesgue. This phenomenon, which
manifests as a change in the nature of the distribution, has shown to occur in
other class of systems within the context of thermodynamical formalism, and
such a transition is commonly referred to as _freezing phase transition_ Bruin
and Leplaideur (2013). Particularly, in Ref. Maldonado and Salgado-García
(2019) the freezing phase transition has been related to the occurrence of
anomalous diffusion. Here in this work, through the analysis of the fraction
of adsorbed particles we show indirectly that this might be the case in a
system of active particles. Certainly, whether or not the distribution passes
from continuous to singular requieres a much more mathematically rigorous
analysis.

###### Acknowledgements.

The author CONACyT by its financial support through grants A1-S-15528 and
FORDECYT-PRONACES/1327701/2020.

##  Appendix A Some statistical features of random vectors

Let us start by characterizing the random vector displacement
$\Delta\tilde{\mathbf{r}}$. As we saw in Sec. II $\Delta\tilde{\mathbf{r}}$ is
the net displacement of the particle from the arriving point and the departing
point on the obstacle surface when the adsorption takes place. The obstacle is
a circle of radius $a$ and therefore the vector displacement can be written as

$\Delta\tilde{\mathbf{r}}=a\left(\mathbf{e}^{\prime}-\mathbf{e}\right),$ (51)

where $a\mathbf{e}$ and $a\mathbf{e}^{\prime}$ are the arriving and departing
points (respectively) on the obstacle surface with respect to a reference
frame whose origin is the center of the obstacle. The vectors $\mathbf{e}$ and
$\mathbf{e}^{\prime}$ are random unitary vectors. These vector are independent
and uniformly distributed on all the possible directions in $d$ dimensions.
The latter implies that both, $\mathbf{e}$ and $\mathbf{e}^{\prime}$, have
zero mean, i.e.,

$\langle\mathbf{e}\rangle=\langle\mathbf{e}^{\prime}\rangle=\mathbf{0}.$ (52)

We are interested in computing the expected value of
$|\Delta\tilde{\mathbf{r}}|^{2}$. Notice that

$|\Delta\tilde{\mathbf{r}}|^{2}=|\mathbf{e}^{\prime}-\mathbf{e}|^{2}=2+\mathbf{e}.\mathbf{e}^{\prime}.$
(53)

Then we have,

$\langle|a\Delta\tilde{\mathbf{r}}|^{2}\rangle=2a+a\langle\mathbf{e}\cdot\mathbf{e}^{\prime}\rangle=2a,$
(54)

a result that is a consequence of the fact that $\mathbf{e}$ and
$\mathbf{e}^{\prime}$ are independent and have zero mean.

Next we will compute the variance matrix $\mathbf{\Sigma}$ of the random
vector $\mathbf{Z}$ appearing in Eq. (17). According to the central limit
theorem for independent and identically distributed random vectors, the
variance matrix of $\mathbf{Z}$ corresponds to the variance of any of the
summands in Eq. (17). If we denote by $\mathbf{\Sigma}_{n,m}$ the $(n,m)$th
entry of $\mathbf{\Sigma}$ (for $1\leq n\leq d$ and $1\leq m\leq d$) , we have
that the components of the variance matrix $\mathbf{\Sigma}$ can be written
as,

$(\mathbf{\Sigma})_{n,m}:=\langle(\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}})_{n}(\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}})_{m}\rangle,$
(55)

where we used the fact that $\Delta{\mathbf{r}}$ and
$\Delta\tilde{\mathbf{r}}$ have both zero mean. In Eq. (55)
$(\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}})_{n}$ and
$(\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}})_{m}$ stands for the $n$th and
$m$th components of $\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}}$,
respectively. Performing some calculations we obtain

$\displaystyle(\mathbf{\Sigma})_{n,m}$ $\displaystyle=$
$\displaystyle\langle(\Delta{\mathbf{r}})_{n}(\Delta{\mathbf{r}})_{m}\rangle+\langle(\Delta\tilde{\mathbf{r}})_{n}(\Delta\tilde{\mathbf{r}})_{m}\rangle,$
(56)

where we used the fact that $\Delta{\mathbf{r}}$ and
$\Delta\tilde{\mathbf{r}}$ are independent and have zero mean, which implies
that
$\langle\Delta{\mathbf{r}}\cdot\Delta\tilde{\mathbf{r}}\rangle=\mathbf{0}$.
Now, using the fact that $\Delta{\mathbf{r}}=\ell_{j}\mathbf{e}_{j}$ we see
that

$\displaystyle(\mathbf{\Sigma})_{n,m}$ $\displaystyle=$
$\displaystyle\langle\ell_{j}^{2}\rangle\langle(\mathbf{e}_{j})_{n}(\mathbf{e}_{j})_{m}\rangle+\langle(\Delta\tilde{\mathbf{r}})_{n}(\Delta\tilde{\mathbf{r}})_{m}\rangle,$
(57)

Which shows that the variance matrix can be put in terms of the underlying
distributions of the model.

Finally we compute the expected value of $|\mathbf{Z}|^{2}$. To this end, we
use the fact that the moments of $\mathbf{Z}$ are related to the moments of
$\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}}$. Specifically, we can write

$\langle|\mathbf{Z}|^{2}\rangle=\langle\big{|}\Delta{\mathbf{r}}+\Delta\tilde{\mathbf{r}}\big{|}^{2}\rangle=\langle\big{|}\Delta{\mathbf{r}}\big{|}^{2}\rangle+\langle\big{|}\Delta\tilde{\mathbf{r}}\big{|}^{2}\rangle,$
(58)

where we used the fact that
$\langle\Delta{\mathbf{r}}\cdot\Delta\tilde{\mathbf{r}}\rangle=0$. Next, we
should notice that

$\displaystyle\langle\big{|}\Delta\tilde{\mathbf{r}}\big{|}^{2}\rangle$
$\displaystyle=$ $\displaystyle 2a^{2},$ (59)

where we used the result given in Eq. (54). On the other hand, since
$\ell_{j}$ is distributed according to the free path distribution (which an
exponential distribution) it is clear that

$\displaystyle\langle\big{|}\Delta{\mathbf{r}}\big{|}^{2}\rangle$
$\displaystyle=$ $\displaystyle\langle\ell_{j}^{2}\rangle=2\lambda^{2}.$ (60)

These results allows us to see that

$\langle|\mathbf{Z}|^{2}\rangle=2\lambda^{2}+2a^{2},$ (61)

which is the result we anticipated in Eq. 31.

##  Appendix B Square displacement for power-law distribution of random
adsorption time

In this Appendix we will compute the asymptotic behavior of the mean square
displacement of an active particle when the PDF of the adsorption time is a
power law as stated in Eq. (35),

$\rho_{\tau}(t)=\left\\{\begin{array}[]{r@{\quad\mbox{ if } \quad}l}\alpha
t^{-\alpha-1}&t\geq 1\\\ 0&t<1.\end{array}\right.$ (62)

We will consider the cases $i$) $0<\alpha<1$, $ii$) $1<\alpha<1$, $iii$)
$\alpha=1$ and $iv$) $\alpha=2$ separately.

Before start to exploring these cases separately, we recall that, in order to
obtain the asymptotic properties of the trajectory of an active particle we
have to obtain an expression for the net displacement $\mathbf{R}(t)$,
explicitly as a function of $t$. The starting point is the equation for the
net displacement in terms of the number of collisions (actually as a sum of
random vectors),

$\mathbf{R}(t)=\sum_{j=0}^{n}\left(\Delta\mathbf{r}_{j}+\Delta\tilde{\mathbf{r}}_{j}\right),$
(63)

and the total time $t$ as well in terms of the number of collisions

$t=\sum_{j=1}^{n}\Delta t_{j}+\sum_{j=1}^{n}\tau_{j},$ (64)

It is clear that changing the statistical properties of the adsorption times
does not affect the statistical properties of the free path between
collisions. Therefore it is clear that we can still made use of the
approximation provided by the central limit theorem for the net displacement
given in Eq. (65),

$\mathbf{R}(t)=\sum_{j=0}^{n}\left(\Delta\mathbf{r}_{j}+\Delta\tilde{\mathbf{r}}_{j}\right)\approx\sqrt{n}\mathbf{Z},\qquad\mbox{for}\quad
n\gg 1,$ (65)

where $\mathbf{Z}$ is a multivariate random normal vector with zero mean and
variance matrix $\mathbf{\Sigma}$.

###  B.1 Case: $0<\alpha<1$

In this case the sum of random adsorption times,

$\sum_{j=1}^{n}\tau_{j},$ (66)

converges to a stable law if normalized according to,

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\tau_{j}}{n^{1/\alpha}}=W,$ (67)

where $W$ is a $\alpha$-stable distribution Gnedenko and Kolmogorov (1968).
According to this result, it is clear that

$\sum_{j=1}^{n}\tau_{j}\approx Wn^{1/\alpha},$ (68)

for large $n$. On the other hand, the sum

$\sum_{j=1}^{n}\Delta t_{j}$ (69)

still converges to a normal distribution because it is related to the sum of
free path between collisions as follows

$\sum_{j=1}^{n}\Delta t_{j}=\frac{1}{v}\sum_{j=1}^{n}\ell_{j}.$ (70)

This implies that

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\Delta
t_{j}-n\lambda/v}{n^{1/2}(\lambda/v)^{2}}=Z_{0}$ (71)

where $Z_{0}$ has a standard normal distribution. Therefore we can made the
following approximation

$\sum_{j=1}^{n}\Delta
t_{j}\approx\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0},$
(72)

for large $n$. Then, using approximations (68) and (72), we have that

$\displaystyle\sum_{j=1}^{n}\Delta t_{j}+\sum_{j=1}^{n}\tau_{j}$
$\displaystyle\approx$
$\displaystyle\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}+Wn^{1/\alpha}$
(73) $\displaystyle\approx$ $\displaystyle Wn^{1/\alpha}.$

In the last equality we used the fact that $0<\alpha<1$, which implies that
$n^{1/\alpha}$ is the leading order in the above equation. Thus, the number of
collisions $n$ occurring up to time $t$ can be obtained (for large $t$) from
the following equation

$t\approx Wn^{1/\alpha}.$ (74)

Solving for $n$ we obtain,

$n\approx\frac{t^{\alpha}}{W^{\alpha}}.$ (75)

Using this result into Eq. (65), we can see that

$\mathbf{R}(t)\approx t^{\alpha/2}\frac{\mathbf{Z}}{W^{\alpha/2}},$ (76)

for asymptotically large $t$. Therefore we have that the square displacement
goes as

$\left|\mathbf{R}(t)\right|^{2}\approx
t^{\alpha}\frac{|\mathbf{Z}|^{2}}{W^{\alpha}}.$ (77)

or equivalently, the mean square displacement behaves as

$\langle\left|\mathbf{R}(t)\right|^{2}\rangle-\big{|}\langle\mathbf{R}(t)\rangle\big{|}^{2}\propto
t^{\alpha},$ (78)

for asymptotically large $t$. This result states that the square displacement
grows in time lower than linear with an exponent $\beta=\alpha$, the same as
the one for the distribution of the random adsorption times. Notice that this
behavior does not depend on the details of the disorder.

###  B.2 Case: $1<\alpha<2$

For $1<\alpha<2$ the mean adsorption time $\bar{\tau}$ is finite, but the sum
of random adsorption times

$\sum_{j=1}^{n}\tau_{j},$ (79)

does not satisfy the central limit theorem because the variance of $\tau_{j}$
is not finite. However, if normalized appropriately, this sum converges to a
stable law. Specifically we have that

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\tau_{j}-n\bar{\tau}}{n^{1/\alpha}}=W,$
(80)

where $W$ is a $\alpha$-stable distribution Gnedenko and Kolmogorov (1968).
This expression allows us to approximate the sum (79) as follows,

$\sum_{j=1}^{n}\tau_{j}\approx n\bar{\tau}+Wn^{1/\alpha},$ (81)

for $n\gg 1$. Using the approximation given in (72) for the sum of $\Delta
t_{j}$, and the above expression for the sum of $\tau_{j}$, we have that

$\displaystyle t$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\tau_{j}+\sum_{j=1}^{n}\Delta t_{j}$ (82)
$\displaystyle\approx$ $\displaystyle
n\bar{\tau}+Wn^{1/\alpha}+\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}$
$\displaystyle=$
$\displaystyle\left(\bar{\tau}+\frac{\lambda}{v}\right)n+Wn^{1/\alpha}+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}.$

Since $1<\alpha<2$, it is clear that $1/2<\alpha^{-1}<1$, which means that the
leading order in the above expression is $n$. Therefor we have that

$t \approx\left(\bar{\tau}+\frac{\lambda}{v}\right)n,$ (83)

for $n\gg 1$. Equivalently we have that

$n \approx\frac{t}{\bar{\tau}+\frac{\lambda}{v}}.$ (84)

Using this last result in Eq. (65) we obtain for the net displacement,

$\mathbf{R}(t)\approx\left(\frac{t}{\bar{\tau}+\frac{\lambda}{v}}\right)^{1/2}\mathbf{Z}.$
(85)

Then, the mean square displacement can be computed from the last relation,
obtaining,

$\langle\left|\mathbf{R}(t)\right|^{2}\rangle-\big{|}\langle\mathbf{R}(t)\rangle\big{|}^{2}\approx\left(\frac{t}{\bar{\tau}+\frac{\lambda}{v}}\right)\langle|\mathbf{Z}|^{2}\rangle.$
(86)

Since $\langle|\mathbf{Z}|^{2}\rangle=2\lambda^{2}+2a^{2}$ it is clear that

$D_{\mathrm{eff}}=\frac{1}{d}\left(\frac{\lambda^{2}+a^{2}}{\bar{\tau}+\frac{\lambda}{v}}\right),$
(87)

which is the expression we have already obtained for the diffusion coefficient
when the central limit theorem takes place.

###  B.3 Case: $\alpha=1$

For $\alpha=1$ the mean adsorption time $\bar{\tau}$ is infinite (as well as
its variance) and the sum of random adsorption times converges to a stable law
in the following form

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\tau_{j}}{n\ln(n)}=W,$ (88)

where $W$ is a $\alpha$-stable distribution for $\alpha=1$ Gnedenko and
Kolmogorov (1968). This limit theorem allows us to approximate the sum of
random adsorption times as follows

$\sum_{j=1}^{n}\tau_{j}=Wn\ln(n).$ (89)

This expression together with Eq. (72) allows us to write

$\displaystyle t$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\tau_{j}+\sum_{j=1}^{n}\Delta t_{j}$ (90)
$\displaystyle\approx$ $\displaystyle
Wn\ln(n)+\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}.$

As we can see the leading order in the above approximation is $n\ln(n)$. In
Ref. Hidalgo-Soria and Salgado-García (2017) it has been proved that $n$ as
function of $t$ can be written approximately as,

$n\approx\frac{t/W}{\ln(t/W)},$ (91)

for large $t$. Using this result in Eq. (65) we obtain

$\mathbf{R}(t)\approx\left(\frac{t/W}{\ln(t/W)}\right)^{1/2}\mathbf{Z}.$ (92)

Then, the square displacement can be written as,

$\left|\mathbf{R}(t)\right|^{2}\approx\left(\frac{t/W}{\ln(t/W)}\right)|\mathbf{Z}|^{2}.$
(93)

This result implies that the mean square displacement grows in time as

$\langle\left|\mathbf{R}(t)\right|^{2}\rangle-\big{|}\langle\mathbf{R}(t)\rangle\big{|}^{2}\propto\frac{t}{\ln(t)},$
(94)

which is a marginal subdiffusion because the growth is lower than linear but
not a power law.

###  B.4 Case: $\alpha=2$

For $\alpha=2$ the mean adsorption time $\bar{\tau}$ is finite but its
variance is infinite. The sum of random adsorption times converges to a normal
distribution in the following form

$\lim_{n\to\infty}\frac{\sum_{j=1}^{n}\tau_{j}-n\bar{\tau}}{\sqrt{n\ln(n)}}=W,$
(95)

where $W$ is a normally distributed random variable with zero mean Gnedenko
and Kolmogorov (1968). This result allows us to write the sum of random
adsorption times approximately as.

$\sum_{j=1}^{n}\tau_{j}\approx n\bar{\tau}+\sqrt{n\ln(n)}W,$ (96)

This expression together with Eq. (72) allows us to write

$\displaystyle t$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\tau_{j}+\sum_{j=1}^{n}\Delta t_{j}$ (97)
$\displaystyle\approx$ $\displaystyle
n\bar{\tau}+\sqrt{n\ln(n)}W+\left(\frac{\lambda}{v}\right)n+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}$
$\displaystyle=$
$\displaystyle\left(\bar{\tau}+\frac{\lambda}{v}\right)n+\sqrt{n\ln(n)}W+\left(\frac{\lambda}{v}\right)n^{1/2}Z_{0}.$

As we can see the two leading orders in the above approximation are $n$ and
$\sqrt{n\ln(n)}$. Retaining the two first leading orders in the above
approximation, we have that $n$ as function of $t$ can be written
approximately as (see Appendix B.3 in Ref. Hidalgo-Soria and Salgado-García
(2017)),

$n\approx\frac{t}{\bar{\tau}+\frac{\lambda}{v}}+O\left((t\ln(t))^{1/2}\right).$
(98)

Thus, the square displacement can be written as,

$\left|\mathbf{R}(t)\right|^{2}\approx\left(\frac{t}{\bar{\tau}+\frac{\lambda}{v}}\right)|\mathbf{Z}|^{2}.$
(99)

This result implies that the square displacement grows linearly in time, i.e.,

$\langle\left|\mathbf{R}(t)\right|^{2}\rangle-\big{|}\langle\mathbf{R}(t)\rangle\big{|}^{2}\approx\left(\frac{t}{\bar{\tau}+\frac{\lambda}{v}}\right)\langle|\mathbf{Z}|^{2}\rangle.$
(100)

The latter means that the effective diffusion coefficient is finite and
results in

$D_{\mathrm{eff}}=\frac{1}{d}\left(\frac{\lambda^{2}+a^{2}}{\bar{\tau}+\frac{\lambda}{v}}\right),$
(101)

which is the expression we have already obtained for the diffusion coefficient
when the central limit theorem takes place.

##  Appendix C Fraction of adsorbed particles

In this Appendix we compute the fraction of adsorbed particles using the
heavy-tailed distribution model for the random adsorption times. According to
Eq. (44) the fraction of adsorbed particles $\gamma$ can be written as

$\gamma=\lim_{t\to\infty}\frac{1}{t}\sum_{j=1}^{n}\tau_{j}.$ (102)

As we saw in Appendix B we have that, for $0<\alpha<1$ the sum of random
adsorption times can be approximated as (see Eq. (68)),

$\sum_{j=1}^{n}\tau_{j}\approx Wn^{1/\alpha},$ (103)

On the other hand, according to Eq. (74), we can write $t$ as

$t\approx Wn^{1/\alpha},$ (104)

for $0<\alpha<1$ and large $t$. Thus, in the limit $t\to \infty$ it is clear
that

$\gamma=\lim_{t\to\infty}\frac{1}{t}\sum_{j=1}^{n}\tau_{j}=\lim_{n\to\infty}\frac{1}{Wn^{1/\alpha}}Wn^{1/\alpha}=1,$
(105)

which proves that, in the average, all the particles get adsorbed for
$0<\alpha<1$.

For $\alpha=1$, a similar behavior occurs. Looking at Eqs. (89) and (90) , we
can see that,

$\displaystyle\sum_{j=1}^{n}\tau_{j}$ $\displaystyle\approx$ $\displaystyle
Wn\ln(n),$ (106) $\displaystyle t$ $\displaystyle\approx$ $\displaystyle
Wn\ln(n),$ (107)

for $t$ going to infinity. Thus it is clear that

$\gamma=\lim_{n\to\infty}\frac{1}{Wn\ln(n)}Wn\ln(n)=1,$ (108)

for $\alpha=1$.

Next, for the case $1<\alpha<2$, Eqs. (81)and (82) gives us approximations for
$t$ and $\sum_{j=1}^{n}\tau_{j}$ for large $t$,

$\displaystyle\sum_{j=1}^{n}\tau_{j}$ $\displaystyle\approx$ $\displaystyle
n\bar{\tau}$ (109) $\displaystyle t$ $\displaystyle\approx$
$\displaystyle\left(\bar{\tau}+\frac{\lambda}{v}\right)n.$ (110)

These two approximations allows us to compute $\gamma$ as follows,

$\gamma=\lim_{n\to\infty}\frac{1}{\left(\bar{\tau}+\frac{\lambda}{v}\right)n}n\bar{\tau}+Wn^{1/\alpha}=\frac{\bar{\tau}}{\bar{\tau}+\frac{\lambda}{v}},$
(111)

for the case $1<\alpha<2$. It is not hard to see that for the case $\alpha\geq
2$ both, $t$ and $\sum_{j=1}^{n}\tau_{j}$ grow linearly in time (because for
the latter, the central limit theorem is valid), and actually they can be
approximated in the same way as shown in Eqs. (109) and (110). Then we can
extend the last result to state that the fraction of adsorbed particles is
given by

$\gamma=\frac{\bar{\tau}}{\bar{\tau}+\frac{\lambda}{v}},$ (112)

for the case $\alpha\geq 2$ and, in general, when the central limit theorem
takes place.

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