\section{Introduction}
\label{sec:intro}

Hawking radiation has invoked some issues which are of relevance to
not only information loss problem in
quantum gravity theory~\cite{Hawking:1974sw, Hawking:1976ra} but
also black hole complementarity~\cite{Susskind:1993if,
  Susskind:1993mu, Stephens:1993an}. The latter states that there are no
contradictory physical observations between a freely falling observer
and a distant observer since the two descriptions are complementary.
The presence of Hawking radiation indicates that the rest observer at
infinity sees the flux of particles. By the way, Unruh has argued
that ``a geodesic detector near the horizon will not see the Hawking
flux of particles"~\cite{Unruh:1980cg} and then showed that the infalling
negative energy flux
can exist near the horizon \cite{Unruh:1977ga}. Moreover, it has been shown that
the finite flux at the horizon can be found even in the freely falling frame
by studying the Green's function
on the Schwarzschild black hole \cite{Candelas:1980zt}.
Recently, it has been claimed
that a freely falling observer finds something special at the
event horizon called the firewall and burns up because of high energy
quanta~\cite{Almheiri:2012rt}, and subsequently much attention has
been paid to the firewall issue \cite{Bousso:2012as, Nomura:2012sw,
  Page:2012zc, Kim:2013fv, Giddings:2013kcj,
  Almheiri:2013hfa}.  A similar prediction referred to as an energetic
curtain has also been done based on different
assumptions~\cite{Braunstein:2013bra}.
However, it has also been proposed that
there is no apparent need for firewalls since the unitary evolution of black hole
entangles a late mode located outside the event horizon with a combination
of early radiation and black hole states, instead of either of them separately \cite{Hutchinson:2013kka},
and argued that the remaining set of nonsingular realistic states do not have
firewalls but yet preserve information
\cite{Page:2013mqa}.

On the other hand, there has
been some interests in radiation in freely falling frames in its own
right and it has been widely believed that the equivalence
principle tells us that the free-falling observer can not see any
radiation.  This fact is based on the classical argument of
locality but it may not be true in quantum regime
such that a freely falling observer can find quantum-mechanical
radiation and temperature~\cite{Brynjolfsson:2008uc, Greenwood:2008zg,
  Barbado:2011dx, Smerlak:2013cha, Smerlak:2013sga}. Recently, it has been claimed that the freely falling observer
 dropped at the horizon necessarily encounters the infinite
negative energy density when the observer passes through the
horizon~\cite{Kim:2013caa}. On general grounds, one may regard this
phenomenon as a very special feature such a simplified model  \cite{Callan:1992rs} that Hawking
temperature happens to be independent of the black hole mass.
Otherwise, is there any special choice of vacuum to give rise to
the infinite energy density at the horizon?
If the existence of the infinite energy density at the
horizon turns out to be true, one may wonder what happens at the horizon
when the frame is dropped far from
the horizon.

Now, we would like to study the quantum-mechanical energy densities
 measured by the
freely falling observer on the two-dimensional Schwarzschild
black hole background where Hawking temperature explicitly depends
on the black hole mass.  The trace anomaly for massless scalar fields
will be employed to calculate the energy-momentum tensors along with
covariant conservation law of the energy-momentum tensors. Then,
the energy density will be characterized by three states \cite{Birrell:1982ix};
the Boulware,
Unruh, and Hartle-Hawking states in order to investigate what state is
relevant to the infinite energy density at the horizon.  If there
exists such a non-trivial effect at the horizon, then this fact will be tantamount to
the failure of no drama condition which has been one of the
assumptions for black hole complementarity, and
this work will be the
quantum field theoretic realization of non-trivial effect at the horizon \cite{Braunstein:2013bra}.

Now, in section~\ref{sec:ff.frame},
we encapsulate how to formulate the freely falling frame by solving
the geodesic equation of motion and present the explicit form of the
corresponding energy density. First,
the simplest Boulware state will be
studied in section~\ref{boulware} where the energy density is always
negative and divergent at the horizon, which is interestingly irrespective of the
initial free-fall position $r_s$.
In section~\ref{sec:unruh}, we
shall find much more non-trivial effects in the Unruh state such that the
observer finds positive radiation during free fall as long as
$r_s > r_0$ where $r_0$ is a point for the free-fall energy density to
vanish.  By the way, there is a critical point $r_c$ for the observer
to see only negative radiation during free fall for
$r_s <r_c$. The closer the initial free-fall position approaches the horizon,
the larger negative energy density can be found,
so that it can be divergent at the horizon eventually.
There is also an intermediate region of $r_c < r_s < r_0$
for the observer to see negative radiation initially and to find
positive radiation finally at the horizon.
Finally, we explain the reason why the infinite energy density at the horizon appears
in our calculations.
In section~\ref{sec:HH}
for the Hartle-Hawking state, some similar behaviors to the case of
the Unruh state will be reproduced; however, the crucial difference
comes from the fact that the energy density at the horizon is
always finite.  After all, the calculation in this work
will show that the non-trivial
effect measured by the freely falling
observer in the semi-classical argument is sensitive to
the initial free-fall position and the black hole states.
Finally, we will discuss
physical consequences of this work in section~\ref{sec:discussion}.

 \section{Freely falling frame}
 \label{sec:ff.frame}

Let us start with the two-dimensional Schwarzschild black hole
governed by~\cite{Unruh:1980cg,Christensen:1977jc},
\begin{equation}
  ds^2 = -f(r)dt^2 +\frac{1}{f(r)} dr^2, \label{metric}
\end{equation}
where the metric function is given by $f(r)=1-2M/r$ and the horizon is
defined at $r_{\rm H}= 2M$. Solving the geodesic equation for the
metric~\eqref{metric}, the proper velocity of a particle can obtained
as~\cite{Ford:1993bw}
\begin{align}
  u^\mu = \left(\frac{dt}{d\tau}, \frac{dr}{d\tau}\right) =
  \left(\frac{k}{f(r)}, \pm \sqrt{k^2 - f(r)} \right), \label{u:k}
\end{align}
where \(\tau\) and \(k\) are the proper time and the constant of
integration, respectively.  The $k$ can be identified
with the energy of a particle per unit mass for \(k > 1\),
which can be written as \(k = 1/\sqrt{1-v^2}\) with
\(v=dr/dt\) at the asymptotic infinity.
In this case, the motion of the particle is unbounded,
so that the particle lies in the range of \(r\ge r_{\rm H}\).
For \(0 \le k \le 1\), the motion of the particle is bounded such that there is a
maximum point \(r_{\rm max}\) where  the particle lies in the range
of \(~r_{\rm H} \le r \le r_{\rm max}\).
We are going to consider a freely falling frame starting at
$r_s= r_{\rm max}$ with zero velocity toward the black hole, which
can be shown to be the latter case by identifying $k =\sqrt{f(r_s)}$ in Eq.~\eqref{u:k},
and thus the proper velocity of a
free-falling observer can be written as
\begin{equation}
  \label{eq:u}
  u^\mu = \left(\frac{dt}{d\tau}, \frac{dr}{d\tau}\right)
  = \left(\frac{\sqrt{f(r_s)}}{f(r)}, -\sqrt{f(r_s)-f(r)}\right).
\end{equation}
If the observer starts to fall into the black hole at the
spatial infinity, then $f(r_s)=1$ while
$f(r_s)=0$ for the observer to fall into the black hole just at the horizon.
Then, the radial velocity with respect to the Schwarzschild time
becomes $v = -f(r)\sqrt{f(r_s) - f(r)} / \sqrt{f(r_s)}$ which
vanishes both at the initial free-fall position and
the horizon, and the maximum
speed occurs at $r= 6M r_s/(4M+r_s)$.  The
proper time from $r_s$ to $r_{\rm H}$ is also obtained as
\begin{equation}
  \label{propertime}
  \tau = 2M \frac{\sqrt{f(r_s)(1-f(r_s))} + \sin^{-1} \sqrt{f(r_s)}}{(1-f(r_s))^{3/2}},
\end{equation}
which is finite except for the case of the initial free fall at the asymptotic
infinity.
So, it will take finite proper time to reach  the
event horizon when free fall begins at finite distance.

Now, in the light-cone coordinates defined by $\sigma^{\pm}=t \pm r^*$ through
$r^*=r+2M \ln(r/M-2)$ the proper velocity
\eqref{eq:u} can be written as
\begin{align}
  u^+ &=\frac{1}{\sqrt{f(r_s)}+\sqrt{f(r_s)-f(r)}}, \label{u:+} \\
  u^- &=\frac{\sqrt{f(r_s)} + \sqrt{f(r_s)-f(r)}}{f(r)}, \label{u:-}
\end{align}
where  $u^\pm=u^t
\pm u^r/f(r)$
and the energy-momentum tensors are expressed as~\cite{Christensen:1977jc}
\begin{align}
  \langle T_{\pm \pm} \rangle&= -\frac{N}{48 \pi} \left(\frac{2M
      f(r)}{r^3} + \frac{M^2}{r^4} \right) + \frac{N}{48}t_{\pm},
  \label{T:++}   \\
  \langle T_{+-} \rangle &= -\frac{N}{48\pi}
  \frac{2M}{r^3}f(r), \label{eq:Tpm}
\end{align}
where $N$ is the number of massless scalar fields and $t_{\pm}$ are
functions of integration to be determined by boundary conditions. The
general covariance is guaranteed by covariant conservation
law of the energy-momentum tensors. The two-dimensional trace
anomaly for scalar fields was employed to
get the non-trivial vacuum expectation value of the energy-momentum
tensors~\cite{Christensen:1977jc}, so that the two differential equations and one anomaly
equation determine the explicit form of the energy-momentum tensors
with two unknowns.

Now, the energy density measured by the free-falling observer can be
calculated as~\cite{pi, Ford:1993bw},
\begin{equation}
  \label{energy}
  \epsilon = \langle T_{\mu \nu} \rangle u^\mu u^\nu,
\end{equation}
by using the proper velocity and the energy-momentum tensor.
In connection with Hawking
radiation, the fields are quantized on the classical background metric
in such a way that non-trivial radiation will appear and the energy
density~\eqref{energy} will not vanish even in the freely falling frame.
Substituting Eqs.~\eqref{u:+}, \eqref{u:-}, \eqref{T:++} and
\eqref{eq:Tpm} into~\eqref{energy}, the energy density
can be
calculated as
\begin{align}
  \label{eq:ed}
  \epsilon(r|r_s) &= -\frac{N}{48 \pi r^4 f(r)} \Bigg[ 8Mrf(r_s) +
  4M^2 \left(\frac{f(r_s)}{f(r)} - \frac{1}{2}\right)  - \pi r^4\left(\sqrt{\frac{f(r_s)}{f(r)}} -
    \sqrt{\frac{f(r_s)}{f(r)}-1}\right)^2t_+ \notag \\
&\quad -\pi r^4\left(\sqrt{\frac{f(r_s)}{f(r)}} +
    \sqrt{\frac{f(r_s)}{f(r)}-1}\right)^2t_-\Bigg],
\end{align}
and
it is reduced to
\begin{align}\label{eq:edr0
  \epsilon\ (r_s|r_s)= & -\frac{N}{48 \pi r_s^4 f(r_s)} [
  8Mr_sf(r_s)+2M^2 -\pi r_s^4(t_++t_-) ],
\end{align}
at the special limit of $r=r_s$
where observation is done at the moment when free fall begins.
Now, let us investigate characteristics
for the energy density measured by the free-falling observer
for the Boulware, Unruh, and Hartle-Hawking states, respectively
in what follows.

 \section{Boulware state}
 \label{boulware}

The Boulware state is obtained by choosing $t_\pm=0$, where the
energy density~\eqref{eq:ed} reads as
\begin{equation}
  \epsilon_B(r|r_s) = -\frac{NM^2}{12 \pi r^4
    f(r)} \left[\frac{2 r f(r_s)}{M} + \frac{f(r_s)}{f(r)} -
    \frac{1}{2} \right], \label{energy:B}
\end{equation}
which is always negative.  So the
freely falling observer encounters more and more negative energy
density and then eventually negative divergent one at the horizon which
is independent of the initial free-fall position.
If the observation is done at the moment when
free fall begins, the energy density is reduced to
\(\epsilon_B(r_s|r_s)=- N[ 4Mr_sf(r_s)+M^2]/[24 \pi
r_s^4 f(r_s)], \) so that the observer who starts  at the horizon
finds the divergent energy immediately
and asymptotically vanishes without Hawking
radiation as shown in Fig.~\ref{fig:EB}.

\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=0.6\textwidth]{EBB1
  \end{center}
  \caption{The energy densities in the Boulware state are plotted by
    simply choosing as $N=12, M=1$. They are
 always negative no matter where the initial free-fall positions are.
    The solid curves represent  $\epsilon_B \ (r|r_s)$,
    and the dotted curve does $\epsilon_B\ (r_s|r_s)$
     where three energy densities at $r_s$ are denoted
    by the black dots (a), (b), and (c).}
  \label{fig:EB}
\end{figure}

 \section{Unruh state}
 \label{sec:unruh}

The Unruh state is characterized by choosing functions of integration as
$t_+=0$ and $t_-=1/(16 \pi M^2)$ in Eq.~\eqref{eq:ed}, which yields the
energy density as
\begin{align}
  \epsilon_U(r|r_s) &= -\frac{N M^2}{12 \pi r^4 f(r)} \Bigg[ \frac{2 r
    f(r_s)}{M} + \frac{f(r_s)}{f(r)} - \frac{1}{2} - \frac{r^4}{64M^4} \left(\sqrt{\frac{f(r_s)}{f(r)}} +
    \sqrt{\frac{f(r_s)}{f(r)}-1} \right)^2 \Bigg]. \label{energy:U}
\end{align}
The energy density measured by the free-falling observer at the horizon
who falls into the black hole from $r_s$ is simplified as $ \epsilon_U (2M|r_s) =( N(63r_s^2-320M
r_s+384M^2 )/[3072 \pi M^2 r_s (r_s-2M )] $,
\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=0.6\textwidth]{EUU1
  \end{center}
  \caption{The energy densities for the Unruh state are plotted by
    setting $N=12,\ M=1$.  The critical point appears at $r_c \approx
    3.1M$ and the energy density vanishes at $r_0 \approx 4.2M$.
    The solid curves are for $ \epsilon_U(r|r_s) $ and the
    dotted curve represents $\epsilon_U (r_s|r_s)$ such that
    there are largely three free-fall cases: (a) is for $r_s < r_c $, (b) is for
    $r_c <r_s<r_0$, and (c) is for $r_s  >r_0$.}
  \label{fig:EUU}
\end{figure
which is not always positive definite.
In other words, the initial free-fall position
is crucial to determine the sign of the energy density at the horizon in contrast to
the Boulware case.
Specifically, the energy density at the horizon is indeed positive for
$r_s > r_0$ where $ r_0$ is the initial free-fall position for the energy
density to vanish. For instance, it is positive finite as seen
from the case (c) in  Fig.~\ref{fig:EUU}, and it becomes
$\epsilon_U(2M| \infty) = 21N/(1024 \pi
M^2)$ where the free-fall frame is dropped at the
spatial infinity.
On the other hand, there is a critical point $r_c =
8(20M+\sqrt{22}M)/63$ which is defined by the point where the observer
finds the zero energy at the horizon so that the observer will
see the positive energy at the horizon as long as $r_s >r_c$.
 For $r_c < r_s
<r_0$, there appears a transition from the
negative energy density to the positive energy density, which can be seen from the
case (b) in Fig.~\ref{fig:EUU}.
For $r_s <r_c$, the observer will see only negative radiation at the
horizon like the case (a).
When the initial free-fall position approaches the horizon closer,
the larger negative energy density appears.

Using Eq.~\eqref{eq:edr0}, the energy density can be obtained
at the moment when the
free fall just begins, then the corresponding energy density is
given as \( \epsilon_U(r_s|r_s) = -NM^2 [ 4 r_s f(r_s)/M + 1 -
r_s^4/(32 M^4) ]/[24 \pi r_s^4 f(r_s)] \) which is described by the
dotted curve in Fig.~\ref{fig:EUU}.  Note that the energy density
at the horizon $\epsilon_{U}(2M| 2M) $ is
negative divergent whereas it is positive finite
$\epsilon_{U}(\infty|\infty) \rightarrow \pi (N/12)T_{\rm H}^2$ at the
asymptotic infinity, where $T_{\rm H}$ is the Hawking temperature.

In particular, we would like to explain why the freely falling observers who are
moving slowly with respect to the black hole when they pass through the horizon should see very high
(negative) energy density. Actually, the conventional wisdom is that the freely falling observer near the
horizon cannot see any outgoing Hawking radiation as $\langle T_{--}\rangle=0$.
It can be easily understood from the Unruh effect \cite{Unruh:1980cg}
which tells us that the frame near the horizon can be described as the local-flat metric in terms of
the Kruskal coordinates on the
Schwarzschild black hole, so that the corresponding observer can be regarded as the freely falling observer
while the fiducial observer sees radiation because the observer is now on the accelerated frame.
Moreover, in the collapsing black hole described by the Unruh state, it was shown that
the energy flow across the future horizon is seen to be negative, $\langle T_{++} \rangle <0 $,
 since the corresponding
positive energy would flow out to infinity \cite{Unruh:1977ga}. All these facts can also be confirmed
by using Eqs.~\eqref{T:++} and \eqref{eq:Tpm}.

In this work, we employed the energy density which consists of three components
of $\langle T_{++} \rangle,~\langle T_{--} \rangle,~\langle T_{+-} \rangle$,
while the energy-momentum tensors related to the fluxes have been discussed in the previous works.
Explicitly, the energy density~\eqref{energy}
can be reduced to $\epsilon = \langle T_{++} \rangle u^+u^+ $
at the horizon since $ \langle T_{--}\rangle=\langle T_{+-}\rangle=0$ there.
Note that it does not vanish but also is negative because of non-vanishing ingoing negative flux as
$\langle T_{++} \rangle =-N/(768 \pi M^2) <0 $ from Eq.~\eqref{T:++}. At the horizon, the non-vanishing energy density
is related to the non-vanishing ingoing energy momentum tensor as it should be.
To explain the reason why the high energy density appears near the horizon for a
very slowly falling frame,
let us rewrite the free-fall energy density \eqref{energy}
 as $\epsilon = \langle T_{tt} \rangle u^tu^t$
in the normal coordinates
where the radial velocity is
fixed as $u^{r}=0$ for convenience when the observer is dropped from rest at $r=r_s$.
Note that the time component of the velocity at the stating point of $r_s$ by definition
becomes $u^t=dt/d\tau=1/\sqrt{f(r_s)}$ which
is larger than one because the function $f(r_s) $ is less than one near the horizon, so that
$dt > d\tau$ where
$dt$ is a time measured by the fiducial observer
 and $d\tau$ is a proper time measured by the freely falling observer.
Moreover, it shows that the gravitational time dilation effect
is much more significant when the observer is dropped close to the horizon.
By the way, as a corollary to this fact, the frequency in the freely falling frame is higher than
that in the fixed frame, so that this factor contributes to the energy density.
Therefore, it eventually becomes the high energy density of $\epsilon(r_s|r_s)  =-N/(768 \pi M^2 f(r_s))$
near the horizon,
where $r_s$ represents the starting position when the observer is dropped from rest.
On the other hand, if the freely falling observer starts with the non-zero initial velocity at
a certain point from the horizon, then the observer can see the positive energy at that instant because
$\langle T_{tr} \rangle$ with $u^r \neq 0$ gives rise to the positive contribution to the energy density.

One more thing to be mentioned in this section is that we could calculate
 the free-fall energy density not only at any finite distance but also near the horizon and at infinity
in the simplified context which is one of  the advantages of the two-dimensional model,
so that we could further discuss
the critical point to characterize the  positive energy zone and the negative energy zone
by solving the exact geodesic equation analytically.
The result shown in Fig.~\ref{fig:EUU} is physically compatible with the previous one that
the positive energy flux would flow out to infinity while a corresponding amount of negative
energy flux would flow down the black hole \cite{Unruh:1977ga},
so that the area of horizon decreases at a rate
expected positive energy flux at infinity \cite{Candelas:1980zt}.

 \section{Hartle-Hawking state}
 \label{sec:HH}

For the Hartle-Hawking state, let us take
$t_\pm=1/(16\pi M^2)$ in Eq.~\eqref{eq:ed}, then the energy density
can be obtained as
\begin{align}
  \label{hh}
  \epsilon_{HH}(r|r_s) = & -\frac{N M^2}{12 \pi r^4 f(r)}
  \left[\frac{2 r f(r_s)}{M}  - \left(\frac{r^4}{16M^4} - 1\right)
    \left(\frac{f(r_s)}{f(r)} - \frac{1}{2}\right)\right].
\end{align}
The free-falling observer at $r_s$ toward the black hole will find the finite
energy density at the horizon of \(\epsilon_{HH}(2M|r_s)=N (r_s - 3M) /
(48\pi M^2 r_s)\). In particular, it becomes
$\epsilon_{HH}(2M|\infty) =
N/(48M^2\pi)$ when the observer is dropped at
spacial infinity at rest.
There is the point $r_0$ where the energy density measured
in the free-falling frame vanishes; however,
the crucial difference from the Unruh case is that the freely falling
observer starting at $ r_s >r_0$ may encounter alternatively the positive energy and
the negative energy density during the free fall as shown in the
case (b) in Fig.~\ref{fig:EHH},
whereas only
the positive energy density appears
in the Unruh state.
\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=0.6\textwidth]{EHH1
  \end{center}
  \caption{The energy densities in the Hartle-Hawking state are plotted
    by setting $N=12,\ M=1$. The sold curves describe $\epsilon_{HH} (r|r_s)$
    and the dotted curve represents $\epsilon_{HH} (r_s|r_s)$.
    There are largely three free-fall cases (a) is for $r_s < r_0$, (b)
    in the box is for $r_0< r_s <r_c$, and (c) is for $r_s >
    r_c$, where $r_0 \approx 2.98M$  and $r_c=3M$.}
  \label{fig:EHH}
\end{figure
 There is also the critical point $r_c$ to
characterize the sign of the energy density at the horizon as has been
done in the Unruh case, so that the freely falling observer at the horizon will see the
positive energy density for $r_s >r_c$ and the negative
energy density for $r_s <r_c$ . Moreover,  the
observer will necessarily find a transition from the positive
energy density to the negative energy density for $r_0<r_s <r_c$.
Note that $r_0$ and $r_c$ in the Hartle-Hawking state are smaller than
those in the Unruh state, respectively, which is shown in
Fig.~\ref{fig:EHH}.

 At $r =r_s$, the energy density in the Hartle-Hawking state
from Eq.~\eqref{eq:edr0} becomes \(
\epsilon_{HH}(r_s|r_s) = - N [8Mr_sf(r_s) + 2M^2 - r_s^4/(8 M^2) ] /
[48 \pi r_s^4 f(r_s)], \) where the behavior of the energy density is
described by the dotted
curve in Fig.~\ref{fig:EHH}.
Explicitly, when $r_s = 2M$, it becomes
negative finite as $\epsilon_{HH}(2M|2M) \rightarrow -N/(96\pi M^2)$
which is contrast to the infinite energy density in the Unruh state.
At the asymptotic infinity, it is finite $
\epsilon_{HH}(\infty|\infty) \to \pi N T_{\rm H}^2/6$, and the energy
density in the Hartle-Hawking state is two times that of the Unruh
state, $i.e.$, $\epsilon_{HH}(\infty|\infty)=2\epsilon_U(\infty|\infty)$.

 \section{Discussion}
 \label{sec:discussion}
At first sight, it seemed that the freely falling observer could not find
any radiation based on the intuitive argument that the surface gravity translated into
the radiation temperature vanishes in local inertial frames;
however, the explicit calculation shows that
radiation can exist even in freely falling frames and moreover
it depends on the free-fall position unequivocally for certain black hole states.
For the
Boulware state, the energy density is always negative divergent at the
horizon, which is independent of the initial free-fall positions.
For the Hartle-Hawking state, the energy density at the horizon is negative finite
when free fall toward the black hole begins at $r_s < r_c$ while it is
positive finite at the horizon for $r_s > r_c$, where $r_c=3M$ is the
critical initial free-fall position to determine the sign of the energy density
at the horizon.  In particular, the Unruh state describing
the collapsing black hole gives a slightly larger critical
value of $r_c \approx 3.1 M$ compared to that of the Hartle-Hawking
state. As expected, the energy density at the horizon in the Unruh state behaves as mixed between
the Boulware state and the Hartle-Hawking state in that the energy density at the horizon
can be either divergent or finite.  Therefore, the energy density
measured by the freely falling observer in the semi-classical black
hole is characterized by mainly two conditions of both the initial free-fall
position and the black hole state.

In connection with black hole complementarity,
let us drop the particle at a finite distance near the
horizon such that it will take finite proper time for the freely falling observer to reach the horizon.
Subsequently, the particle will reach
the origin without any drama at the horizon.  On the other hand, the
distant observer using the Schwarzschild coordinates feels that the
particle is still at rest at the horizon so that the observer cannot
see the particle crossing the horizon forever. But, the particle will
appear awkwardly through Hawking radiation eventually.  To resolve
this problem, the notions of the membrane paradigm and the
stretched horizon with black hole complementarity are helpful in
understanding the relation between the infalling particle and Hawking
radiation \cite{Susskind:1993if,
  Susskind:1993mu, Stephens:1993an}.
 However, the present calculation will modify black hole
complementarity since the freely falling observer can perceive radiation
throughout free fall especially at the event horizon. It means
that when the observer passes through the horizon, he/she sees that
the infalling particle may be either excited or annihilated by the
energy density around the black hole.

 \acknowledgments %
 WK was indebted to Shailesh Kukarni, Andrei Smilga, Donghan Yeom, and
 Sang-Heon Yi for exciting discussions.
This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIP) (2014R1A2A1A11049571).


\section{Introduction}
 Molecular communication (MC) is a promising solution for enabling the transfer of information in several scenarios considered impractical for conventional electromagnetic wave-based communication (EMC). For example, MC is expected to perform better than EMC inside tunnels, in saline environments, and inside the human body \cite{farsad23,guo2015}. In MC, the transmitter conveys the information to the receiver by utilizing molecules, termed as \textit{information molecules} (IMs), as the carrier of the information. Among different propagation mechanisms, the diffusion-based propagation mechanism is the most studied one in the literature due to the ease in mathematical modeling, energy efficiency, and no requirement of communication infrastructure for carrying IMs from the transmitter to the receiver.
In MC, the transmitters and receivers are usually nanomachines with nanoscale functional parts. The functionality of a single nanomachine is limited to simple actuation, sensing, and storage \cite{nakano2013}. However, complex tasks like target detection and event sensing can be performed via the cooperation of multiple nanomachines. Such systems with multiple nanomachines working together can help to realize the internet of nano \cite{akyildiz2010}, and bio-nano things (IoNT and IoBNT) \cite{akyildiz2015}.

 \subsection{Related work}
For an MC system with a point transmitter and a fully-absorbing receiver (FAR), the analytical expression for the hitting probability was derived in \cite{yilmaz2014,schulten2000lectures}. As mentioned above, multiple receiver nanomachines can exist in an MC to enable complex tasks. Therefore, the channel model for an MC system when multiple receivers are present is of crucial importance. Many works \cite{koo2016, damrath2018,lee2017,ahuja2021,gursoy2019a} employ fitting models, machine learning-based approaches, and particle-based simulations to obtain the channel model. However, an analytical equation for the hitting probability for MC systems involving multiple fully absorbing receivers in 3-D is not present in the past literature. In our past work \cite{sabu2020a}, we have derived
an analytical expression for the hitting probability of an IM on each of the FAR of the same size in a $ 2-$FAR system. The approximate expression was simple enough so that several design insights were discussed in \cite{sabu2020a}. Later this work was extended to derive the hitting probability of IM on each FAR with different size \cite{sabu2021}. For MC systems involving more than two FARs, an approximate expression for the hitting probability of an IM on \textit{any one} of the FARs was derived in \cite{sabu2020}. However, the hitting probability on each of the FARs is still missing from the current literature.
Systems with multiple receivers may improve the performance of the MC systems \cite{koo2016, damrath2018,lee2017,ahuja2021,gursoy2019a}. In MC systems with multiple receivers, cooperative detection can be used to improve the error performance  \cite{yutingfang2017}. In a cognitive MC system, multiple primary and secondary receivers may be present.  The analysis of the systems with multiple FARs will require a channel model to start. Therefore, it is essential to analyze and characterize the channel in MC systems with multiple receivers.
According to the authors knowledge, no work in the literature provides an analytical expression for the 3-D channel model of systems with multiple FARs (\textit {i.e., } more than two FARs) using the same type of IM. Obtaining hitting probability using machine learning and particle-based simulation methods \cite{koo2016, damrath2018,lee2017,ahuja2021,gursoy2019a}  is time-consuming and inconvenient. In this work, we bridge this gap by deriving the analytical hitting probability on each FAR in a multi FAR system and characterizing the channel in this system.

 \subsection{Contributions}
In this work, we consider a 3-D diffusion-based MC system with multiple FARs and derive the hitting probability of an IM on each of the  FARs. We also validate the accuracy of the derived equations using particle-based simulations. In contrast to \cite{sabu2020a} which only considers two FARs, this work considers $N$ FARs and include the necessary modifications in the differential equations to incorporate the mutual influence of all the FARs. We also analyze a single-input multiple-output (SIMO) system comprising a UCA of FARs based on cooperative detection schemes. Several important design insights are also provided based on the derived expressions.
The novel contributions of this work are listed below.
\begin{itemize}
	\item First, we consider a 3-D diffusion-based MC system with 3 FARs and derive an approximate analytical expression for the hitting probability on each FAR. We validate the derived expressions using particle-based simulations.
\item We then extend it to derive the general approximate analytical expression for the hitting probability on each FAR in a system with $ N $ FARs for an arbitrary value of $N$. We present another method that is recursive and can be convenient to use when hitting probability for subsystems that are already known.  We also develop several simplified expressions under special cases, including symmetric systems.
\item  To demonstrate the applicability of the derived expressions, we  consider a single-input multiple-output (SIMO) communication system comprising of a point transmitter and a UCA of FARs as a compound receiver and present the analytical expressions for the hitting probability.
\item Using these expressions, we now derive the channel model for the considered SIMO system and present various design insights, including the asymptotic array gain of UCA.
\item Using the derived channel model, we analyze the communication performance of the considered SIMO system with a cooperative detection scheme and compare the performance for different fusion rules. We also quantify the impact of increasing the number of FARs on the performance.
\end{itemize}

\section{System Model}

	In this work, we consider an diffusion based MC system with a point transmitter and $N$ spherical fully absorbing receivers (FARs) located  at different positions in a 3D medium as shown in Fig. \ref{Fig:sm}. Let the transmitter be located at the origin and $N$ FARs of the same radius $ a $ be at positions $ \pmb{x}_{1}, \pmb{x}_{2} $, $ \pmb{x}_{3} $,....,\ $ \pmb{x}_{N} $ respectively. The distance between the transmitter and the $i$th FAR FAR$_i$ located at $ \pmb{x}_i $ is represented by $ r_i $. The angular difference between the center of the
	FAR$_i$ and FAR$_j$
	is denoted by $ \phi_{ij} $. Let $ \pmb{y}_i $ denote the closest point of FAR$ _i $ from the origin.  Also, $ R_{ij} $ represents the distance between the closest point of FAR$_i  $ from origin (\textit {i.e., } $ \pmb{y}_i $) and the center of FAR$ _j$. \textit {i.e., } $ R_{ij}=\sqrt{(r_i-a)^2+r_j^2-2(r_i-a)r_j\cos\phi_{ij}} $.
	 The transmitter emits information molecules (IMs), which propagates through the medium via Brownian motion \cite{einstein2011}.  The diffusion coefficient of the IM is represented by $ D $, which is assumed to be constant over space and time. The IMs are detected at a FAR when they hit the surface of that particular receiver.

	\begin{figure}[ht]
		\centering
		\includegraphics[width=3.4in]{NrxSM}
		\caption{An illustration showing w 3D MC  system with a point transmitter at the origin and FARs located at positions $\pmb{x}_{1},\ \pmb{x}_{2},\cdots, \pmb{x}_{N}$ in $ \mathbb{R}^3 $.
		}
		\label{Fig:sm}
	\end{figure}

	When there are no other FARs present except the $i$th FAR, the hitting probability $ \overline{\mathsf{p}}(t,r_i) $ of an IM at the surface of the $ i $th FAR within time $t$

is given by \cite{yilmaz2014,schulten2000lectures}
	\begin{align}
		\overline{\mathsf{p}}(t,r_i)=\frac{a}{r_i}\mathrm{erfc}\left(\frac{r_i-a}{\sqrt{4Dt}}\right).\label{eq1rx}
	\end{align}

	Now, when there is other FARs present, it causes competition in capturing the IMs among the FARs. Therefore, the hitting probability reduces in the presence of other FARs. For the case when there are only two FARs located at $ \pmb{x}_1 $ and $ \pmb{x}_2 $  present in the medium  (\textit {i.e., }
	  $ N=2 $), an approximate analytical hitting probability until time $ t $ of an IM on FAR$ _1 $
 was given in our previous work \cite{sabu2020a} as
	\begin{align}
		\hp{1}(t,\pmb{x}_1\mid \pmb{x}_2)

		&\sum_{n=0}^{\infty} \frac{a^{2 n}}{R_{12}^{n} R_{21}^{n}}\left[\frac{a}{r_{1}} \mathrm{erfc}\left(\frac{r_{1}{-}a+n\left(R_{21}{-}a\right)+n\left(R_{12}{-}a\right)}{\sqrt{4 D t}}\right)\right. \nonumber\\
		&\left.-\frac{a^{2}}{r_{2} R_{21}} \mathrm{erfc}\left(\frac{r_{2}{-}a+(n+1)\left(R_{21}{-}a\right)+n\left(R_{12}{-}a\right)}{\sqrt{4 D t}}\right)\right]\le \overline{\mathsf{p}}(t,r_i).
		\label{eq2rx}
	\end{align}

	The hitting probability reduces further with an increase in $N$, due to the increase in the mutual influence of FARs, which is caused by the increasing competition in capturing the IMs by the FARs. However, the expression for the general $N$ case is not available in the literature.
The hitting probability expressions for $ N>2 $ can be used to analyze SIMO, multiple-input multiple-output (MIMO), and cognitive molecular communication systems. In the following sections, we first derive the hitting probability (or fraction of IMs absorbed) at each of the FARs for a $ 3-$FAR system, and later it is extended to $ N- $FAR system. To demonstrate the applicability, the derived equations are then applied to a SIMO system based on co-operative detection, and its error performance is analyzed.

Let us denote the hitting probability until time $ t $ of an IM on FAR$ _i $ in the presence of the other $ N-1 $ FARs by $ \hp{}\left(t,\pmb{x}_i\mid \{\pmb{x}_j\}_{j=1,\ j\neq i}^N\right) $ or just $ \hp{i}(t) $ when there is no ambiguity. Without the loss of generality, we will take $ i=1 $ and derive the hitting probability at FAR$ _1 $ denoted by $ \hp{}\left(t,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^N\right) $ or just $ \hp{1}(t) $. Let
$ \mathcal{L}\left[.\right] $ and
$ \mathcal{L}^{-1}\left[.\right] $ denote the Laplace transform and inverse Laplace transform respectively.

	\section{Hitting Probability for a $3-$FAR system}
In this section, we derive the  hitting probability for a $ 3-$FAR system.
Let us consider a MC system with  three FARs located at $ \pmb{x}_1,\ \pmb{x}_2 $ and $ \pmb{x}_3 $ in $ \mathbb{R}^3 $. In the $3-$FAR scenario, the hitting probability of an IM at FAR$ _1$ would be less than the hitting probability for 2 FAR scenario as given by
 \eqref{eq2rx}
 due to the increased competition in capturing the IM by the additional FAR.
An approximate analytical equation for $\hp{1} (t)$ is given in the following theorem.
\begin{theorem}\label{t3rx}
	In an MC system with three FARs located at $\pmb{x}_1,\ \pmb{x}_2$ and $\pmb{x}_3$, the probability that an IM emitted from the point source at the origin hits  $\mathrm{FAR} _1 $ within time $ t $ in the presence of other two FARs is approximately given by
	\begin{align}
	\hp{1}(t)&=\mathcal{L}^{-1}\left[\lhp{1}(s)\right],\label{eq3rx}
\text{ where }
	\lhp{1}(s)= \frac{\overline{\mathcal{P}}(s,r_i) -s\alpha_1(s)+s^2\beta_1(s)}
	{1-s^2\gamma(s)+s^3\delta(s)}, \\
 	\text{with }\alpha_1(s)&=\overline{\mathcal{P}}(s,r_2)\overline{\mathcal{P}}(s,R_{21}){+}\overline{\mathcal{P}}(s,r_3)\overline{\mathcal{P}}(s,R_{31})\nonumber\\
 	\beta_1(s)&=-\overline{\mathcal{P}}(s,r_1)\overline{\mathcal{P}}(s,R_{23})\overline{\mathcal{P}}(s,R_{32})+\overline{\mathcal{P}}(s,r_2)\overline{\mathcal{P}}(s,R_{23})\nonumber\\
 	 &\qquad\qquad\overline{\mathcal{P}}(s,R_{31})+\overline{\mathcal{P}}(s,r_3)\overline{\mathcal{P}}(s,R_{32})\overline{\mathcal{P}}(s,R_{21})\nonumber\\
 	\gamma(s)&=\overline{\mathcal{P}}(s,R_{12})\overline{\mathcal{P}}(s,R_{21}) + \overline{\mathcal{P}}(s,R_{32})\overline{\mathcal{P}}(s,R_{23}

	+\overline{\mathcal{P}}(s,R_{13}) \overline{\mathcal{P}}(s,R_{31})\nonumber\\
 	\text{and }\
 	\delta(s)&=\overline{\mathcal{P}}(s,R_{12}) \overline{\mathcal{P}}(s,R_{23})\overline{\mathcal{P}}(s,R_{31})

	+ \overline{\mathcal{P}}(s,R_{13})\overline{\mathcal{P}}(s,R_{32})\overline{\mathcal{P}}(s,R_{21})\nonumber.\\
\text{Here, }	\overline{\mathcal{P}}(s,x)&=\laplace{}{\overline{\mathsf{p}}(t,x)}=\frac{a}{sx}\exp\left(-\left(x-a\right)\sqrt{\frac{s}{D}}\right)\nonumber\hspace{3.5in}\ \quad
\end{align}
with $\ x\in\{r_1,R_{ij}\}$ for all $i,j\in\{1,2,3\}, i\neq j
$
\begin{proof}
	See Appendix \ref{a3rx}.
\end{proof}
\end{theorem}
As explained in Appendix \ref{a3rx}, the only source of approximation in the above result is due to the approximation of the hitting point of an IM at a FAR to the closest point on the surface of the FARs as seen from the transmitter at the origin. The approximation is minimal, and does not affect the accuracy much, as shown in Section \ref{acc3rx}. Further, the approximation can be easily removed and the exact expression for hitting probability can be obtained if the distribution of hitting point at the surface of FAR$_i$ is known. In this case, there would be an expectation with respect to the distribution of the hitting point around terms involving $R_{ij}$.

The derived  hitting probability equation can be computed using numerical simulations. Note that, in Theorem \ref{t3rx}, $ \alpha_1(s) $ corresponds to the motion of IMs via FAR$ _2 $ to FAR$ _1 $ and via FAR$ _3 $ to FAR$ _1 $. This term double counts the IMs that went to both FAR$ _2 $ and FAR$ _3 $ (including IMs first went to FAR$ _2 $ and IMs first went to FAR$ _3 $). This double counting is negated by  $ \beta_1(s) $. The first term in $ \beta_1(s) $ corresponds to the IMs that went to FAR$ _1 $ first and then goes to FAR$ _2 $ and FAR$ _3 $. Now, the IMs can go in loops between FAR$ _2 $ and FAR$ _3 $ before hitting FAR$ _1 $ and such IMs are factored in by the term $ \gamma(s) $, whereas the double counting in $ \gamma(s) $ is negated using the term $ \delta(s) $.

When the FARs are equidistant from the transmitter at the origin and equidistant from each other, Theorem \ref{t3rx} simplifies, and the simplified expression is given in the Corollary \ref{c3rxeq}.

\begin{coro}\label{c3rxeq}

	In an MC system with three FARs that are equidistant from the transmitter at the origin and equidistant from each other (\textit {i.e., } $ r_i=r \text { and } R_{ij}=R,\ i,j\in{1,2,3},\ i\neq j$), the probability that an IM emitted from the point source at the origin  hits  $\mathrm{FAR} _1 $ within time $ t $ in the presence of other two FARs is

	\begin{align}
		\hp{1}(t)=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-2a)^n}{R^n}\mathrm{erfc}\left(\frac{r-a+n(R-a)}{\sqrt{4Dt}}\right).\label{eq3rxeq}
	\end{align}
\end{coro}
\begin{proof}
	See Appendix \ref{a3rxeq}.
\end{proof}
\subsection{Evaluation of hitting probability in a 3- FAR System}\label{acc3rx}
\begin{figure}
	\centering
	\includegraphics[width=3.4in]{Figures/threerx}
	\caption{Variation of the hitting probability $\hp{1}(t)$ versus the time $t$ for a $ 3-$FAR system. Parameters: $\pmb{x}_1= [20,0,0],\ \pmb{x}_2=[-20,10,0],\ \pmb{x}_3=[20,-15,0],\ a=3\ \mu m,\ D=100\mu m^2/s,\ \Delta t=10^{-4}\ s $. The solid lines represent the analytical value, and markers represent the simulation values.  The dashed lines show the value in the absence of other FARs.}
	\label{fig:3rxf1}
\end{figure}
Theorem \ref{t3rx} is validated using particle-based simulation using MATLAB \cite{suzuki2017} and is shown in Fig. \ref{fig:3rxf1}. The numerical inverse Laplace transform function from \cite{mcclure2021} is used to evaluate the analytical results. In the particle-based simulation, an IM is generated in each iteration, and the Brownian motion of the IM is tracked. The location of the IM in $ \mathbb{R}^3 $ is tracked and checks whether it hits any of the FAR. If it hits any FAR, it is removed from the environment. After many iterations, the average hitting events at each FAR are taken to get the hitting probability.  The step size $ \Delta t $ taken for the simulation is $ 10^{-4}\ s $ . The dashed lines in the Fig. \ref{fig:3rxf1} corresponds to the hitting probability of IMs in the absence of other FARs (\textit {i.e., } \eqref{eq1rx}). \par
We can verify that the approximate analytical equation is in a good match with the particle-based simulation results. The growing deviation between the dashed and solid lines with time $ t $ accounts for the increasing mutual influence of FARs with time $ t $.

\section{Hitting probability for a $N-$FAR System}
Now, we extend the analysis to derive
the  hitting probability expression for a  $ N-$FAR system for an arbitrary $N$.
The hitting probability of an IM on each of the $ N-$FARs located in $ \mathbb{R}^3 $ in a diffusion-based MC system  is presented in Theorem \ref{tnfar}.

\begin{theorem}\label{tnfar}

	In an MC system with $N$ FARs, the probability that an IM emitted from the point source at the origin  hits  $\mathrm{FAR} _1 $ within time $ t $ in the presence of other $N-1$ FARs is

	 \begin{align}
	 	\mathsf{P}(t)=\mathcal{L}^{-1}\left(\mathcal{A}^{-1}(s)\mathbf{P}(s)\right),\label{eqnxr}
	 \end{align}
 where
 \begin{align*}
 	&	\mathcal{A}(s)

		\begin{bmatrix}
 		1 & s\overline{\mathcal{P}}(s,R_{21}) & s\overline{\mathcal{P}}(s,R_{31}) & \cdots & s\overline{\mathcal{P}}(s,R_{N1}) 	\\
 		s\overline{\mathcal{P}}(s,R_{12}) & 1 & s\overline{\mathcal{P}}(s,R_{32}) & \cdots & s\overline{\mathcal{P}}(s,R_{N2})\\
 		\vdots &\vdots &\vdots&\ddots&\vdots \\
 		s\overline{\mathcal{P}}(s,R_{1N}) & s\overline{\mathcal{P}}(s,R_{2N}) & s\overline{\mathcal{P}}(s,R_{3N}) & \dots & 1 \\
 	\end{bmatrix}\nonumber\\
 	&	\mathsf{P}(t)=\left[ \begin{array}{c} \hp{1}(t) \\ \hp{2}(t)\\ \vdots \\\hp{N}(t) \end{array} \right] \text{ and } \mathbf{P}(s)=\left[ \begin{array}{cc} \overline{\mathcal{P}}(s,r_1) \\ 	\overline{\mathcal{P}}(s,r_2)  \\ \vdots \\ \overline{\mathcal{P}}(s,r_{N}) \end{array} \right].
 \end{align*}
\end{theorem}
\begin{proof}
	See Appendix \ref{anxr}.
\end{proof}

The mutual influence of FARs in capturing the IM can be  characterized by the decrease in the hitting probability due to the presence of additional FARs as given in the following Lemma.
\begin{lemma}\label{cdiff}
	The fraction of IMs that are supposed to hit FAR$ _i $ within time $ t $, but are hitting the other FARs first due to their presence in the same communication medium  is given by
	\begin{align}
		\hpdiff{i}(t)=\overline{\mathsf{p}}(t,r_i)-\hp{i}(t).\label{ediff}
	\end{align}
\end{lemma}

It is worth mentioning that $ \hpdiff{i}(t) $ quantifies the mutual influence of FARs in capturing the IM.

\begin{coro}
In a MC system with $N$ FARs that are far from each other (\textit {i.e., } $ R_{ij}\rightarrow 0,\forall\  i,j\in\{1,2,\cdots N\},\ i\neq j $), the mutual influence of FARs perishes, and the hitting probability for an IM on each FAR is the same as \eqref{eq1rx}.
\end{coro}
\begin{proof}
	When $R_{ij}\rightarrow \infty $, $ s\overline{\mathcal{P}}(s,R_{ij})\rightarrow 0 $. Therefore, $ \mathcal{A}(s)=\mathrm{I} $.\\
	Now,  from \eqref{eqnxr}, $ \mathsf{P}(t)= \mathcal{L}^{-1}(\mathbf{P}(s))$.
\end{proof}

\subsection{Accuracy of the Hitting Probability Equations}
\begin{figure}
	\centering
	\includegraphics[width=3.4in]{Figures/AbErrnew}
	\caption{Variation of the absolute error of the hitting probability of FAR$ _4 $ in a $ 4- $FAR system. Parameters: $ a=3\mu m,\ D=100\mu m^2/s,\ t=1s,\ \Delta t=10^{-4}s,\ \pmb{x}_1=[10,\ 20,\ 0],\ \pmb{x}_2=[10,\	14.14,\	14.14],\  \pmb{x}_3=[10,\	14.14,\	-14.14],\  \pmb{x}_3=[10,\ y,\	z], \text{where } y,z$ is varied in $ y,z $ axis.}
	\label{fig:aberr}
\end{figure}
Fig. \ref{fig:aberr} shows the variation of the absolute error ($= \mid $Analytical result $ - $ Simulation result$ \mid $) of hitting probability at FAR$ _4 $ for a $4-$ FAR system. Here, the transmitter is at the origin, and the centers of the other FARs (blue circles) are located in a circle (dotted circle). The gray circles represent the area where FAR$_4$ cannot be located (as the FARs will overlap). It can be verified that the absolute error is minimal in most regions. Also, note that the derived equation is a good approximation for the UCA arrangement.

Similar to the 2-FAR system \cite{sabu2020a}, the accuracy of the derived equation can be bad if the FARs are too close to each other and the target FAR hinders the other FARs from the line of sight of the transmitter. In such scenarios, the approximating hitting point to a single point $ \pmb{y}_i $ is not a good approximation because the IM may hit the FARs at a different random point.

\subsection{Recursive model}
We now present one more method to derive the hitting probability.
This method is a recursive method that gives the same expression for the hitting probability; however, it can be convenient when expressions for the sub-systems are known.
In this method, the Laplace transform of the hitting probability for an $ N-$FAR system is expressed in terms of the Laplace transform of the hitting probability for an $ N-1 $ FAR system. Taking its inverse Laplace transform gives the corresponding hitting probability in the time domain as shown in Theorem \ref{threc}.

\begin{theorem}\label{threc}
	In an MC system with $N- $FARs, the  probability that an IM emitted from the point source at the origin hits  FAR$ _1 $  within time $ t $ is given as
	\begin{align}
	\hp{}\left(t,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^N\right)=\mathcal{L}^{-1}\left[\lhp{}\left(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^N\right)\right],
	\end{align}
where the Laplace transform $\lhp{}\left(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^N\right)$ of the hitting probability in $N-$FAR system can be recursively written in terms of  the Laplace transform  of the hitting probability in $N-1-$FAR systems $\{x_j\}_{j=1}^{N-1}$ and $\{x_j\}_{j=2}^{N}$ as
	\begin{align}
		&\lhp{}\left(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^N\right)=\frac{\lhp{}\left(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^{N-1}\right){-} s\lhp{}\left(s,\pmb{x}_N{\mid} \{\pmb{x}_j\}_{j=2}^{N-1}\right)\lhp{}\left(s,\pmb{x}_1-\pmb{y}_N{\mid} {\{\pmb{x}_j\}_{j=2}^{N-1}-\pmb{y}_N}\right)}{
		1{-}s^2\lhp{}\left(s,\pmb{x}_N{-}\pmb{y}_1{\mid} {\{\pmb{x}_j\}_{j=2}^{N-1}{-}\pmb{y}_1}\right)\lhp{}\left(s,\pmb{x}_1{-}\pmb{y}_N{\mid} {\{\pmb{x}_j\}_{j=2}^{N-1}{-}\pmb{y}_N}\right)}.\label{rexnresub}
	\end{align}
Here $ \lhp{}\left(s,\pmb{x}_N{-}\pmb{y}_1{\mid} \{\pmb{x}_j \}_{j=2}^{N-1}{-}\pmb{y}_1\right)$ represents the Laplace transform of the  probability that an IM with its starting point  $ \pmb{y}_1 $ at FAR$ _1 $ hits FAR$ _N $ when the medium consists of $N-1$ FARs at $\pmb{x}_2,\pmb{x}_3,\cdots,\pmb{x}_N$.
\end{theorem}
\begin{proof}
	See Appendix \ref{arec}.
\end{proof}

The  general expressions for the hitting probability obtained using the direct method and the recursive method can be simplified under several special cases as given in Corollary \ref{cnrx1} and Section \ref{sec5uca}.

\begin{coro}\label{cnrx1}
		In an MC system with $N=4$ FARs that are equidistant from the transmitter at the origin and also equidistant from each other (\textit {i.e., } $ r_i=r \text { and } R_{ij}=R,\forall\ i,j\in{1,2,3,4},\ i\neq j$), the hitting probability of the IM on each of the FAR is given by
			\begin{align}
			\hp{1}(t)=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-3a)^n}{R^n}\mathrm{erfc}\left(\frac{r-a+n(R-a)}{\sqrt{4Dt}}\right).\label{eqnrxeq}
		\end{align}
	\begin{proof}
		See Appendix \ref{anrx1}.
	\end{proof}
\end{coro}
The above scenario is equivalent to placing the center of FARs at the vertex of a tetrahedron, and the transmitter is placed at the centroid of the tetrahedron.

  To demonstrate the importance and applicability of the derived channel model, we now describe a  system with a receiver consisting of multiple FARs arranged in a UCA fashion \cite{gursoy2019,tang2021,ahuja2021,gursoy2019a}.  We first derive the hitting probability expressions for this SIMO system in the next section. Using these  derived  expressions, we  analyze the performance of this system   in terms of the probability of bit error in Section \ref{sec:comm_UCA}.

    \section{Hitting Probability in a SIMO System with UCA of FARs}\label{sec5uca}
        \begin{figure}[ht!]
    	\centering
    	\includegraphics[width=3.4in]{Figures/UCA1}
    	\caption{An illustration showing a SIMO system with a point transmitter and a receiver consisting of $N $ FARs arranged as a UCA.}
    	\label{fig:uca}
    \end{figure}

    We now study a SIMO system with a point transmitter at the origin and multiple FARs arranged as UCA,  as shown in Fig. \ref{fig:uca}. The UCA center is located at $ x -$ axis at $ [w,0,0] $ and UCA is in $ y-z $ plane. The location of FAR$ _i $ is at $x_i= [w,\ d\cos(2\pi n/N),\ d\sin(2\pi n/N)],\ n\in\{1,2,\cdots, N\} $.
The distance between the transmitter and each FAR is $r=\sqrt{w^2+d^2}$.
Without loss of generality, denote any one of the FAR in the UCA by FAR$_1 $ and the rest of the FARs by FAR$ _2 $ to FAR$ _N $ in the clockwise direction.
From the perspective of the  distance from FAR$ _1 $ to the rest FARs,
there are $\delta= \text{ceil}((N-1)/2) $ types of neighbors of FAR$_1$.  Let $ R_1 $ is the first neighbor distance, which is equal to $ R_{12} $ and $ R_{1N} $, $ R_2 $ is the second neighbor distance, which equal to $ R_{13} $ and $ R_{1(N-1)} $ and so on.

The hitting probability of an IM in a SIMO system with FARs forming a UCA  and a single transmitter can be computed in closed form for any value of $N$ from Theorem \ref{tnfar}.
We first derive the hitting probability  for up to 5 FARs as

given by Theorem \ref{tuca}, and extend it for the general $N$ in Section \ref{sec:ucagen}.

\begin{theorem}\label{tuca}
    	In an SIMO system with $N$ receiver FARs arranged as a UCA, the  probability that an IM emitted from the point source at the origin hits a FAR is given as

    	\begin{align}
    		\hp{1}(t)&=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-a)^n}{R_1^n}\mathrm{erfc}\left(\frac{r-a+n(R_1-a)}{\sqrt{4Dt}}\right),\label{UCA2}&&\text{if } N=2\\
    		\hp{1}(t)&=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-2a)^n}{R_1^n}\mathrm{erfc}\left(\frac{r-a+n(R_1-a)}{\sqrt{4Dt}}\right),\label{UCA3}&&\text{if } N=3\\
    		\hp{1}(t)&=\sum_{n=0}^{\infty}(-1)^n\frac{a^{n+1}}{r}\sum_{k=0}^{n}{n \choose k}\frac{2^k}{R_1^kR_2^{n-k}}&&\nonumber\\ &\times\mathrm{erfc}\left(\frac{r-a+k(R_1{-}a)+(n-k)(R_2{-}a)}{\sqrt{4Dt}}\right)&&\text{if } N=4\label{UCA4}\\
    		\text{and}\nonumber\\
    		\hp{1}(t)&=\sum_{n=0}^{\infty}(-2)^n\frac{a^{n+1}}{r}\sum_{k=0}^{n}{n \choose k}\frac{1}{R_1^kR_2^{n-k}}\nonumber&&\\ &\times\mathrm{erfc}\left(\frac{r-a+k(R_1{-}a)+(n-k)(R_2{-}a)}{\sqrt{4Dt}}\right),&&\text{if } N=5\label{UCA5}
    	\end{align}
    	respectively.
    \end{theorem}
    \begin{proof}
    	See Appendix \ref{aUCA}.
    \end{proof}

\subsection{Numerical evaluation}
\begin{figure}
	\centering
	\includegraphics[width=3.4in]{Figures/ucaf3}
	\caption{Variation of the hitting probability $\hp{1}(t)$ with the radius $a$ of the FAR for different values of $D$. Here, $d=20 \mu m, w=10\mu m,\ r=22.36\mu m,\

		\Delta t=10^{-4}\ s,\ t=1s $.}
	\label{fig:ucaf3}
\end{figure}
The variation of the hitting probability with the radius $a$ of the FARs for $ N=4,\, 5 $ and $ D=100,\ 200\mu m^2/s $ is shown in Fig. \ref{fig:ucaf3}. We can verify that the approximate analytical results given in \eqref{UCA4} and \eqref{UCA5} are in good match with the particle-based simulation results.
We can observe that the hitting probability increases with the radius of the FARs. This is because an increase in $a$ causes an increase in the surface area of the FARs, which allows more IMs to hit the FARs.

When $D$ increases, the hitting probability raises due to the faster motion of the molecules. An increase in  $ D $ causes molecules to wander more, causing them to bump to other FARs with a higher chance. This can increase the mutual influence of FARs. When the number of FARs increases, we can see a slight drop in the hitting probability, and the drop increases when $ a $ increases. The widening gap between the hitting probabilities for $ N=4 $ and $ N=5 $ with $ a $ is due to the increase in the mutual influence by adding one extra FAR and the reduction in the mutual distance between adjacent FARs.

\subsection{Array gain in the received IMs due to multiple FARs}
Now, we discuss the array gain $g(t)$ in the number of IMs absorbed by all FARs when multiple FARs are present in the communication channel, compared to an MC system with only one FAR.
Ignoring the noise molecules and considering equal gain combining (EGC) with gain factor $ 1 $, $ g(t) $ is the same as the ratio of the summation of hitting probabilities of IMs on each of the $ N $ FARs to hitting probability on the FAR when only one FAR is present.
In other words,
\begin{align}
	g(t) = \sum_{i=1}^{N}\frac{\hp{i}(t)}{\overline{\mathsf{p}}(t,r)
	.\label{eqgain}
\end{align}
In symmetric systems with two and three FARs, $ g(t) $ can be derived as given in Theorem \ref{thdg2rx}.
\begin{coro}\label{thdg2rx}
	For an MC system with FARs located at a distance $ r $ from the transmitter, and FARs are equidistant from each other, the array gain $ g(t) $
	is given by
	\begin{align}
		g(t)&=\frac{N\hp{1}(t)}{\overline{\mathsf{p}}(t,r)}=
		\begin{cases}
		\displaystyle 2-\frac{2\lambda(t,r,1)}{\overline{\mathsf{p}}(t,r)},& \text{if } N=2
	\\
	\displaystyle 3-\frac{3\lambda(t,r,2)}{\overline{\mathsf{p}}(t,r)},& \text{if } N=3
	\end{cases}\\
	\label{dg3}

	\text{where }	\lambda(t,r,b)&=\frac{a}{r}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(ba)^n}{R^n}\mathrm{erfc}\left(\frac{r-a+n(R-a)}{\sqrt{4Dt}}\right).
	\end{align}
	\begin{proof}
		See Appendix \ref{adg2rx}.
	\end{proof}
\end{coro}

From \eqref{eqgain}  we can see that the gain is equal to $N$ if the mutual influence is ignored. However, due to the mutual influence the array gain is less than $N$. For example, from Corollary \ref{thdg2rx}, we can verify that the gain $ g(t)\leq 2 $ and $ g(t)\leq 3 $ for $ N=2 $ and $ N=3 $ respectively.

	Further, when all the FARs are far apart (\textit {i.e., } $ R_{ij}\rightarrow\infty,\forall \ i,j, i\neq j$), then $\lambda(t,r_1,b) \rightarrow 0 $. This implies that $ g(t)\rightarrow 2 $ and  $ g(t)\rightarrow 2 $ for $N=2$ and $N=3$ respectively.
	This is true for any $N$ as the mutual influence of FARs decreases with the increase in the mutual distance of FARs. Therefore the hitting probability on each FAR in $N$ receiver scenario will tend to be equal to hitting probability on FAR in single receiver case, and hence $ g(t)\rightarrow N $.

The following corollary presents the asymptotic array gain  \eqref{eqgain} for the symmetric systems.

\begin{coro}\label{DGtinf}
	The asymptotic array gain for an SIMO system with $N$ FAR UCA
	at $ t\rightarrow\infty $ is given by
	\begin{align}
		g_\infty=\lim_{t\rightarrow\infty}g(t)&=
		\begin{cases}
		\qquad 1& \text{if } N=1\\
		\displaystyle\frac{2}{1+a/R_{1}
		&\text{if } N=2\\
		\displaystyle\frac{3}{1+2a/R_{1}
		&\text{if } N=3
		\end{cases},
		&&
		\begin{aligned}

		&\frac{4}{1+2a/R_{1}+a/R_2
		&\text{if } N=4\\

		&\frac{5}{1+2a/R_{1}+2a/R_2}&\text{if } N=5
		\end{aligned}
		\label{asymgain}
	\end{align}

\begin{proof}
	Applying the final value theorem for the Laplace transform (\textit {i.e., } $ \lim_{t\rightarrow\infty} f(t)=\lim_{s\rightarrow0}sF(s)$) \cite{spiegel1992} in \eqref{l2rx}, \eqref{l3rx}, \eqref{l4rx}, and \eqref{l5rx}, gives Corollary \ref{DGtinf}.
\end{proof}
\end{coro}
The above results show that the maximum gain that can be achieved eventually using $ N $ UCA FARs is always less than $ N $, unless FARs are located far away from each other (\textit {i.e., } $ R_1,R_2\rightarrow \infty $).

\begin{figure}[ht!]
	\centering
	\includegraphics[width=3.4in]{Figures/DGt}
	\caption{Variation of array gain with time for different values of $ N $ in an SIMO system with $N$ FARs UCA.
	Here, $d=20\mu m, r=20 \mu m, w=0\mu m,\ a=4 \mu m,\ D=100\mu m^2/s,\
		$. }
\label{fig:dgt}

\end{figure}
Fig. \ref{fig:dgt} shows the variation of $ g(t) $ with $ t $ for different values of $ N $ in a system for $N$ UCA FARs.
When $ t $ increases, the $ g(t) $ reduces from $ g(t)\approx N $ to a lower value. The reduction in the $ g(t) $ with $ t $ is due to the increase in the mutual influence between FARs with time as seen in Fig. \ref{fig:3rxf1}.  When $ t\rightarrow \infty $, from \eqref{asymgain}
$g_\infty$ is given by $ g_\infty=\frac{2}{1+(a=4)/(R_1=36)}=1.8,\ g_\infty=\frac{3}{1+2(a=4)/(R_1=31.24)}=2.39 $ and $ g_\infty=\frac{4}{1+2(a=4)/(R_1=25.61)+(a=4)/(R_2=36)} =2.81$, respectively for $ N=2,3 $ and $ 4 $.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=3.4in]{Figures/DGangle}
	\caption{Variation of array gain with the angular distance between two FARs for different values of $ a $ for FARs arranged in a circular array ($N=2$). Here, the two FARs are equidistant from the origin and, $\ D=100\mu m^2/s,\ t=1 s,\ \pmb{x}_1=[d,0,0],\ \pmb{x}_2=[d\cos(\theta),d\sin(\theta),0] ,\ d=20\mu m$.}
	\label{fig:dgangle}
\end{figure}

Fig. \ref{fig:dgangle} shows the variation of gain with the angle between two FARs for different values of $ a $ in a circular array arranged $ 2-$FAR system.
The increase in $ g(t) $ with the increase in the angle between the two FARs is due to the decrease in mutual influence between the FARs. Also, we can see that an increase in $ a $ increases the $ g(t) $. The result further shows that with an increase in $N$, the angular distance between the FARs decreases, causing the mutual influence to decrease the array gain significantly.

\subsection{Hitting probability for an SIMO system with $N$ UCA for the general value of  $N$}\label{sec:ucagen}
Similar to the $N=2,3,4$ and $5$ case, the hitting probability and the array gain can be computed in closed form for any value of $N$ from Theorem \ref{tnfar}. However, due to lack of space and time, it was not possible for us to derive these metrics for each value of $N$. However, a close look at the pattern of the values of the hitting probability $p_1(t)$ for the $N=2,3,4$ and $5$ case as given in Theorem \ref{tuca} gives us the following conjecture (hypothesis) for its general expression for any value of $N$.

\begin{conj}\label{NUCA}
	In an SIMO system with $N$ FARs arranged as UCA, the  probability that an IM emitted by the point source hits one FAR  is given as\\
	For odd N:
	\begin{align}
	\hp{1}(t)&=\invlaplace{}{
	\frac{\overline{\mathcal{P}}(s,r)}{1+\sum_{i=1}^{\delta}2s\overline{\mathcal{P}}(s,R_i)} }{}\label{lNrxodd}\\
	&\stackrel{(a)}{=}\invlaplace{}{\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-2)^n\sum_{k_1+k_2+\cdots+k_\delta=n} {n \choose k_1, k_2, \ldots, k_m}

	\prod_{m=1}^\delta \left(s\overline{\mathcal{P}}(s,R_m)\right)^{k_m}}\nonumber\\
	&=\sum_{n=0}^{\infty}(-2)^n\frac{a^{n+1}}{r}\sum_{\mid \mathbf{k}\mid=n} {n \choose \mathbf{k}}\frac{1}{\mathbf{R^k}}

	\mathrm{erfc}\left(\frac{r-a+\sum_{m=1}^{\delta}k_m(R_m-a)}{\sqrt{4Dt}}\right),\label{NUCAO}
	\end{align}
For even N:
		\begin{align}
	\hp{1}(t)&=\invlaplace{}{\frac{\overline{\mathcal{P}}(s,r)}{1+\sum_{i=1}^{\delta-1}2s\overline{\mathcal{P}}(s,R_i)+s\overline{\mathcal{P}}(s,R_\delta)} }\label{lNrxeven}\\
	&\stackrel{(b)}{=}\invlaplace{}{\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-2)^{n-k_\delta}\mkern-32mu\sum_{k_1+k_2+\cdots+k_\delta=n} {n \choose k_1, k_2, \ldots, k_m}

		\prod_{m=1}^\delta \left(s\overline{\mathcal{P}}(s,R_m)\right)^{k_m}}\nonumber\\
	&=\sum_{n=0}^{\infty}(-1)^n\frac{a^{n+1}}{r}\sum_{\mid \mathbf{k}\mid=n} {n \choose \mathbf{k}}\frac{2^{n-k_\delta}}{\mathbf{R^k}}

	\mathrm{erfc}\left(\frac{r-a+\sum_{m=1}^{\delta}k_m(R_m-a)}{\sqrt{4Dt}}\right).\label{NUCAE}
	\end{align}
 Here, $ \mathbf{k}=(k_1, k_2, \ldots, k_\delta) $, $ |\mathbf{k}|=k_1+k_2+\cdots +k_\delta $, ${n \choose \mathbf{k}}={n \choose k_1, k_2, \ldots, k_\delta}  $
and $ \mathbf{R^k}=\prod_{i=1}^\delta (R_i)^{k_i}$ based on multi-index notation \cite{bricogne2010}. Also, the second sum is taken over all combinations of non-negative integer indices $ k_1 $ through $ k_\delta $ such that $ \sum_{i=1}^\delta k_i=n $.
\end{conj}
In the above, steps $ (a)  $ and $ (b) $ are obtained by applying multinomial theorem, \textit {i.e., }
$ (x_1 + x_2  + \cdots + x_m)^n
= \sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m}
\prod_{i=1}^m x_t^{k_i} $.
The following corollary presents the asymptotic array gain  \eqref{eqgain} for the symmetric systems.

\begin{coro}\label{cornrx}
The asymptotic array gain for an SIMO system with $N$ FAR UCA 	at $ t\rightarrow\infty $ is given by
	\begin{align}
		g(t)=\begin{cases}
			\displaystyle\frac{N}{1+\sum_{i=1}^{\delta}2a/2R_i}& \text{for odd N} \\
			\displaystyle \frac{N}{1+\sum_{i=1}^{\delta-1}2a/R_i+a/R_\delta}& \text{for even N}.
		\end{cases}
	\end{align}
\end{coro}
\begin{proof}
	Applying the final value theorem for the Laplace transform in  \eqref{lNrxodd} and \eqref{lNrxeven} gives the desired result.
\end{proof}

  \begin{figure}[ht!]
    	\centering
    	\includegraphics[width=3.4in]{Figures/NUCA}
    	\caption{Variation of the hitting probability $p_1(t)$ with $ N $ for different values of $ D $ for an SIMO system with $N$ FAR UCA. Here, $d=20 \mu m, w=10\mu m,\ r=22.36\mu m,\ a=4\mu m,\

    		t=10^{-4}\ s,\ t=1s $.}
    	\label{fig:nuca}
    \end{figure}

    Fig. \ref{fig:nuca} validates the  analytical expressions derived in Conj. \ref{NUCA} for $ D=100\mu m^2/s $ and $ D=200\mu m^2/s $. The particle-based simulation results are in good match with the analytical results, thus confirming that the conjecture is true. Further, we can observe that the hitting probability drops with $N$ due to the increase in the mutual influence of FARs.

   Now, we use the derived hitting probability expressions for the UCA arrangement of FARs to analyze the performance of the SIMO system with different cooperative detection schemes in the next section.

    \section{Communication Performance in a SIMO System with UCA of $N$ Co-operative FARs}\label{sec:comm_UCA}

Let us consider a SIMO communication system consisting of a point transmitter and the receiver consisting of a UCA of $N$ FARs as described in the previous section.
Let the duration of each time slot be $ T_{\mathrm{b}} $. The point transmitter impulsively emits molecules at the beginning of the time slot corresponding to the transmit information bit  $ b[k] $ at that time slot.

 $ b[k] $ is Bernoulli distributed with parameter $ q $. We consider on-off keying (OOK) based modulation, in which the transmitter emit $ u[k]=M $ number of molecules for bit $ b[k]=1 $ and  no molecules for bit $ b[k]=0 $.  The fraction of IMs transmitted at the time slot $ k $ that reach FAR$ _i$ in the slot $ l $ is given by
 \begin{align}
   	h_{i}[l-k]=&\hp{i}((l{-}k{+}1)T_{\mathrm{b}})-\hp{i}((l{-}k)T_{\mathrm{b}}).\label{simoccoeff}
   \end{align}
Let $ z_{i}[k;l] $ denote the number of IMs received at FAR$ _i $ at the $ l $th time slot due to
the $ k $th time slot transmission. The indicator that an IM emitted at the $ k $th slot hits the FAR$ _i $ in the $ l $th slot is Bernoulli random variable with parameter $ h_{i}[l-k]  $. Therefore, $z_{i}[k;l] $ is Binomial distributed with parameters $ b[k]u[k] $ and $ h_{i}[l-k] $, \textit {i.e., } $ z_{i}[k;l]\sim\mathcal{B}( b[k]u[k],h_{i}[l-k] ) $. For mathematical tractability,
the distribution of $ z_{i}[k;l] $ can be approximated as the  Poisson distribution having mean $ b[k]u[k]h_{i}[l-k] $ (\textit {i.e., } $  z_{i}[k;l]\sim\mathcal{P}\left( b[k]u[k]h_{i}[l-k]\right)$) \cite{jamali2019}. The total number of molecules received at the FAR$_i $ in the $ l $th time slot (\textit {i.e., } $ z_{i}[l] $) is given as
   \begin{align}
   	z_{i}[l]=z_{i}[l;l]+\sum_{k=1}^{l-1} z_{i}[k;l].\label{simorxmolwith}
   \end{align}
   where $z_{i}[l;l]  $) denotes the number of signal molecules and $ \sum_{k=1}^{l-1} z_{i}[k;l]$ denotes the inter symbol interference (ISI) molecules.
 %
From \eqref{simorxmolwith}, conditioned on the transmit bit $b[l]$, the mean number of IMs received at FAR$ _i $
is given by
\begin{align}
	\lambda_{0i}[l]&=\expectA{z_{i}[l]|b[l]=0}=qM\sum_{k=1}^{l-1}h_i[l-k

	=qM\left(\hp{}(lT_{\mathrm{b}},r_i\mid \pmb{x}_{1:N})-\hp{}(T_{\mathrm{b}},r_i\mid \pmb{x}_{1:N})\right)\label{emu0}\\
	\lambda_{1i}[l]&=\expectA{z_{i}[l]|b[l]=1}=M\hp{}(T_{\mathrm{b}},r_i\mid \pmb{x}_{1:N})+\lambda_{0i}[l]\label{emu1}.
	\end{align}%

\subsection{Cooperative detection in SIMO Systems}
 Only a fraction of the emitted IMs reaches  the FARs, and the FARs count the IMs received in each time slot. The FARs individually make a \textit{hard local decision} based on the IMs it counted in each time slot which is independently forwarded to a \textit{fusion center} (FC) \cite{yutingfang2017} . The fusion center is assumed to be transparent to the motion of IMs. We consider perfect reporting of the local decision at the FARs to the FC.  A fully transparent FC with perfect reporting of local decisions can be challenging to implement in real scenarios. Hence the derived bit error probability expression can be used as the upper bound to any practical implementation. The FC employs the $ K $ out of $ N $ rule to make the \textit{global decision} of the transmitted information bit.  That is, the FC decides that the transmitted bit is $ 1 $, if at least $ K $ of the $ N $ FAR's local decision is $1 $, otherwise $ 0 $. Some special cases of $ K $ out of $ N $ fusion rule are
	\begin{itemize}
		\item \textit{Majority rule}: The global decision is $ 1 $ if at-least the local decision of $ K=\floor{N/2}+1 $ out of  $ N $ FARs is 1.
		\item \textit{OR rule}: The global decision is $ 1 $ if at-least the local decision of $ K=1 $ out of  $ N $ FARs is 1.
		\item \textit{AND rule}: The global decision is $ 1 $ if the local decision of $ K=N $ out of  $ N $ FARs is 1.
	\end{itemize}
	The detection scheme mentioned above is a co-operative detection scheme.

\subsection{The Probability of Bit Error}
	To make a local decision of the transmitted bit in each time slot, each FAR
 consider the null hypothesis $ \mathcal{H}_{0} $ and alternate hypothesis $ \mathcal{H}_{1} $ corresponding to the bits $ 0 $ and $ 1 $ respectively. The number of the IMs received at each of the FAR is compared with the detection threshold   $ \eta[l] $. The decision rule for the detection process can be represented as
	\begin{align}
		z_i[l] \underset{\mathcal{H}_0[l]}{\overset{\mathcal{H}_1[l]}{\gtrless}}\eta[i;l]\label{edetrule}
	\end{align}

The local probability of miss detection $ \mathrm{P}_{\mathrm{m} i} [l]$ and probability of miss detection $ \mathrm{P}_{\mathrm{f} i} [l]$
of FAR$_i  $ are given as
\begin{align}
	\mathrm{P}_{\mathrm{m} i} [l]&=\mathrm{P}\left(z_{i}[l]<\eta[i ; l] \mid \mathcal{H}_{1}[l]\right)=\exp\left(-\lambda_{1 i}[l]\right)\sum_{n=0}^{\eta[i;l]}\frac{\lambda_{1 i}[l]^n}{n!},\label{lpm}\\
\text{and  }
	\mathrm{P}_{\mathrm{f} i} [l]&=\mathrm{P}\left(z_{i}[l]\geq \eta[i ; l] \mid \mathcal{H}_{0}[l]\right

	=1-\exp\left(-\lambda_{0 i}[l]\right)\sum_{n=0}^{\eta[i;l]}\frac{\lambda_{0 i}[l]^n}{n!}.\label{lpf}
\end{align}

Note that, due to the symmetric arrangement of FARs in the UCA, the distribution of the received numbers of molecules at each FAR are identically distributed.
Therefore, the detection threshold are the same for each FARs (\textit {i.e., } $ \eta[i;l]=\eta[l] $). Thus the local miss detection and false alarm probabilities of each FARs are  the same. Let us denote $ \mathrm{P}_{\mathrm{m} i} [l]= \mathrm{P}_{\mathrm{m}} [l]$ and $ \mathrm{P}_{\mathrm{f} i} [l]= \mathrm{P}_{\mathrm{f}} [l]$ \cite{yutingfang2017}. Hence, at the $ l $th time slot, the global miss detection and false alarm probability based on $ K $ out of $ N $ fusion rule are given by \cite{varshney1997,yutingfang2017}
\begin{align}
	\mathrm{Q}_{\mathrm{m}} [l]&=1-\sum_{k=K}^{N}\binom{N}{k}\left(1-\mathrm{P}_{\mathrm{m}} [l]\right)^k\mathrm{P}_{\mathrm{m}} [l]^{N-k},\label{gpm}\\
\text{and }
	\mathrm{Q}_{\mathrm{f}} [l]&=\sum_{k=K}^{N}\binom{N}{k}\mathrm{P}_{\mathrm{f}} [l]^{k}\left(1-\mathrm{P}_{\mathrm{f}} [l]\right)^{N-k},\label{gpf},
\end{align}
respectively.
Therefore, the the probability of bit error is given as
\begin{align}
	\mathrm{P_e}[l]=q\mathrm{Q_m} [l]+(1-q)\mathrm{Q_f} [l].
\end{align}
For the OR-rule (\textit {i.e., } $ K=1 $), the  probability of bit error is given by
\begin{align}
	\mathrm{P_e}[l]=q\mathrm{P}_\mathrm{m}[l]^N +(1-q)\left(1-(1-\mathrm{P}_\mathrm{f}[l])^N\right),
\end{align}
and for the AND-rule (\textit {i.e., } $ K=N$), the  probability of bit error is given by
\begin{align}
	\mathrm{P_e}[l]=q\left(1-(1-\mathrm{P}_\mathrm{m}[l])^N\right) +(1-q)\mathrm{P}_\mathrm{f}[l]^N.
\end{align}

\subsection{Numerical Results
}
We now present numerical results for the performance of the SIMO system.
Here, the value of  $ l $ is fixed as $ 9  $ (\textit {i.e., } current time-slot is the $ 9 $th slot) because the average ISI molecules were observed to be contained in $ 8 $ slots for the considered values of the parameters  $ a,\ D,\ M,\ T_{\mathrm{b}},\ r,\ w  $ and $ d $. Therefore, the BER will be the same for any $ l>9 $.

Fig. \ref{fig:pvsetauca} shows the variation of the probability of bit error $\mathrm{P_e}[l]$ with the detection threshold $ \eta[l] $ for different fusion rules. We also include the values for the single FAR case.
$\mathrm{P_e}[l]$ first reduces with $ \eta[l] $. After reaching the minimum value, it  increases confirming the presence of an optimal detection threshold. The optimal threshold is different for different fusion rules and has to be selected according to the implemented fusion rule. The following results correspond to the optimal threshold value, which is computed by varying $ \eta[l] $ for the specific set of parameter-values and selected fusion rule to find the value for which the probability of bit error is the minimum.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=3.4in]{Figures/PvsEtaUCA}
	\caption{Variation of the probability of bit error with the detection threshold. Here, $ d=10 \mu m, w=25\mu m,\ r=26.92\mu m,\ a=4\mu m,\ D=100\mu m^2/s, \ M=200,\ N=4,\ q=0.5,\ T_{\mathrm{b}}=5 s,\

		l=9$.}
	\label{fig:pvsetauca}
\end{figure}

\begin{figure}[ht]
	\centering
	\includegraphics[width=3.4in]{Figures/Pvsd}
	\caption{Variation of the probability of bit error with the radius of UCA. Parameters; $ a=4\mu m,\ D=100\mu m^2/s, \ M=200,\ N=4,\ q=0.5,\ T_{\mathrm{b}}=5 s,\ \pmb{x}_i=[25,d\cos\left(\frac{2\pi (i-1)}{N}\right),d\sin\left(\frac{2\pi (i-1)}{N}\right)],\ i=1\cdots, N  ,\ l=9,\ \eta[l]=\text{optimal value}$.}
	\label{fig:pvsd}
\end{figure}

Fig. \ref{fig:pvsd} shows the variation of the probability of bit error with respect to the radius of the UCA arrangement (\textit {i.e., } $ d $). Since the distance between the transmitter and the FARs increases with $d$
resulting in the fall in the received IMs at each of the FARs, the probability of bit error increases with $d$.

\begin{figure}[ht]
	\centering
	\includegraphics[width=3.4in]{Figures/PvsM}
	\caption{Variation of the probability of bit error with number of molecules emitted for bit $ 1 $. Parameters; $d=10 \mu m, w=25\mu m,\ r=26.92\mu m, a=4\mu m,\ D=100\mu m^2/s,,\ N=4,\ q=0.5,\ T_{\mathrm{b}}=5 s,\

		l=9,\ \eta[l]=\text{optimal value}$.}
	\label{fig:pvsm}
\end{figure}
Fig. \ref{fig:pvsm} shows the variation of the probability of bit error with respect to the transmitted number of IMs. As the transmitted number of IMs is increased, the probability of bit error reduces.
The gap between bit error probability increases with $ M $ for the different fusion rules.\par

\begin{figure}[ht]
	\centering
	\includegraphics[width=3.4in]{Figures/PvsN1}
	\caption{Variation of the probability of bit error with the number of FARs in an SIMO with a UCA receiver. Parameters; $d=15 \mu m, w=25\mu m,\ r=29.15\mu m,\ a=4\mu m,\ D=200\mu m^2/s,\ q=0.5,\ M=500,\ T_{\mathrm{b}}=5 s,\

		l=9,\ \eta[l]=\text{optimal value}$. The dashed lines shows the value ignoring the presence of other FARs. }
	\label{fig:pvsn1}
\end{figure}

Fig. \ref{fig:pvsn1} shows the variation of the probability of bit error when $ N $ is increased when different fusion rules are employed.
 Fig. \ref{fig:pvsn1} also shows  the probability of bit error when other FARs are not present (\textit {i.e., } using the hitting probability 	corresponding to \eqref{eq1rx}) using dashed lines for comparison. With no mutual influence, the probability of bit error reduces significantly with $N$ due to  an increase in the number of IMs captured.
 However, when other FARs are present, it results in a different probability of bit error due to their mutual influence. With an increase in $N$, the individual hitting probability decreases at each FAR degrading the overall performance for AND fusion rule. For majority rule, the BER improvement with $N$ decreases due to the mutual influence of FARs.
Further,  it can be observed that an increase in the mutual influence degrades the probability of bit error (solid lines with markers) for each fusion rule.
Therefore, it is essential to consider the effect of the mutual influence of FARs while designing the system.

 Fig. \ref{fig:pvsetauca}-\ref{fig:pvsn1}
 show that the majority-rule gives a better performance compared to others. From the figures, we can verify that, in a SIMO system, among the fusion rules, OR-rule outperforms AND-rule and Majority-rule outperforms both of them. Also, we can verify that  the bit error performance of the SIMO system is much better compared to a single input single output (SISO) system.

\section{Conclusions}
Molecular communication systems may consist of multiple receivers that are added to increase the communication systems' reliability and data rate. The exact analytical channel model for an MC system with multiple FARs does not exist in the literature due to the mathematical intractability. This work first presented an approximate equation for the hitting probability of an IM emitted from a point transmitter on each of the FARs in a system with three FARs. We then derived the hitting probability expressions for a general $ N-$FAR case. The derived expressions are further simplified for various special cases, for example, when FARs are arranged in UCA, and the transmitter is equidistant from the FARs. We also considered a SIMO system with one point transmitter and a receiver with UCA of $N$ FARs. Using the hitting probability expressions derived for the UCA case, we subsequently analyzed the error performance of cooperative detection in the considered SIMO system. We also compared the error performance under different decision rules like the majority, OR, and AND rules. The derived hitting probability equations can be easily used to analyze SIMO, MIMO, and cognitive systems and the interference analysis of MC systems with multiple FARs.

\appendices
	\section{Derivation of Fraction of IMs Absorbed at Each FAR in a $3-$FARs System}\label{a3rx}

The probability that an IM emitted from the point source at the origin first hits the surface of FAR$_{2} $ in the interval $[\tau,\tau+\mathrm{d}\tau] $ is  $\frac{\partial \hp{2}(\tau)}{\partial \tau} \mathrm{d}\tau$. The probability that this IM, if not absorbed by FAR$ _2 $,
would have eventually hit the FAR$ _1 $ in the remaining $ t-\tau $ time is $\overline{\mathsf{p}}(t-\tau,R_{21})$, where $R_{21}$ is the distance between the nearest point ($ \pmb{y}_2 $) on the surface of FAR$_{2} $  from the transmitter and the center of FAR$_{1} $. Here, we have approximated the hitting point of IMs at FAR$_2$ by  $\pmb{y}_2$. Similarly, the probability that an IM emitted from the point source at the origin first hits the surface of FAR$ _3 $ in the interval $[\tau,\tau+\mathrm{d}\tau] $ is  $\frac{\partial \hp{3}(\tau)}{\partial \tau} \mathrm{d}\tau$. The probability that this IM, if not absorbed by FAR$_3 $, would have eventually hit the FAR$ _1 $ in the remaining $ t-\tau $ time  is $\overline{\mathsf{p}}(t-\tau,R_{31})$, where $R_{31}$ is the distance between the nearest point ($ \pmb{y}_3 $) on the surface of FAR$_{3} $  from the transmitter and the center of FAR$_{1} $. Therefore,
the hitting probability of an IM that was supposed to hit FAR$ _{1} $ within time $ t $ but is hitting either FAR$ _{2} $ or FAR$ _{3} $ is given by
\begin{align}
	\overline{\mathsf{p}}(t,r_{1}) - \hp{1}(t)   =&
	\int_{0}^{t} \frac{\partial \hp{2}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{21}) 	\mathrm{d}\tau

	+
	\int_{0}^{t} \frac{\partial \hp{3}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{31}) \mathrm{d}\tau , \label{eqn1}
\end{align}
Similarly, the hitting probability of an IM that was supposed to hit FAR$ _{2} $ within time $ t $ but is hitting either FAR$ _{1} $ or FAR$ _{3} $, and the hitting probability of an IM that is supposed to hit FAR$ _{3} $ within time $ t $ but is hitting either FAR$ _{1} $ or FAR$ _{2} $  first is given by
\begin{align}
	\overline{\mathsf{p}}(t,r_{2}) - \hp{2}(t)   = &
	\int_{0}^{t} \frac{\partial \hp{3}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{32}) \mathrm{d}\tau

	+ \int_{0}^{t} \frac{\partial \hp{1}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{12}) \mathrm{d}\tau ,
	\label{eqn2}\\
\text{and }
	\overline{\mathsf{p}}(t,r_{3}) -  \hp{3}(t)   = &
	\int_{0}^{t} \frac{\partial \hp{1}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{13}) \mathrm{d}\tau

	+ \int_{0}^{t} \frac{\partial \hp{2}(\tau)}{\partial \tau}\overline{\mathsf{p}}(t-\tau,R_{23}) \mathrm{d}\tau
	\label{eqn3}  ,
\end{align}
respectively. Now, taking Laplace transform on both sides of \eqref{eqn1}, \eqref{eqn2} and \eqref{eqn3} gives
\begin{align}
	\overline{\mathcal{P}}(s,r_1) -\lhp{1}(s)  =
	s\lhp{2}(s) \overline{\mathcal{P}}(s,R_{21}) +
	s\lhp{3}(s)\overline{\mathcal{P}}(s,R_{31}),\label{eqn4}\\
	\overline{\mathcal{P}}(s,r_2) -\lhp{2}(s)   =
	s\lhp{1}(s) \overline{\mathcal{P}}(s,R_{12})
	+ s \lhp{3}(s) \overline{\mathcal{P}}(s,R_{32}),\label{eqn5}\\
\text{and }
	\overline{\mathcal{P}}(s,r_3) - \lhp{3}(s)   =
	s\lhp{1}(s)\overline{\mathcal{P}}(s,R_{13})
	+ s\lhp{2}(s) \overline{\mathcal{P}}(s,R_{23}).	\label{eqn6}
\end{align}
Here $\lhp{i}(s)$ and $\overline{\mathcal{P}}(s,x),\ x\in\{r_i,R_{ij}\}\ i,j\in\{1,2,3\}, i\neq j
$ are the Laplace Transforms of $\hp{i}(t)$ and $\overline{\mathsf{p}}(t,x)$ respectively. Note that,
\begin{align}
	\overline{\mathcal{P}}(s,x)=\frac{a}{sx}\exp\left(-\left(x-a\right)\sqrt{\frac{s}{D}}\right).\label{lapeq}

\end{align}

Solving \eqref{eqn4}, \eqref{eqn5} and \eqref{eqn6} for the value of $\lhp{1}(s)$ gives \eqref{eq3rx}.
\section{Derivation of Corollary \ref{c3rxeq}}\label{a3rxeq}
On substituting $ r_i=r,\ \forall i  $ and $ R_{ij}=R,\ \forall i,j,\  i\neq j $ in \eqref{eqnxr}, we get
\begin{align}
	\lhp{1}(s)   =
	\frac{\overline{\mathcal{P}}(s,r) -s\mathsf{a}(s)+s^2\mathsf{b}(s)}
	{1-s^2\mathsf{c}(s)+s^3\mathsf{d}(s)}, \label{abeq19}
\end{align}
where $\mathsf{a}{=}2\overline{\mathcal{P}}(s,r)\overline{\mathcal{P}}(s,R) ,\ \mathsf{b}(s){=}\overline{\mathcal{P}}(s,r)\overline{\mathcal{P}}^2(s,R),\ \mathsf{c}(s)=3\overline{\mathcal{P}}^2(s,R),\  \mathsf{d}(s){=}2\overline{\mathcal{P}}^3(s,R) $.
Simplifying \eqref{abeq19} further gives
\begin{align}
	\lhp{1}(s)  &=\overline{\mathcal{P}}(s,r)
	\frac{1 -2s\overline{\mathcal{P}}(s,R)+s^2\overline{\mathcal{P}}^2(s,R)}
	{1-3s^2\overline{\mathcal{P}}^2(s,R)+2s^3\overline{\mathcal{P}}^3(s,R)

	=\overline{\mathcal{P}}(s,r)\frac{1}{1+2s\overline{\mathcal{P}}(s,R)}\label{l3rx}\\
	&\stackrel{(a)}{=}\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-2)^n\left(s\overline{\mathcal{P}}(s,R)\right)^n,
	\label{abeq20}
\end{align}
where  $ (a) $ is due to the identity $ (1+2x)^{-1}=\sum_{n=0}^{\infty}(-2)^nx^n $. Now substituting \eqref{lapeq} in \eqref{abeq20} gives
\begin{align}
	\lhp{i}(s)=\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-2)^n\left(s\overline{\mathcal{P}}(s,R)\right)^n.\label{abeq21n}
\end{align}
Substituting \eqref{lapeq} in \eqref{abeq21n} gives
\begin{align}
	\lhp{1}(s)=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-2a)^n}{sR^n}\exp\left(-(r-a+n(R-a)\sqrt{\frac{s}{D}})\right).\label{eqcor}
\end{align}
Applying the inverse Laplace transform on both sides of \eqref{eqcor} gives Corollary \ref{c3rxeq}.
	\section{Derivation of Fraction of Molecules Absorbed in a $ N-$FAR System}\label{anxr}
	For $ N=2 $,  \cite[Eq. 19]{sabu2020a} and \cite[Eq. 20]{sabu2020a} can be modified into a system of equations of the form  $\mathcal{A}(s)\mathsf{P}(s)=\mathbf{P}(s)$. \textit {i.e., }
	\begin{align}
		\underbrace{\begin{bmatrix}
			1 & s\overline{\mathcal{P}}(s,R_{21})  \\
			s\overline{\mathcal{P}}(s,R_{12}) & 1 \\
		\end{bmatrix} }_{\mathcal{A}(s)} \underbrace{\left[ \begin{array}{c}\lhp{1}(s) \\\lhp{2}(s) \end{array} \right]}_{\mathsf{P}(s)} =\underbrace{\left[ \begin{array}{cc} \overline{\mathcal{P}}(s,r_1) \\ 	\overline{\mathcal{P}}(s,r_2) \end{array} \right]}_{\mathbf{P}(s)}.\label{mat2}
	\end{align}
For $ N=3 $, \eqref{eqn4}, \eqref{eqn5} and \eqref{eqn6} can be modified with
\begin{align}
	\mathcal{A}(s)=\begin{bmatrix}
		1 & s\overline{\mathcal{P}}(s,R_{21}) & s\overline{\mathcal{P}}(s,R_{31}) \\
		s\overline{\mathcal{P}}(s,R_{12}) & 1 & s\overline{\mathcal{P}}(s,R_{32})\\
		s\overline{\mathcal{P}}(s,R_{13}) & s\overline{\mathcal{P}}(s,R_{23}) & 1 \\
	\end{bmatrix} \nonumber\\
\mathsf{P}(s)= \left[ \begin{array}{c}\lhp{1}(s) \\\lhp{2}(s)\\\lhp{3}(s) \end{array} \right] \text{ and } \mathbf{P}(s)=\left[ \begin{array}{cc} \overline{\mathcal{P}}(s,r_1) \\ 	\overline{\mathcal{P}}(s,r_2) \\ 	\overline{\mathcal{P}}(s,r_3) \end{array} \right].\label{mat3}
\end{align}
Similarly, for any arbitrary $ N $, we can generalize \eqref{eqn4} - \eqref{eqn6} from the following system of equations $\mathcal{A}(s)\mathsf{P}(s)=\mathbf{P}(s)$ with $\mathcal{A}(s)\mathsf{P}(s)=\mathbf{P}(s)$ given as Theorem.
The solution of the above system of equations is given as
\begin{align}
		\mathsf{P}(s)=\mathcal{A}^{-1}(s)\mathbf{P}_{}(s).\label{mateqn}
\end{align}
 Finally the hitting probability can be given by taking the inverse Laplace transform of both sides of \eqref{mateqn}.
\section{Derivation of the Recursive Hitting Probability equation}\label{arec}
First notice that, for $ N=2 $, \cite[Eq. 19]{sabu2020a} and \cite[Eq. 20]{sabu2020a} state that
 \begin{align}
 	\overline{\mathcal{P}}{}(s,r_1)-\lhp{}(s,\pmb{x}_1\mid \pmb{x}_{2})=s\lhp{}(s,\pmb{x}_2\mid \pmb{x}_{1})\overline{\mathcal{P}}{}(s,R_{21}),\label{eq2rx1}\\
\text{and }

	\overline{\mathcal{P}}{}(s,r_2)-\lhp{}(s,\pmb{x}_2\mid \pmb{x}_{1})=s\lhp{}(s,\pmb{x}_1\mid \pmb{x}_{2})\overline{\mathcal{P}}{}(s,R_{12}).\label{eq2rx2}
\end{align}
Solving \eqref{eq2rx1} and \eqref{eq2rx2} gives
\begin{align}
	\lhp{}(s,\pmb{x}_1\mid \pmb{x}_{2})=\frac{\overline{\mathcal{P}}{}(s,r_1)-s\overline{\mathcal{P}}{}(s,r_2)\overline{\mathcal{P}}{}(s,R_{21})}{1-s^2\overline{\mathcal{P}}{}(s,R_{12})\overline{\mathcal{P}}{}(s,R_{21})}.\label{lp2rx}
\end{align}

 The proof argument is the same for any value of $N$. However,
for simplicity, we present the argument for $N=3$. Here, the hitting probability of an IM that was supposed to hit FAR$ _{1} $ within time $ t $ but is hitting either FAR$ _{2} $ or FAR$ _{3} $ is given by
\begin{align}
	&\hp{}(t,\pmb{x}_1{\mid} \pmb{x}_{2}) -\hp{}(t,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^3)   =

	\int_{0}^{t} \frac{\partial\hp{}(\tau,\pmb{x}_3{\mid} \{\pmb{x}_j\}_{j=1}^2)}{\partial \tau}\overline{\mathsf{p}}{}(t-\tau,\pmb{x}_1{-}\pmb{y}_3{\mid} \pmb{x}_{2}{-}\pmb{y}_3) \mathrm{d}\tau , \label{req3rxeqn1}
\end{align}
Similarly, the hitting probability of an IM that was supposed to hit FAR$ _{3} $ within time $ t $ but is hitting either FAR$ _{1} $ or FAR$ _{2} $ is given by
\begin{align}
	&\hp{}(t,\pmb{x}_3{\mid} \pmb{x}_{2}) -\hp{}(t,\pmb{x}_3{\mid} \{\pmb{x}_j\}_{j=1}^2)   =

	\int_{0}^{t} \frac{\partial \hp{}(\tau,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^3)}{\partial \tau}\overline{\mathsf{p}}{}(t-\tau,\pmb{x}_3{-}\pmb{y}_1{\mid} \pmb{x}_{2}{-}\pmb{y}_1) 	\mathrm{d}\tau   \label{req3rxeqn2}
\end{align}
 Now, taking Laplace transform on both sides of \eqref{req3rxeqn1} and \eqref{req3rxeqn2} gives
 \begin{align}
	\lhp{}(s,\pmb{x}_1{\mid} \pmb{x}_{2})-\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^3)=&s\lhp{}(s,\pmb{x}_3{\mid} \{\pmb{x}_j\}_{j=1}^2

	\overline{\mathcal{P}}{}(s,\pmb{x}_1{-}\pmb{y}_3{\mid} \pmb{x}_{2}{-}\pmb{y}_3),\label{eq3rx1}\\
\text{and }
	\lhp{}(s,\pmb{x}_3{\mid} \pmb{x}_{2}){-}\lhp{}(s,\pmb{x}_3{\mid} \{\pmb{x}_j\}_{j=1}^2){=}&s\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^3

	\overline{\mathcal{P}}{}(s,\pmb{x}_3{-}\pmb{y}_1{\mid} \pmb{x}_{2}{-}\pmb{y}_1).\label{eq3rx2}
\end{align}
Solving \eqref{eq3rx1} and \eqref{eq3rx2} gives
\begin{align}
	\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^3)=&
	\frac{
	\left(\lhp{}\left(s,\pmb{x}_1\mid {\pmb{x}}_{2}\right)\right.-\left. s\lhp{}\left(s,\pmb{x}_3\mid {\pmb{x}}_{2}\right)\lhp{}\left(s,\pmb{x}_1-\pmb{y}_3\mid {\pmb{x}_{2}-\pmb{y}_3}\right)\right)}
	{1{-}s^2\lhp{}\left(s,\pmb{x}_3{-}\pmb{y}_1{\mid} {\pmb{x}_{2}{-}\pmb{y}_1}\right)\lhp{}\left(s,\pmb{x}_1{-}\pmb{y}_3{\mid} {\pmb{x}_{2}{-}\pmb{y}_3}\right)}

	.\label{lp3rx}
\end{align}
The same argument can be extended  for $N-$FAR system. Here, we get the following two equations
 \begin{align}
	\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^{N-1})&-\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^N)=s\lhp{}(s,\pmb{x}_{N}{\mid} \{\pmb{x}_j\}_{j=1}^{N-1}

	\lhp{}(s,\pmb{x}_1-\pmb{y}_{N}{\mid} \{\pmb{x}_j\}_{j=2}^{N-1}-\pmb{y}_N),\label{eqn_1rx1}\\
\text{and }
	\lhp{}(s,\pmb{x}_{N}{\mid }\{\pmb{x}_j\}_{j=2}^{N-1})&-\lhp{}(s,\pmb{x}_{N}{\mid} \{\pmb{x}_j\}_{j=1}^{N-1})=s\lhp{}(s,\pmb{x}_1{\mid} \{\pmb{x}_j\}_{j=2}^N
\lhp{}(s,\pmb{x}_N-\pmb{y}_{1}{\mid} \{\pmb{x}_j\}_{j=2}^{N-1}-\pmb{y}_1).\label{eqn_1rx2}
\end{align}
Solving \eqref{eqn_1rx1} and \eqref{eqn_1rx2} gives \eqref{rexnresub}. Taking the inverse Laplace transform of \eqref{rexnresub} gives Theorem \ref{threc}.

\section{Derivation of Corollary \ref{cnrx1}}\label{anrx1}
On substituting $ r_i=r,\ \forall i  $ and $ R_{ij}=R,\ \forall i,j,\  i\neq j$ in \eqref{rexnresub} for $ N=4 $, owing to symmetry $ \lhp{}\left(s,\pmb{x}_i\mid \{\pmb{x}_j\}_{j=i,j\neq 1}^{N-1}\right){=} \lhp{}\left(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^{N-1}\right), \forall i$  and $ \lhp{}\left(s,\pmb{x}_i{-}\pmb{y}_j{\mid} {\{\pmb{x}_j\}_{j=i,j\neq 1}^{j-1}{-}\pmb{y}_j}\right){=}\\ \lhp{}\left(s,\pmb{x}_1{-}\pmb{y}_N{\mid} {\{\pmb{x}_j\}_{j=2}^{N-1}{-}\pmb{y}_N}\right), \forall i,j, \ i\neq j $. Therefore, the common terms cancel out in \eqref{rexnresub}, and the Laplace transform of the hitting probability is given as
\begin{align}
	\lhp{}(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^4)=\frac{\lhp{}(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^3)}{1+s\lhp{}(s,\pmb{x}_1-\pmb{y}_4\mid \{\pmb{x}_j\}_{j=2}^3-\pmb{y}_4)},\label{tetra1}
\end{align}
where
\begin{align}
	\lhp{}(s,\pmb{x}_1\mid \pmb{x}_{2})&=\frac{\lhp{}(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^3)}{1+s\lhp{}(s,\pmb{x}_1-\pmb{y}_3\mid \pmb{x}_{2}-\pmb{y}_3)

	=\frac{\overline{\mathcal{P}}{}(s,r)}{1+2s\overline{\mathcal{P}}{}(s,R)},\label{tetra2}
\end{align}
and
\begin{align}
\lhp{}(s,\pmb{x}_1-\pmb{y}_4\mid \{\pmb{x}_j\}_{j=2}^3-\pmb{y}_4)=\frac{\overline{\mathcal{P}}{}(s,R)}{1+2s\overline{\mathcal{P}}{}(s,R)}.\label{tetra3}
\end{align}
Substituting \eqref{tetra2} and \eqref{tetra3} in \eqref{tetra1} gives

\begin{align}
	\lhp{}(s,\pmb{x}_1\mid \{\pmb{x}_j\}_{j=2}^4)&=\frac{\overline{\mathcal{P}}(s,r)}{1+3s\overline{\mathcal{P}}(s,R)

	=\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-3)^n\left(s\overline{\mathcal{P}}(s,R)\right)^n.
\end{align}
Substituting the value of $ \overline{\mathcal{P}}(s,x),\ x\in\{r,R\} $ in the above equation using \eqref{lapeq} and taking its inverse Laplace transform gives Corollary \ref{cnrx1}.

\section{Derivation of Hitting Probability of an IM on an FAR in UCA}\label{aUCA}
The hitting probability for $ N=2 $ (when two FARs are in the UCA) is given by \eqref{eq2rxhp1}, and the derivation is given in Appendix \ref{adg2rx}.
The hitting probability for $ N=3 $ is the same as given in Corollary \ref{c3rxeq}. For derivation, please refer to Appendix \ref{a3rxeq}.

\textbf{$ 4-$FAR case}:
For $ N=4 $, $ \mathcal{A}^{-1}(s) $ in \eqref{eqnxr} is given by,
\begin{align}
	\mathcal{A}^{-1}(s)&=\frac{1}{4s^2\overline{\mathcal{P}}(s,R_1)-(1+s\overline{\mathcal{P}}(s,R_2))

	\begin{bmatrix}
		\mathsf{e}(s)& s\overline{\mathcal{P}}(s,R_1) & \mathsf{f}(s)& s\overline{\mathcal{P}}(s,R_1) 	\\
		s\overline{\mathcal{P}}(s,R)  & \mathsf{e}(s)& s\overline{\mathcal{P}}(s,R)  & \mathsf{f}(s) \\
		 \mathsf{f}(s)  & s\overline{\mathcal{P}}(s,R_1) & \mathsf{e}(s) &s\overline{\mathcal{P}}(s,R_1) \\
		s\overline{\mathcal{P}}(s,R_1)  &  \mathsf{f}(s)  &s\overline{\mathcal{P}}(s,R_1) &\mathsf{e}(s)
	\end{bmatrix}\label{ainvuca4}\\
\text{where }
	\mathsf{e}(s)&=-\frac{-2s^2\overline{\mathcal{P}}^2(s,R_1)+s\overline{\mathcal{P}}(s,R_2)+1}{1-s\overline{\mathcal{P}}(s,R_2)} \nonumber
\text{ and }\\
	\mathsf{f}(s)&= -\frac{2s^2\overline{\mathcal{P}}^2(s,R_1)-s^2\overline{\mathcal{P}}^2(s,R_2)-s\overline{\mathcal{P}}(s,R_2)}{1-s\overline{\mathcal{P}}(s,R_2)}\nonumber.
\end{align}
  Substituting \eqref{ainvuca4} in $ \mathsf{P}(s)=\mathcal{A}^{-1}(s)\mathbf{P}_{}(s) $ and solving further gives
  \begin{align}
  	\lhp{1}(s)&=\frac{\overline{\mathcal{P}}(s,r)}{1+2s\overline{\mathcal{P}}(s,R_1)+s\overline{\mathcal{P}}(s,R_2)} \label{l4rx}\\
  	&\rem{32}=\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-1)^n\left(2s\overline{\mathcal{P}}(s,R_1)+s\overline{\mathcal{P}}(s,R_2)\right)^n.\label{bbin}
  \end{align}
Now applying binomial theorem \cite{abramowitz2013} in \eqref{bbin} gives,
\begin{align}
	\lhp{1}(s)&=\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-1)^n\sum_{k=0}^{n}{n \choose k

	\left(2s\overline{\mathcal{P}}(s,R_1\right)^k\left(s\overline{\mathcal{P}}(s,R_2)\right)^{n-k}.
\end{align}
  Substituting the value of $ \overline{\mathcal{P}}(s,x),\ x\in\{r,R_1,R_2\} $ in the above equation using \eqref{lapeq} and taking its inverse Laplace transform gives \eqref{UCA4}.\par

\textbf{$ 5-$FAR case}:
For $ N=5 $, $ \mathcal{A}^{-1}(s) $ is given by,
  \begin{align}
  	\mathcal{A}^{-1}(s)=\frac{1}{\mathsf{g}(s)}\times\begin{bmatrix}
  		\mathsf{h}(s)& \mathsf{i}(s) & \mathsf{j}(s)&\mathsf{j}(s)& \mathsf{i}(s) 	\\
  		\mathsf{i}(s) & \mathsf{h}(s)& \mathsf{i}(s) &\mathsf{j}(s)& \mathsf{j}(s) 	\\
  		\mathsf{j}(s)& \mathsf{i}(s) & \mathsf{h}(s)&\mathsf{i}(s)& \mathsf{j}(s) 	\\
  		\mathsf{j}(s)& \mathsf{j}(s) & \mathsf{i}(s)&\mathsf{h}(s)& \mathsf{i}(s) 	\\
  		\mathsf{i}(s)& \mathsf{j}(s) & \mathsf{j}(s)&\mathsf{i}(s)& \mathsf{h}(s)
  	\end{bmatrix},\label{ainvuca5}
  \end{align}
  where $ \mathsf{g}(s)=(1+2s\overline{\mathcal{P}}(s,R_1)+2s\overline{\mathcal{P}}(s,R_2))\times(s^2\overline{\mathcal{P}}^2(s,R_1)-3s^2\overline{\mathcal{P}}(s,R_1)\overline{\mathcal{P}}(s,R_2)+s\overline{\mathcal{P}}(s,R_1)+s\overline{\mathcal{P}}(s,R_2)+s^2\overline{\mathcal{P}}^2(s,R_2)-1),\ \mathsf{h}(s)= s^2\overline{\mathcal{P}}^2(s,R_1)+s^2\overline{\mathcal{P}}(s,R_1)\overline{\mathcal{P}}(s,R_2)-s\overline{\mathcal{P}}(s,R_1)-s\overline{\mathcal{P}}(s,R_2)+s^2\overline{\mathcal{P}}^2(s,R_2)-1,\ \mathsf{i}(s)=s^2\overline{\mathcal{P}}^2(s,R_1)-s^2\overline{\mathcal{P}}(s,R_1)\overline{\mathcal{P}}(s,R_2)+s\overline{\mathcal{P}}(s,R_1)-s^2\overline{\mathcal{P}}^2(s,R_2)$ and $ \mathsf{j}(s)= -s^2\overline{\mathcal{P}}^2(s,R_1)-s^2\overline{\mathcal{P}}(s,R_1)\overline{\mathcal{P}}(s,R_2)+s\overline{\mathcal{P}}(s,R_2)+s^2\overline{\mathcal{P}}^2(s,R_2)$.
  Now, substituting \eqref{ainvuca5} in $ \mathsf{P}(s)=\mathcal{A}^{-1}(s)\mathbf{P}_{i}(s) $ and solving further gives
  \begin{align}
  	\lhp{1}(s)&=\frac{\overline{\mathcal{P}}(s,r)}{1+2s\overline{\mathcal{P}}(s,R_1)+2s\overline{\mathcal{P}}(s,R_2)} \label{l5rx}\\
  	&=\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-2)^n\sum_{k=0}^{n}{n \choose k

	\left(s\overline{\mathcal{P}}(s,R_1\right)^k\left(s\overline{\mathcal{P}}(s,R_2)\right)^{n-k}.
  \end{align}
  Finally, substituting the value of $ \overline{\mathcal{P}}(s,x),\ x\in\{r,R_1,R_2\} $ in the above equation using \eqref{lapeq} and taking its inverse Laplace transform gives \eqref{UCA5}.

\section{Diversity Gain of a two and three FAR system}\label{adg2rx}
  \textbf{$ 2-$FAR case}:
  For a $ 2-$FAR system, the diversity gain is given by
  \begin{align}
  	g(t) = \frac{2\hp{1}(t)}{\overline{\mathsf{p}}(t,r)},\label{dg2rx}
  \end{align}
  When the FARs are arranged as UCA and $ r_i=r $, \eqref{lp2rx} becomes
  \begin{align}
  	\lhp{1}(s)   &=\overline{\mathcal{P}}(s,r)\frac{1}{1+s\overline{\mathcal{P}}(s,R)} \label{l2rx}\\
  	&\stackrel{(a)}{=}\overline{\mathcal{P}}(s,r)\times\sum_{n=0}^{\infty}(-1)^n\left(s\overline{\mathcal{P}}(s,R)\right)^n.\label{lap2rx}
  \end{align}
  Note that $ (a) $ is from the identity $ (1+x)^{-1}=\sum_{n=0}^{\infty}(-1)^nx^n $. Substituting \eqref{lapeq} in \eqref{lap2rx} and taking the inverse Laplace transform gives
  \begin{align}
  	\hp{1}(t) &=\frac{a}{r}\sum_{n=0}^{\infty}\frac{(-a)^n}{R^n}\mathrm{erfc}\left(\frac{r-a+n(R-a)}{\sqrt{4Dt}}\right).\label{eq2rxhp1}
  \end{align}
  Rearranging \eqref{eq2rxhp1} gives
  \begin{align}
  	&\hp{1}(t)=\overline{\mathsf{p}}(t,r)-\lambda(t,r,1).\label{eq2rxhp2}
  \end{align}
  Now substituting \eqref{eq2rxhp2} and \eqref{eq1rx} in \eqref{dg2rx} gives the desired result.

  \textbf{$ 3-$FAR case}:
 The derivation  of $ g(t) $ for $ N=3 $ FAR case is similar to that of $ g(t) $ for $ N=2 $ FAR case given above. Rearranging the terms in \eqref{eq3rxeq} and substituting it in \eqref{eqgain} gives the desired result
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